Physics exam
USFP: PHYSICS A
MOTION IN OUR WORLD
WORKBOOK
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Contents
UNIVERSITY OF SYDNEY .......................................................... Error! Bookmark not defined. FOUNDATION PROGRAM ......................................................... Error! Bookmark not defined.
Introduction ................................................................................................................................... 4
Scalars and Vectors. ...................................................................................................................... 5 Worksheet 1: Vectors .............................................................................................................. 6
Motion ............................................................................................................................................ 7 Answering graph questions .......................................................................................................... 7
Worksheet 3: Interpreting position – time graphs for one-dimensional motion ...................... 9 Worksheet 4: Interpreting velocity – time graphs for one-dimensional motion .................... 10
Worksheet 5: Displacements from velocity – time graphs for one-dimensional motion ...... 11 Worksheet 6: “Falling Cat” Video ......................................................................................... 12
Solving problems ......................................................................................................................... 13 Worksheet 7: Equations of motion in 1 dimension ............................................................... 14
Projectile Example .................................................................................................................... 16 Worksheet 8: Projectile motion ............................................................................................. 19
Uniform circular motion ............................................................................................................. 22 Worksheet 9: Uniform circular motion.................................................................................. 22
Newton’s Laws ............................................................................................................................. 25 Reaction force .............................................................................................................................. 25
Free body diagrams ..................................................................................................................... 25 Worksheet 10: Free-body diagrams for a particle ................................................................. 26 Worksheet 11: Free-Body diagrams for a system of bodies .................................................. 28
Worksheet 12: Newton’s Second Law: One-dimensional problems ..................................... 30 Worksheet 13: More problems on Newton’s laws ................................................................ 35 Worksheet 14: Forces problems ............................................................................................ 42
Worksheet 15: Newton’s 2nd Law: Two-dimensional problems ........................................... 46
Simple Harmonic Motion (SHM) ............................................................................................... 50 Worksheet 16: Simple Harmonic Motion .............................................................................. 51 Worksheet 17 ......................................................................................................................... 54
Worksheet 18: Problems on momentum and energy ............................................................. 56 Worksheet 19: Using work and energy to solve problems .................................................... 59 Worksheet 20: Collisions in one dimension .......................................................................... 63
Worksheet 21: Collision problems in 1 dimension................................................................ 65
Power, work and energy ............................................................................................................. 68 Power, force and velocity ............................................................................................................ 69
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Introduction
Motion in our world consists of the following topics:
Scalars and vectors
Motion
Graphs of motion
Equations of motion
Projectile motion
Uniform circular motion
Newton’s laws
Forces and components
Simple Harmonic Motion
Collisions in 1 and 2 dimensions
Work, energy and momentum
Practical experiences are an essential part of this topic.
Now do Practical 1: The pendulum
Now do the Lesson “Skills” on
Studysmart
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Scalars and Vectors.
Activity
Fill in the table below identifying the scalar and vector equivalents for types of motion.
Scalar Vector Symbols
distance
velocity
a, a
, a
Adding vectors.
When the vectors are placed head to tail in the triangle method, the resultant vector can be found
By using the cosine rule:
c2 = a2 + b2 – (2ab cos c)
to calculate the magnitude of the resultant.
The direction (as a bearing) can be found using the sine rule:
c
c
b
b
a
a
sinsinsin
Now do the Lesson “Scalars and
Vectors” on Studysmart
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Worksheet 1: Vectors
1. State which measurements are scalar and which are vector quantities: (a) 12.3 s (b) 100 km h-1 North (c) 3.8 m East (d) 87 km along a winding track (e) 1.8 x 102 km West (f) 3.2 m upwards
2. Does a car’s speedometer and odometer indicate a scalar quantity, a vector quantity or both?
3. A girl runs exactly twice around a circular track 100 m in diameter. At the end of the run what is:
(a) her total distance travelled? (b) her displacement from her original position?
4. A man runs 550 m North, then turns and runs 220 m South before stopping for a short rest. What is the man’s
(a) total distance travelled? (b) displacement from his original position?
5. ship is sailing South at 20.3 km h-1 and comes across a current of 5.4 km h-1 from the East. What is the resultant velocity of the ship? (Note – this is a vector addition).
6. A yacht in a race is heading on a bearing of S 24o E at a velocity of 45 km h-1. Calculate the Easterly and Southerly components of the yacht’s velocity.
7. A ball is kicked into the air and at a particular instant, its velocity is 27.2 m s-1 at an angle of 15º to the horizontal. Calculate the vertical and horizontal components of the ball’s velocity
at this instant.
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Motion
Answering graph questions When you are asked to calculate quantities from a graph you need to do the following things:
look at the axes, check that time is on the horizontal axis and note what is on the vertical axis - some values can be read directly from the axes
use the gradients of sections of the graph to give you other values
use the areas-under-the-graph to give you additional values Worksheet 2.
The graph represents the motion of an object along a straight line, not necessarily in the one
direction.
-10
-5
0
5
10
15
0 1 2 3 4 5 6 7 8
time (s)
v e
lo c it y (
m s -1
)
(i)
At what times is the velocity zero?
(ii)
What is the highest speed of the object?
(iii)
When is the acceleration zero?
(iv)
What is the greatest positive acceleration? Does this have a larger size than that of the
negative acceleration?
(v)
What is the change in position over the first three seconds? …over the time interval from
t = 5 to t = 8 s?
Now do the Lesson “Linear Motion” on
Studysmart
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Sample Answers
(i) The velocity is zero at t = 4.2 s and again at t = 8.0 s.
(ii) The highest speed of the object is 15 m s-1, this is
during the third second of the motion.
(iii) The acceleration is zero during the third second of
the motion, i. e. from t = 2.0 s until t = 3.0 s; it is
instantaneously zero at t = 5.0 s.
(iv) There are two intervals of positive acceleration,
from t = 0.0 s to t = 2.0 s and from t = 5.0 s to
t = 8.0 s.
The gradients here are:
a 0 to 2 s = (15 - 5) / 2 = + 5.0 m s -2
and a 5 to 8 s = {0 -(- 10} / 3 = + 3.3 m s -2
The greatest positive acceleration is thus 5.0 m s-2.
The negative acceleration is over the 4th and 5th
seconds:
a 3 to 5 s = {- 10 - (+15)} / 2 = - 12.5 m s -2
No, the greatest positive acceleration does not
have a larger size than the negative acceleration.
(v) The change in position over the first three
seconds is the area under the graph:
(5 x 2) + ½ (10 x 2) + (15 x 1)
That is, the displacement over the first three
seconds is + 35 m.
From t = 5 to t = 8 s the area under the graph
is
½ {3 x (- 10)} = - 15 m
The displacement here is -15 m.
The graph is a velocity-time graph - the velocities and times can be read directly
from the axes
The ‘speed’ is the size of the velocity, thus it could be in either the positive or
negative direction - here it happens to be
in the positive direction
The acceleration is zero when the gradient of the velocity-time graph is
zero, this occurs very clearly over the
time interval. There is a sharp point in the
graph, corresponding to a sudden change
in acceleration at the instant t = 5 s. In
reality this change takes a finite, if very
small time interval and we understand
“instantaneously zero at t = 5.0 s” to
mean this.
The acceleration is obtained from the gradient of the velocity-time graph, you
must remember to include the sign
convention and so indicate whether it is
positive or negative.
On the velocity-time graph changes in position, or displacements, are obtained
from the area under the graph, here too
you must remember to include the sign
indicating the direction.
Now do Practical 4: Motion of a basketball
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Worksheet 3: Interpreting position – time graphs for one-dimensional motion
Look at the graph and then answer the
questions.
Displacement
(m)
1. What is the position at t = 0.0 s?
2. What is the position at t = 1.5 s?
3. What is the change in position from t = 0.0 s to t = 1.5 s?
4. What is the average velocity between t = 0.0 s to t = 1.5 s?
5. When is the body at rest?
6. What is the velocity at t = 6.5 s?
7. What is the displacement between t = 2.0 s and t = 8.0 s?
8. What is the average velocity between t = 2.0 s and t = 8.0 s?
9. What is the total displacement from t = 0.0 s and t = 8.0 s?
10. What is the total distance travelled from t = 0.0 s and t = 8.0 s?
0
1
2
3
4
5
0 1 2 3 4 5 6 7 8 time (s)
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Worksheet 4: Interpreting velocity – time graphs for one-dimensional motion
Look at the graph opposite and then
answer the questions.
-5
0
5
10
15
20
25
0 1 2 3 4 5 6 7 8 time (s)
v e
lo c it y (
m s
-1 )
1. What are the velocities at t = 0.0, 5.0 and 7.5 s?
2. What is the change in velocity from t = 1.0 s to t = 3.0 s?
3. What is the average acceleration from t = 1.0 s to t = 3.0 s?
4. What is the change in velocity from t = 2.0 s to t = 6.0 s?
5. What is the average acceleration from t = 2.0 s to t = 6.0 s?
6. What is the acceleration at t = 7.5 s?
7. At what times is the velocity zero?
8. At what times is the acceleration zero?
9. There are three time intervals during which the particle is being
uniformly accelerated. What are
they and what are the constant
accelerations during these three
periods?
10. What is the average acceleration from t = 0.0 s and t = 8.0 s?
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Worksheet 5: Displacements from velocity – time graphs for one-dimensional motion
Look at the graph and then answer the
questions
-5
0
5
10
15
20
25
0 1 2 3 4 5 6 7 8 time (s)
v e
lo c it y (
m s
-1 )
1. What is the displacement between the times t = 0.0 s and t = 1.0 s?
2. What is the displacement between the times t = 1.0 s and t = 3.0 s?
3. What is the displacement between the times t = 3.0 s and t = 4.0 s?
4. What is the total displacement over the time interval from t = 0.0 s to t
= 4.0 s?
5. What is the average velocity over the time interval from t = 1.0 s to t
= 3.0 s?
6. What is the average velocity over the whole time interval from
t = 0.0 s to t = 4.0 s?
7. Over what time interval is the object going in the opposite direction to that in which it is going at t = 0.0 s?
8. Describe the whole motion of the object in
terms of its time
intervals and
accelerations.
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Worksheet 6: “Falling Cat” Video
While watching the video answer the following questions in the space provided.
1. What is meant by “terminal velocity”?
2. What is a human’s terminal velocity? A cat’s terminal velocity?
3. Why are they different?
4. What kind of resistance slows you down?
5. What was the most dangerous height for a cat to fall from?
6. What does a cat do within the first metre of a fall?
7. What is the difference between fact and theory?
8. What does a cat do when it realises that it is not going to fall any faster?
9. Which parts of the cats body stretch to absorb the energy of the impact?
10. Would it be possible to form the same conclusions in this video from examining only ten cats that had fallen from high-rise buildings? Why/why not?
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Solving problems
1. Think about the Physics
Think about the physical situation described in the problem and what laws or principles might be involved – it helps to write these down.
2. Draw a Diagram
Visualise the situation and draw a simple, clear diagram.
Choose a sign convention – the usual convention is that ‘up’ and ‘to the right’ are both positive but this is not always the case.
3. Identify the Data
Label ‘knowns’ and ‘unknowns’ on the diagram. This clears your mind to think more effectively about the problem.
4. Generate Equations
Once you have worked out the principles and laws involved and have the relevant information write down the equations you think will help you solve the problem.
5. Substitute into the Equation(s)
Put the known values, including the correct ‘ + ’ or ‘ – ’ sign, into the equation(s).
6. Solve the Equation(s)
Use whatever mathematics you know to find the unknown quantities.
You should have as many equations as there are unknowns.
7. State the Answer
State the answer to the problem clearly in words. The value of any quantity should be given to an appropriate number of significant figures and include
appropriate units.
Include any indication of direction if this is needed.
8. Think about your answer
Does your answer make sense; is it of the correct size? For example if you are asked to calculate the mass of Jupiter and get an answer of 1.9 x 10-3 kg then you should realise
that this is far too small and so you should check your answer again.
Remember that you are trying to solve a problem in Physics, not do
a maths problem!
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Worksheet 7: Equations of motion in 1 dimension
Assume that air resistance is negligible and the acceleration due to gravity is 9.8 m s– 2.
1. A car is travelling at a constant velocity of 60 m s-1 along a straight road. When the brakes are applied it slows down uniformly at 15 m s-2, how far does it travel while slowing from
its initial velocity to rest?
Answer: 120 m
2. An object is projected vertically upward from the ground and it takes 10.5 s to reach the ground again. What is its launch velocity?
Answer:51.5 m s-
3. A stone is thrown vertically upwards at 36 m s-1 from the top of an 80 m high cliff. What is its velocity just before it hits the ground at the bottom of the cliff
Answer: 53.5 m s-1
4. On a planet in another solar system a man projects an object at 40 m s-1 from the surface up to a spaceship vertically above him. If it takes 10.0 s to just reach the spacecraft,
(a) what is the acceleration due to gravity on the planet?
(b) how high is the spaceship above the ground Answers: (a) 4 m s-2 (b) 200 m
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5. The acceleration due to gravity on another planet is 25 m s-2. A ball is projected vertically upward with a velocity of 75 m s-1 from a platform 100 m above the surface, how long does
it take to reach the surface. Answer: 7.1 s
6. Two Physics students arrange to do an experiment with steel balls projected from the top of a 42 m high cliff. Sam projects his vertically upward at 24 m s-1 and Sally projects hers
vertically down at the same speed of 24 m s-1.
(a) How long does each take to reach the ground at the base of the cliff? (b) What is the velocity of each just before it hits the ground at the base of the cliff.
Answers: t sam = 6.3 s t sally = 1.4 s final velocity 37 m s-1
Optional
7. You are driving 20 m behind a truck on a straight flat road, both travelling at a constant speed of 40 m s–1. The time between you seeing the truck’s brake light and applying your
car’s brakes is 0.250 s. Assuming that both the truck and the car slow down at the same rate
and the truck takes 3.2 s to stop, discuss what happens.
Answer: car stops 10 m behind the truck
8. A brick is dropped from a helicopter when it is 160 m from the ground and moving upwards with a constant velocity of 20 m s-1.
(a) How long does the brick take to reach the ground? (b) How high is the helicopter when the brick hits the ground?
Answers: (a) = 8.1 s (b) = 322 m
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9. 10. A ball is projected vertically upward from the ground with an initial velocity of 50 m s-1.
How long will it take to land on top of the roof of a house that is 8.0 m above the ground?
Answer: 10
Projectile Example
Example 1
A ball is thrown horizontally at 5 m s-1 from a height of 7 m. Calculate:
(a) the time of flight (how long it is in the air)
(b) the range (the horizontal distance)
Draw a diagram to illustrate this question.
Answer:
We know that for projectiles thrown horizontally the time of flight depends only on the vertical
height. If we drop a ball from 7 m or throw it horizontally at any velocity then the time it takes
to reach the ground depends only on its height and can be found as follows:
(a) The time is found by taking vertical components
Data
y = uy t + ½ay t² uy = 0 m s -1
7 = 0 + ½ x 9.8 x t² y = 7 m
t = 1.2 s ay = 9.8 m s -2
t = ?
The time of flight = 1.2 s
(b) The range is found by taking horizontal components
Data
x = uxt + ½ax t² x = horizontal displacement
x = 5 x 1.2 + 0 x = ?
x = 6 m ax = 0 m s -2
t = 1.2 s
ux = 5 m s -1
The range = 6 m
Now do the Lesson “Projectile Motion”
on Studysmart
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Projectile Example 2
A golfer hits a ball with a velocity of 48 m s-1 at an angle of 30° to the horizontal. If air
resistance is neglected, find
(a) the time of flight, (b) the range, (c) the maximum height.
Answer:
(Draw a sketch of the situation)
u = 48 m/s
Now we find the horizontal
and vertical components. u
Ux
Uy
30 0
ux = u cos 30° = 48 x cos 30° = 41.8 m s -1
uy = u sin 30° = 48 x sin 30° = 24.0 m s -1
(a) The time of flight (this is twice the time to get to the top of the flight)
To find the time we only have to consider the vertical components.
Taking vertical components:
Convention: take down as positive (this is a decision you have to make)
Data ay = + 9.8 m s
-2
positive vy = 0 m s -1 (at the top)
uy = - 24 m s -1 (- means up)
t = ?
y = ?
Now look for the simplest equation of motion to get the time, write it down and substitute in.
v = u + at
0 = - 24 + 9.8 t
24 = 9.8 t
t = 2.5 s
The total time of flight = 2.4 x 2 = 4.9 s
(b) The range
The horizontal velocity (ux = 41.8 m s -1) does not change if we ignore air resistance.
x = uxt + ½ ax t² ax = 0
x = 41.8 x 4.9 m ux = 41.8 m s -1
x = 205 m t = 4.9 s
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(c) Maximum height
(this is at the top of its flight when Vy = 0 m s -1)
y = uy t + ½ ay t² y = ?
y = - 24 x 2.4 + ½ x 9.8 x 2.4² uy = - 24 m s -1
y = - 57.6 + 28.2 ay = + 9.8 m s -2
y = - 29.4 m ( - means up) t = 2.4 s
The maximum height = 29.4 m
Projectile Example 3
A ball is thrown from the top of a cliff at 20 m s-1 at angle of 25.
Find the time of flight, range and final velocity of the ball if the cliff is 40 m high.
To get the time: divide the problem up into sections. Find:
the time to reach the highest point
how high it goes
the time it takes to fall from the highest point.
To calculate the range: find the horizontal velocity and then use x = uxt to find the range.
(t = total time of flight here)
To get the final velocity: find the final vertical velocity and then add this to the horizontal
velocity by a vector addition. Remember both magnitude and direction are required.
40 m
25
20 m s -1
Now watch Demonstration 2: Projectile motion
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Worksheet 8: Projectile motion
1. A pellet leaves an air gun with a velocity of 100 m s-1 at an angle of 30° above the horizontal. Calculate
(a) the maximum height reached (127 m) (b) the total time of flight (10.2 s) (c) the horizontal distance travelled (883 m)
2. A projectile is thrown horizontally from a height of 10 m with a velocity of 20 m s-1. Calculate
(a) the time of flight (1.43 s) (b) the range of the projectile (28.6 m) (c) the velocity of the projectile just before it hits the ground (24.4 m s-1 at 35 below the
horizontal)
3. A projectile is launched horizontally at 10 m s-1 from a height of 5 m. Calculate
(a) the time of flight (1 s) (b) the range (10 m)
(c) the velocity just before hitting the ground (14.4 m s-1 at 45 below the horizontal)
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4. A projectile is launched to have a time of flight of 41 s and a range of 12000 m. Calculate
(a) the maximum height reached (2059 m) (b) the launch velocity of the projectile (355 m s-1 at 35 above the horizontal)
5. A projectile has a range of 80 m and reaches a maximum height of 34 m. Calculate
(a) the time of flight (5.2 s) (b) the launch velocity (30 m s-1 at 59.5 above the horizontal)
6. A bullet is fired with a velocity of 350 m s-1 at 20 to the horizontal from the top of a cliff 50 m high.
Calculate
(a) the maximum height reached above the bottom of the cliff (781 m) (b) the time of flight (25 s) (c) the range (8178 m)
7. A projectile is fired from the top of a 40 m high cliff with a velocity of 200 m s-1 at an angle
of 30 above the horizontal.
Calculate
(a) the greatest height reached above the bottom of the cliff (550 m) (b) the time of flight (20.8 s) (c) the range (3600 m)
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8. A projectile is fired from the top of a 10 m high cliff. It has a range of 152 m and reaches a height of 8.2 m above the cliff top.
Calculate
(a) the time of flight (3.2 s) (b) the angle to the horizontal at which the projectile was fired. (15 above horizontal)
9. A plane flying straight and level at an altitude of 200 m with a speed of 600 m s-1 drops a bomb.
(a) How long does it take the bomb to fall? (6.4 s) (b) How far horizontally from the target should the bomb be released? (3833 m)
10. A toy cannon has standard charges (a fixed amount of gunpowder in them) supplied with it designed to propel the small cannon balls of mass 100 g. The cannon was set up on a table
with the barrel horizontal. The cannon was then fired and the distance travelled by the
cannon ball was measured. The following diagram is a sketch of the result.
This result was collected by a student attempting to solve the problem, “How much of the
energy stored in a standard charge, as supplied by the manufacturers of the cannon, is
transferred to the cannon ball when fired?”
(a) Outline the basic physics the student is going to apply to enable him to obtain an answer to his problem.
(b) Suggest possible techniques that might allow the student to be more sure of his result. (c) Using the student’s sketch of his results, calculate the velocity the cannon ball left the
barrel of the cannon. (16.2 m s-1)
1.40 m.
8.60 m.
Not to scale Path of cannon ball
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Uniform circular motion
Note: With uniform circular motion although the speed remains the same, an object travelling
in a circular path must be accelerating since its velocity is changing continually.
Worksheet 9: Uniform circular motion
1. An angle is produced at the centre of a circle of radius 3.2 m by an arc of length 1.1 m. How many radians are there in this angle?
2. An angle is produced at the centre of a circle of radius 5 m by an arc of length 6.1 m. How many radians are there in this angle?
3. What length of arc would produce an angle of 2.1 radians at the centre of a circle of radius 4 m?
4. Convert the following angles to radians: (a) 45° (b) 90° (c) 270°
(d) 35° (e) 52° (f) 232°
5. Convert the following radians to degrees: (a) 5.1 radians (b) 1 radian
(c) 3.2 radians (d) 4.1 radians
(e) 1.5 radians (f) 0.5 radians
6. Calculate the linear velocity of an object carrying out uniform circular motion with: (a) angular velocity 1.3 rad s-1, radius 4 m
(b) angular velocity 2.1 rad s-1, radius 3.2 m
(c) angular velocity 3.6 rad s-1, radius 2.7 m
7. Calculate the angular velocity of an object carrying out uniform circular motion with: (a) radius 5.1 m, linear velocity 2.1 m s-1
(b) radius 13.4 m, linear velocity 6.2 m s-1
(c) radius 14.2 m, linear velocity 3.1 m s-1
Now do Practical 5: Video analysis of projectiles
Now do the Lesson “Uniform Circular
Motion” on Studysmart
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8. An object carrying out uniform circular motion has a linear velocity of 10 m s-1. At one point it is travelling due north and a short time later travelling due west. What is the change in
velocity between these two points?
9. An object travels in a circular path of radius 15 m. If it travels with a steady speed of 10 m s-1: (a) How long does it take to complete one circuit?
(b) What is its angular velocity in radians per second?
10.An object travels around a circular path of radius 12 metres with an angular velocity of 1.2
radians per second.
(a) What is the circumference of the circle?
(b) How long does it take to complete one circuit?
(c) What is the speed of the object?
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11.The earth orbits the sun with a radius of 150 000 000 km once every 365 days.
(a) What is the angular velocity of the earth in radians per second?
(b) What is the linear velocity in m s-1?
(c) What is the centripetal acceleration in m s-2?
12.An object travels at a constant speed of 15 m s-1 around a circular path of radius 25 m.
Calculate:
(a) the time it takes to travel a complete revolution (called the period T)
(b) the angular velocity
Answers
1. 0.34 rad
2. 1.22 rad
3. 8.4 m
4. (a) 0.79 (b) 1.57 rad (c) 4.7 rad (d) 0.61 rad (e) 0.91 rad (f) 4.05 rad
5. (a) 292 (b) 57 (c) 183 (d) 235 (e) 86 (f) 28.6
6. (a) 5.2 m s-1 (b) 6.7 m s-1 (c) 9.7 m s-1
7. (a) 0.41 rad s-1 (b) 0.46 rad s-1 (c) 0.22 rad s-1
8. 14.14 m s-1 at SW
9. (a) 9.4 s (b) 0.67 rad s-1
10 (a) 75.4 m (b) 5.2 s (c) 14.4 m s-1
11. (a) 1.99 x 10-7 rad s-1 (b) 2.99 x 104 m s-1 (c) 5.96 x 10-3 m s-2
12. (a) T = = 10.5 s. (b) = 0.6 rad s-1
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Newton’s Laws
Reaction force The reaction force, also called the normal, is a force perpendicular to the surface. Consider a
mass placed on a table. The force acting down is the weight which acts through the centre of
gravity of the mass (the middle of the mass). The weight = mg. This has to be accompanied by
an opposite force acting upwards of the same size but opposite in direction. This force is called
the reaction force. We can represent this as follows.
Question
What is the net force in this case?
Free body diagrams When working out problems with objects in lifts etc accelerating it is useful to use a free body
diagram.
We use the following method to solve these types of problems.
1. Sketch a diagram representing the general problem.
2. Choose one body whose motion is to be analysed, and draw a free body diagram for that body. Show clearly the direction of all the forces acting on this body.
3. Show on this free body diagram the direction of acceleration of the body selected.
4. Use Newton's second law ∑F = ma for the weights
mass
weight
reaction
Now do the Lesson “Newtons Laws” on
Studysmart
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Worksheet 10: Free-body diagrams for a particle
For each of the situations shown draw:
1. A free-body diagram showing all of the individual forces acting on the particle 2. A vector diagram and/or a word description of the addition of the individual forces to give a
resultant force that is appropriate for the particle.
Free body diagram Vector diagram
Particle at rest
Particle falling freely with no
air resistance
Particle at rest
Particle at rest
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Particle at rest
Particle falling with a constant
terminal velocity against air
resistance
Particle tied to a rigid post and
moving on a horizontal
frictionless surface in a circle at
constant speed
Particle in free flight
at the top of its trajectory
Particle oscillating on a spring at
its lowest point
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Worksheet 11: Free-Body diagrams for a system of bodies
For each of the situations shown draw:
1. A free-body diagram showing all of the individual forces acting on the shaded body 2. A vector diagram and/or a word description of the addition of the individual forces to give a
resultant force for the body
Free body diagram Vector diagram
Three blocks being pushed by a
constant force along a horizontal
frictionless table top
Three blocks being pushed by a
constant force up an inclined
frictionless slope
A trolley being pulled along a
horizontal frictionless table top by a
hanging mass
A trolley being pulled up along a flat
frictionless incline by a hanging mass
Two blocks being pulled along a
horizontal table at constant speed
against friction
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Free Body diagram of bodies connected by strings
Consider a mass m1 connected by a string to a second mass m2 as shown below on a frictionless
table.
The string transfers the weight of m2 to become a horizontal force acting on m1. We say that
there is a tension T in the string that is constant throughout the string. When m1 is released the
system will begin to accelerate.
Activity
Work out expressions for the acceleration and the tension in the string by using free body
diagrams on masses 1 and 2. Start by drawing all the forces acting on the two masses above.
One way of solving these problems (for example calculate tensions and acceleration):
1. Draw diagram and on it draw forces acting on the masses
2. Label the positive direction as from one end of the string to the other
3. Write F = ma for EACH MASS ALONE
4. Solve those F = ma equations for the unknown values you are asked to calculate.
Note: for a simple case like question 1 in worksheet 12, it might be simple to add the
masses and apply F = ma to get the acceleration first.
m1
m2
T
T
a acceleration of the system
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Worksheet 12: Newton’s Second Law: One-dimensional problems
1. The diagram shows two blocks being pulled along a smooth horizontal surface by a constant force of 20 N. The two blocks are joined together by a string.
20 N
(a) What is the acceleration of the two blocks?
(b) What is the tension in the string?
2. (a) The diagram shows two masses hanging on strings from the ceiling of a lift that is accelerating upwards at
2.0 m s–2. What is the tension in each of the strings?
(b) Suppose the lift’s motion changes to be a constant
velocity of 5.0 m s-1 upwards. What are the tensions in the
two strings?
(c) Suppose the motion changes so that that the tension in the
lower string is 36 N. What is the acceleration of the lift
and the tension in the upper string?
2.5 kg 7.5 kg
7.6 kg
4.2 kg
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3. The trolley and mass in the diagram are joined together by a string over a smooth pulley and the system is allowed to move freely.
(a) What is the acceleration of the system?
(b) What is the tension in the string?
(c) What force must be applied to the trolley to hold the system at rest?
(d) What force must be applied to the hanging mass to hold the system at rest?
(e) When the system is held at rest, what is the tension in the string?
3.6 kg
0.8
kg
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4. Two masses are suspended by a string over a smooth frictionless pulley as shown in the diagram.
(a) Write equations of motion for each mass
(b) By eliminating the acceleration of the system ‘a’ write an expression for the size of the string
tension ‘T’ as a function of the masses and the
acceleration due to gravity.
(c) By eliminating the string tension ‘T’ write an expression for the size of the acceleration of
the system ‘a’ as a function of the masses and
the acceleration due to gravity.
(d) Using the two expressions derived in (b) and (c) above explain what happens when the two
masses are equal, that is, when m = M.
(e) What ratio of masses gives the system an acceleration of 0.5 g?
5. A third mass is added to one of two masses that are suspended by a string over a smooth frictionless pulley as shown in the diagram.
(a) What is the acceleration of the system? (b) What is the tension in each string?
m
M
2 kg
4 kg
4 kg
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Non-parallel forces which are in equilibrium
Consider the following problem:
A mass of 125 kg is supported by two ropes, one making an angle of 45° with the vertical and
the other making an angle of 30° to the vertical as shown below. We have to find the tension in
each rope.
Answer
The mass is not moving i.e. the net force is zero.
Hence the three forces, the two tensions and the weight of the
object, are in equilibrium.
To solve this problem we can use either a vector diagram, trigonometry or components.
Trigonometry
This type of vector diagram is described as being closed as the net (resultant) force is zero.
Work out T1 and T2.
Using components
We split up the two tensions into components and then add them together. You should get the
same answer as above.
45 30
weight = mg = 1225 N
T1 T2
m = 125 kg
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Another example
A mass of 8.5 kg is suspended by two strings that make an angle of 35 and 50 to the vertical.
Find the tension in each of the strings.
Answer
As always draw a diagram so you can fully understand the question.
Horizontally The horizontal components of the tensions must equal each other as there is no acceleration
horizontally.
TH35 = TH55
T35 sin 35 = T55 sin 55
Vertically The vertical components of the tensions must add up to equal the weight as there is no
acceleration vertically.
TV35 + TV55 = mg
T35 cos 35 + T55 cos 55 = 8.5 x 9.8
These equations can then be solved to find the answer.
Activity
Use trigonometry to check the answer you get above.
8.5 kg
35 55
T35 T55
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Worksheet 13: More problems on Newton’s laws
1. A boy of mass 50 kg is in a lift ascending to the top floor. As the lift approaches the top floor, it decelerates at 2 m s-2. The reaction force between the boy and the lift floor is then:
(a) 590 N (b) 390 N (c) 490 N (d) 290 N
2. In the diagram below two masses are pushed by a 24 N force. Which of the following is true?
(a) acceleration of both masses is 1 m s-2 (b) acceleration of the 4 kg mass is greater (c) 4 kg mass is pushed with a force of 8 N (d) 8 kg mass is pushed with a force of 8 N
3. An 80 kg man slides vertically down a rope. His grip on the rope causes a frictional force of 240 N opposing his motion.
The approximate value of the acceleration of the man down the rope is:
(a) 13 m s-2 (b) 10 m s-2 (c) 9 m s-2 (d) 7 m s-2
4. A person holds a rope that hangs vertically and supports a mass of 6 kg. (a) Calculate the tension in the rope. (b) Calculate the force exerted by the man on the rope.
4 kg 8 kg
frictionless sufrace
24 N
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5. Find the force that must be applied upwards parallel to an inclined plane in order to hold a mass of 140 g at rest on a smooth plane inclined at 24° to the horizontal.
6. A 240 N vehicle is to be pulled up a 30° incline at constant speed. How great a force parallel to the incline is needed if friction effects can be ignored?
7. A mass 280 g rests on a smooth plane inclined at 30° to the horizontal. Find the force parallel to the plane which will keep the mass at rest.
8. Three blocks of wood are joined together by string as shown in the diagram. The blocks are
pulled to the left by a constant force of 8 N.
(a) Calculate the acceleration of the blocks. (b) Calculate the tension at Y.
8 N Y 2 kg 2 kg
frictionless
4 kg
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9. A 5 kg mass is held by a spring balance. Calculate the reading on the balance when the mass is
(a) at rest (b) moving upwards with a constant velocity of 3 m s-1 (c) accelerating downward at 4 m s-2 (d) accelerating downward at 9.8 m s-2
10. A trolley loaded with four 50 g masses and attached to a 50 g mass carrier by a string is set up in an experiment to investigate the relationship between force, mass and acceleration.
The trolley is released and after several runs the average acceleration is calculated. The
experiment is repeated 4 times, with one mass from the trolley being transferred to the
mass carrier each time. This varies the accelerating force while keeping the mass of the
system constant.
The results are shown in the table.
(a) Plot a graph of accelerating force against average acceleration. Draw a straight line of best fit (put force on the y axis.)
(b) What does the gradient of the straight line you have drawn represent? (c) Use your graph to find the mass of the unloaded trolley.
hanging mass (g) average acceleration (m s-2)
50.0 0.34
100.0 0.64
150.0 0.99
200.0 1.28
250.0 1.60
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11. A 10 kg mass is supported on a frictionless inclined plane and is connected to a second mass M by means of a string and frictionless pulley as shown in the diagram.
The acceleration of the mass M is 3 m s-2 upward.
Find
(a) the tension in the string.
(b) the value of the mass M
(c) If the string breaks calculate the acceleration of the two masses.
10 kg
30
M
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12. Two blocks, one of mass 3 kg and one of mass 2 kg, are connected by a string over a frictionless pulley.
The blocks are released and move. Find:
(a) the acceleration of the 2 kg block
(b) the tension in the string
13. Three toy trucks with masses 1, 2 and 3 kg are connected by strings and pulled in a straight line along a horizontal surface as shown in the diagram below. Assume there is no
frictional resistance to the motion.
1 kg 2 kg
15 N 3 kg
X Y
Find
(a) the magnitude of the acceleration of the trucks
(b) the tension in string X
(c) the tension in string Y.
.
2 kg3 kg
frictionless pulley
string
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14. Two tugs are pulling on a ship, one with a force of 6000 N and the other with a force of 8000 N. The cables from the tugs to the ship are at right angles to each other. Find the
magnitude of the resultant force on the ship, and the angle that its direction makes with the
cable in which the force is 8000 N.
15. The resultant of two forces acting at right angles to each other is 35 N. One of the component forces is 21 N. Find the magnitude of the other component force, and the angle
that this force makes with the direction of the resultant.
16. A box is pulled along the floor by means of a rope making an angle of 30° with the horizontal, the pull in the rope being 80 N. Calculate the effective part of the pull in
moving the box, and also the force trying to lift it off the floor.
17. A small object of mass 50 kg is held at rest on a smooth ramp inclined at 30° to the horizontal by means of a light rope parallel to the ramp. Find the tension in the rope and
the reaction of the ramp on the object.
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18. A railway truck is moving on a level track with a velocity of 3 m s-1. It is brought to rest in a distance of 45 m when it moves up an incline. Find the angle of inclination.
19. A smooth incline rises 3 m vertically for every 5 m of its inclined length. A body weighing 100 N is placed on the incline. Find:
(a) the resolved parts of its weight parallel and perpendicular to the incline
(b) the least force parallel to the plane necessary to hold the body in position
(c) the velocity of the body after sliding freely from rest 3 m down the incline.
Answers
1. b 2. c 3. d 4. (a) 58.8 N (b) 58.8 N 5. 0.56 N 6. 120 N 7. 1.37N
8. (a) 1 m s-2 (b) 6 N 9. (a) 49 N (b) 49 N (c) 29 N (d) 0 N
10. (c) 1.28 kg
11. (a) 18.9 N (b) 1.48 kg (c) 9.8 m s-2 for the mass M and 4.9 m s-2 for the 10 kg mass.
12. (a) 1.96 m s-2 (b) 23.5 N 13. (a) 2.5 m s-2 (b) 12.5 N (c) 7.5 N
14. 10000 N 37 15. 28 N 37 16. FV = 40 N FH = 69 .3 N
17. 245 N up the slope 424 N reaction 18. 0.58
19. (a) parallel = 60 N perpendicular = 80 N (b) 60 N (c) 5.9 m s-1
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Worksheet 14: Forces problems
1. The diagram shows a mass M being acted on by two forces F1 and F2. In which direction will the mass move?
(a) A (b) B (c) C (d) D
2. In the diagram below a feather of mass m is suspended in a vacuum, below a ball of mass 2 m. The string attached to the ball is cut at the point P.
The ball will fall:
(a) with an acceleration of 2g and quickly overtake the feather (b) and reach a terminal velocity (c) with the same acceleration as the feather (d) continues to accelerate because a = Fm
3. A mass is suspended by two strings, each making an angle of 30° with the vertical. The tension in each string is 3.9 N. Draw a vector diagram and hence calculate the value of the
mass.
M F1
F2
A
B
C
D
P
feather mass m
ball of mass 2m
vacuum
fixed support
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4. An electric lamp of weight 56 N is suspended over a table by two wires that make angles of 60° and 30° to the vertical respectively. Draw a vector diagram and hence find the tensions in
each wire.
5. A lamp of mass 4.8 kg is suspended over the centre of a 15 m wide road by two wires running to the tops of two 7 m high poles on either side of the road. If the lamp is suspended 6.5 m
above the road calculate the tension in the wires.
6. A large vertical pole has three cables connected to it running in the South, East and North - West directions. The pole is kept vertical only by these three cables which are horizontal. The
tension in the North - West cable is 2800 N.
(a) Draw a diagram of the problem. (b) Find the tensions in the other two cables. (c) If the south cable breaks in what direction will the pole begin to fall?
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7. The diagram below shows a picture hanging from a wall. The picture has a mass of 1.5 kg.
(a) On the diagram, show all the forces acting on the picture. (b) Calculate the tension in the supporting rope. (c)
(d) For heavy pictures, is it best to use a rope with a big angle or a small angle to the vertical? Explain your answer.
8. Three blocks A, B and C of mass 50 kg, 20 kg and 30 kg respectively, are at rest on a frictionless surface and touch each other as shown. A force of 600 N is applied from the left
to block A as shown.
Calculate
(a) the acceleration of block B (b) the magnitude and direction of the two horizontal forces acting on block B.
1.5 kg
30 30
A
50 kg
B
20 kg
C
30 kg
600 N
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9. A student with his hand above his head whirls a mass of 100 g in a horizontal circle on the end of a 1.5 m string. His hand is 0.7 m above the horizontal circle.
(a) Draw a diagram showing the two forces acting on the mass. (b) What is the radius of the circular path? (c) What is the value of the resultant force acting on the mass? (d) Find the tension in the string (e) Calculate the period of revolution of the mass
10. A body of mass 2.0 kg is suspended by a string of length 1.3 m. A horizontal force is applied to the body, displacing it 0.50 m from the vertical as shown in the diagram.
(a) Calculate the magnitude of the horizontal force.
(b) Calculate the magnitude of the tension in the string
Answers
1. b 2. c 3. 6.7 N 4. T30 = 48.5 N T60 = 28 N 5. 355 N 6. 1980 N each
7. b) 8.5 N 8. (a) 6 m s-2 (b) 300 N A on B 180 N C on B
9. (b) 1.3 m (c) 1.84 N (d) 2.09 N (e) 1.68 s
10. (a) F = 8.16 N (b) 21.2 N
1.2 m
0.50 m m
string
mass
F
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Worksheet 15: Newton’s 2nd Law: Two-dimensional problems
1. The diagram shows a block of mass 1.64 kg sliding down a smooth (frictionless) surface inclined at an angle of 350 to the horizontal.
35 0
(a) What are the normal and parallel components of the weight of the block? (b) What is the block’s acceleration down the slope?
Answers: normal = 13.2 N parallel = 9.2 N acceleration 5.6 m s-2
2. The block in question 1 is now placed on a rough surface that is inclined at the same angle and slides down this with a constant speed of 0.46 m s-1. What is the friction force acting on
the block?
Answers: friction force = 9.2 N
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3. Now consider a block of mass 5.2 kg being pulled up a smooth frictionless slope by a constant force of 40 N as shown in the diagram.
(a) If the acceleration of the block is 1.25 m s-2, what is net force on the block? (b) Write expressions for the normal and parallel components of the block’s weight in
terms of the angle of the slope.
(c) What is the angle of the slope the angle of the slope?
Answers: (a) = 6.5 N (b) weight normal = 51 cos Ө weight parallel = 51 sin Ө (c) 41º
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4. Consider the large flat television screen hung from two wires as shown in the diagram.
30 0 60 0
What is the tension in each wire? Answer: T30 = 73 N T60 = 127 N
5. The diagram shows a trolley of mass 3.6 kg and the hanging mass is 0.8 kg, there is no friction and the angle of the incline is 250.
(a) What is the acceleration of the system? (b) What is the tension in the string? (c) What must the angle of the incline be to give the same sized acceleration in the opposite
direction?
(d) What then is the tension in the string? Answer: (a) 1.61 m s-2 down the slope (b ) 9.1 N (c) 1.23◦ (d) 6.44 N
15.0 kg
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6. A student sets up an experiment with two trolleys and a hanging mass as shown in the diagram.
The lower trolley has a mass of 2.4 kg, the upper trolley a mass of 1.8 kg, the hanging mass
is 2.6 kg and the slope is set at 300.
(a) What is the acceleration of the system when it is allowed to move freely?0.72 m s-1 (b) What is the tension in the lower string? 13.5 N up the slope (c) What is the tension in the upper string? 23.6N up the slope (d) What do you think would be the effect of the pulley not being smooth – that is there is
friction between the string and the pulley?
Now do the Lesson “Simple Harmonic
Motion” on Studysmart
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Simple Harmonic Motion (SHM) A mass on a spring that is pulled down and then released is an example of simple harmonic
motion (SHM). The mass will freely oscillate up and down which we describe as SHM.
Draw graphs of displacement, velocity and acceleration against time for the SHM motion. Make
t = 0 the point where the displacement is zero, and then make the displacement positive.
displacement
positive
negative
time
positive
negative
time
velocity
positive
negative
time
acceleration
mass which is pulled down and
then released.
spring
motion sensor
connected to
data logger
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Worksheet 16: Simple Harmonic Motion
1. Explain what is meant by simple harmonic motion. Use diagrams to explain your answer and give an example of an object undergoing simple harmonic motion.
2. An object moving with simple harmonic motion has an amplitude of 2 cm (maximum displacement from the mean position) and a frequency of 20 Hz (20 complete up and down
motions in one second)
(a) Calculate the period of oscillation - that is the time for one complete oscillation.
(b) Sketch a graph of displacement against time for the first 2 cycles (2 complete oscillations).
(c) Sketch a graph of velocity against time for the first 2 cycles.
(d) Sketch a graph of acceleration against time for the first 2 cycles.
3. A pendulum undergoes simple harmonic motion as shown below. The displacement is given
by . If the period T of oscillation is 0.5 s then draw graphs of displacement, velocity and
acceleration versus time. Take the starting point when t = 0 at position A.
4. When a particle undergoing simple harmonic motion is at maximum displacement from its mean position, it has:
(a) minimum acceleration and maximum velocity
(b) minimum velocity and minimum acceleration
(c) maximum acceleration and maximum velocity
(d) maximum acceleration and minimum velocity
A
B
C
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5. The diagram below is of a trolley, held by a spring on each end, oscillating between positions X and Y.
At the moment shown above the trolley is midway between XY and its velocity is shown by
the arrow.
At a later moment the trolley is as shown below.
If point Z on the graph shown below represents the first diagram, the point on the graph
which represents the second diagram is
(a) A
(b) B
(c) C
(d) D
time
displacement
6. When a pendulum is given a small displacement, the magnitude of the restoring force is
proportional to the displacement.
A pendulum is given a small displacement and then released at time t = 0.
(a) Sketch graphs showing how the subsequent displacement, velocity, and acceleration vary
with time.
(b) Show clearly on your graph of displacement against time from part (a), the displacement
when:
X Y
positive movement of trolley
X Y
positive
negative
Z
A B
C D
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(i) the velocity is maximum (label as point P);
(ii) the acceleration is maximum (label as point Q);
(iii) the kinetic energy is maximum (label as point R);
(iv) the potential energy is maximum (label as point S).
7. A racing car of mass 1450 kg is travelling at a constant speed of 60 m s-1 around a circular section of a track of radius 60 m.
(a) What is its acceleration and what direction is this in?
(b) What is the resultant force on the car?
(c) What is the car’s weight?
8. A student whirls a stone of mass 0.50 kg around in a horizontal circle on a string of length 1.20 m so that it 25 revolutions in 10 seconds.
(a) What is the period of this motion?
(b) What is its orbital speed?
(c) What is its acceleration and in what direction is this?
(d) What is the tension in the string? Explain any approximations you have made here.
(e) If the string breaks in what direction will the stone move? – draw a diagram showing clearly this direction in relation to its circular path and the acceleration.
Selected Answers
2. (a) 1/20 s 4. (d) 5. (c)
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WORK (W) , ENERGY (E) and MOMENTUM. (p)
Worksheet 17
1. A force of 25.0 N acts for a time of 10.0 seconds on a body of mass 5.0 kg moving in one dimension.
(a) What is the impulse of the force?
(b) What is the change in momentum of the body?
(c) Suppose the body had an initial velocity of 20.0 m s-1 eastwards and the force was applied eastwards, what is the body’s final velocity?
(d) What if the body had an initial velocity of 30.0 m s-1 westwards and the same force was applied for the same time, what now is the body’s final velocity?
Answers: (a) 250 N s (b) 250 kgm s-1 (c) 70 m s-1 East (d) 20 m s-1 East
2. A golf club applies a force to a golf ball of mass 0.0455 kg for 5.0 x 10-4 s so that the ball is given a velocity of 48.0 m s-1 as it leaves the ground.
(a) What is the change in the ball’s momentum?
(b) What is the average force applied by the club to the ball?
(c) Suppose that the average force applied by the club is 3640 N for 4.0 x 10-4 s, what velocity does the ball have as it leaves the ground?
Answers: (a) 2.18 N s (b) 4368 N (c) 32 m s-1
3. A car of mass 1600 kg is travelling at 30 m s-1 when it hits a wall and comes to rest in 0.25 seconds.
(a) What is the change in the car’s momentum in the crash?
(b) What average force does the wall exert on the car while it is brought to a stop?
(c) What average force does the car exert on the wall while it is brought to a stop?
Answers: (a) 48000 N s against its motion (b) 192 000 N (c) 192 000 N
Now do the Lesson “Work, Energy and
Momentum” on Studysmart
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Impulse and momentum change for a changing force
t is the time interval over which the force acts so we can use calculus to calculate the total
impulse of a changing force over some time interval from t1 to t2. We consider the time interval
to be divided into little sections each t long and then add up the products F t as t approaches
zero – this is the process of integration and we get:
The integral in this equation is the area under the graph when force is graphed as a function of time. We can calculate the impulse of an applied force and the change in momentum of the body
if we measure the area under the force-time graph.
Example
The graph shows a varying force applied to a body over a time interval of 10 seconds
(a) What is the impulse of the force over the time from zero to 10 seconds?
(b) What is the final momentum of the body if it had an initial momentum of 25 N s in the same direction as the force?
(c) What is the average force applied to the body over the ten-second period?
(d) If the force is applied to a mass of 2.5 kg initially at rest, what speed does this mass have because of the force?
I = p = F dt
0
2
4
6
8
10
12
0 1 2 3 4 5 6 7 8 9 10
Time (s)
F o rc
e (
N )
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Worksheet 18: Problems on momentum and energy
1. A 50 kg mass is dropped from a height of 0.45 m to the ground. The impact lasts 0.04 s and the mass does not rebound.
Calculate:
(a) the velocity of the mass just before it hits the ground
(b) the momentum of the mass just before it hits the ground
(c) the impulse on the mass as it comes to rest
(d) the average force exerted on the mass by the ground in bringing the mass to rest.
(a) 2.97 m s-1 (b) 148.5 kg m s-1 down (c) 148.5 kg m s-1 up (d) 3712.5 N
2. According to the rules, tennis must be played with a ball of mass 70 g which bounces more than 1.3 m and less than 1.5 m when dropped 2.5 m onto a concrete floor.
(a) How much kinetic energy has the ball just before hitting the floor when dropped from
2.5 m?
(b) If it bounces to 1.4 m how much energy did it lose on bouncing?
(c) When the ball hits the ground it is stationary for an instant. Into what forms has its
kinetic energy been changed at this time?
(a) 1.7 J (b) 0.75 J
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3. A brick of mass 1 kg falls from rest and moves freely under gravity for 5 m to the ground.
(a) Plot a graph on the axis below of gravitational potential energy against distance,
taking its potential energy as zero after it has fallen 5 m.
(b) On the same axes, plot its kinetic energy against distance, up to 5 m.
(c) What is the coordinate of the point where the graphs cross?
(d) What quantity is represented by the magnitude of the slope of each graph?
(c) 24.5 J, 2.5 m (d) weight in both cases
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4. The diagram below represents a track which is used to demonstrate the energy of a body. A mass of 4 kg moves with a speed of 10 m s-1 along AB. Friction may be ignored.
(a) Find the kinetic energy of the mass as it moves along AB.
(b) Find the change in the potential energy of the mass between the points D and B
(c) Find the kinetic energy of the mass as it passes through the point C.
(d) Find the velocity of the mass at D.
(a) 200 J (b) 78.4 J (c) 396 J (d) 7.8 m s-1
5. A mass of 90 kg on a slope which is not frictionless, travels 50 m in 5.25 s, starting from
rest, accelerating uniformly. If the change in vertical height is 20 m, find:
(a) the change in potential energy,
(b) the speed at the end of the 50 m
(c) the change in kinetic energy
(d) the work done overcoming friction
(e) the average frictional force acting.
(a) 17.6 kJ (b) 19 m s-1 (c) 16.3 kJ (d) 1.31 kJ (e) 26.2 N
B
2 m
5 m
A
D E
C
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Worksheet 19: Using work and energy to solve problems
1. The diagram shows a block being pulled along a horizontal bench top against friction by a constant force of 40 N. The friction force is 16 N.
(a) What is the acceleration of the block
Answer 3 m s-2 to the right
(b) What is the speed of the block after it has moved 4.2 m, assuming it starts from rest?
Answer 5.0 m s-1
(c) What is the work done by the 40 N force on the block as it moves the 4.2 m?
Answer 168 J
(d) What is the energy ‘lost’ due to friction? Answer 67 J
(e) What is the change in gravitational potential energy? Answer 0J
(f) What is the block’s final kinetic energy? Answer 101 J
(g) What is the block’s speed after it has moved 4.2 m, assuming it starts from rest?
Answer 5.0 ms-2
8.0 kg
16 N 40 N
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2. Now consider a 4.0 kg block and the same forces on a slope at some angle to the horizontal.
(a) Suppose that in being pulled 4.2 m up the slope it was a height 1.6 m above its starting point. What is the increase in gravitational potential energy?
Answer: 62.7 J
(b) What is the work done against the friction force?
Answer: 67.2 J
(c) What is the final speed of the block if it started from rest?
Answer 4.4 m s-1
16 N
40 N
4.0 kg
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3. The diagram shows a 7.5 kg block sliding from rest down a smooth straight 8.0 m slope from a height of 6.0 m with no friction.
Find the speed of the block at the bottom of the slope.
Answer 10.8 m s-1
Now suppose that the box slides, without friction, down another slope through the same vertical
height:
What does the conservation of mechanical energy tell us about the speed at the bottom?
What if the slope was not straight but wavy, still without friction?
7.5 kg
h = 6.0 m
L = 8.0 m
7.5 kg
h = 6.0 m
L = 11.5 m
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4. Consider the same 7.5 kg block as in question 3, but now acted against by a constant friction force of 20 N as it slides 8.0 m down the straight slope through a vertical height of 6.0 m.
Find the speed at the bottom of the slope. Answer 8.6 m s-1
7.5 kg
20N h = 6.0 m
L = 8.0 m
Now do the Lesson “Momentum and
Collisions” on Studysmart
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Worksheet 20: Collisions in one dimension
Here we apply the two conservation laws, momentum and kinetic energy, to some one-
dimensional collisions, both “perfectly sticky” (or “inelastic”) and “perfectly bouncy” (or
“elastic”) collisions between two bodies – think of trolleys on a frictionless track.
The first table shows a number of collisions where the two bodies stick together after the
collision. In these momentum is conserved but kinetic energy is not conserved, some of it is
‘lost’ in damaging or deforming the bodies as they collide.
Work out what % KE is lost in each case.
Inelastic Collisions
Before the collision After the collision
Optional
Optional
Optional
m m u
at rest
m 2 m u
at rest
2 m m u
at rest
m m u – u
m 2 m u – u
m m 2u – u
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Elastic Collisions
The second table shows a number of collisions where the two bodies bounce apart after the
collision. In these assume that they are “perfectly elastic collisions” – that is, both momentum
and kinetic energy are conserved. Find the velocities after collision in each case.
Before the collision After the collision
Optional
Optional
Optional
m m u
at rest
m 2 m u
at rest
2 m m u
at rest
m m u – u
m 2 m u – u
m m 2u – u
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Worksheet 21: Collision problems in 1 dimension
1. Tony, mass 60 kg, has a new pair of roller blades and is skating along a horizontal footpath at a constant speed of 8.0 m s– 1. When he passes Susanna, she puts her pet puppy, Snoopy
with a mass of 25 kg, into his arms.
(a) What is Tony’s speed immediately after he takes Snoopy? 5.7 m s-1
(b) What is the change in kinetic energy involved in this “collision”? 539 J
2. A railway carriage of mass 20,000 kg is rolling along a flat horizontal track at a constant speed of 3.0 m s– 1 when it collides and couples with a stationary railway carriage so that the
two continue to move at 1.2 m s– 1.
(a) What is the mass of the second carriage? 30 000 kg
(b) How much kinetic energy is ‘lost’ in the collision? 54 kJ
3. A 75 kg football player tackles a 105 kg forward who is running at him at 8.0 m s – 1.
(a) In a head-on tackle what speed must the lighter player have if they both end up stationary after the tackle? 11.2 m s-1
(b) How much energy is used up in ‘re-arranging’ the two players’ bodies in the tackle? . 8064 J
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4. A 1500 kg car travelling at 20.0 m s – 1 collides head-on with a truck of mass 4500 kg travelling at 10.0 m s– 1 in the opposite direction. In the collision the truck is brought to a
stop and the car bounces back.
(a) What assumptions are needed to apply conservation of momentum to the collision?
(b) What is the speed at which the car bounces back? 10 m s-1
(c) How much energy is ‘lost’ in damaging the two vehicles? 450 000 J
(d) Suppose the actual collision took only 0.20 s, what is the size of the forces involved in the impact? 225 000 N
(a) asteroid 1 P = 3000 Ns asteroid 2 P = - 2000 Ns
(b) P = 1000 Ns in the direction of asteroid 1
(c) P = 400 Ns
(e) P = 1200 kN
-3000
-2000
-1000
0
1000
2000
3000
4000
Time
M o m
e n
tu m
( N
.s )
5. The graph shows the momentum of two asteroids before they collide head-on,
and one of them after the collision.
(a) Calculate the momentum of each before the collision?
(b) What is the total momentum before the collision?
(c) What is the momentum of the second asteroid after the collision?
(d) Complete the graph for the second asteroid.
(e) If the collision took 0.0020 s, what
are the forces involved?
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6. Two trolleys on a horizontal frictionless track collide and experience a perfectly elastic collision. Before the collision, one with mass 0.25 kg, is moving at 0.80 m s – 1 to the right
and the other, with mass 0.40 kg is moving at 0.50 m s– 1 to the left.
(a) What are their velocities after the collision?
v of 0.25 kg object = 0.25 m s-1 to the left
v of 0.40 kg object = 0.50 m s-1 to the right
(b) Sketch a velocity-time graph showing their motions before, during and after the collision.
(c) Sketch a momentum-time graph showing their momenta before, during and after the collision.
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Power, work and energy :
The SI unit of power is the watt, W; one watt is one joule per second 1 W = 1 J s-1
Power is a measure of how ‘difficult’ it is to do the work – if you can spread the work over a
longer period of time, then the power needed is less.
Example
1. Suppose that you have to move 100 concrete blocks, each with a mass of 2.5 kg, from the footpath up 12 m to a balcony.
(a) What is the change in the potential energy of the 100 blocks when they have all been moved up to the balcony?
(b) How much work has been done in moving the blocks up onto the balcony?
(c) Suppose that it took you one hour to move all of the bricks, what was your average power doing this work?
(d) The method you used to move the blocks was to carry them up a ladder two at a time. Suppose you had carried them up five at a time over half an hour, what would have been
the work done and your average power then? Explain your reasoning.
(e) You decide to work out whether it would be better to have walked up the long wheelchair ramp from the footpath to the balcony carrying the blocks one at a time.
What would be the work done in this case and what of the power you would have to
supply in doing it?
P = W/t
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Power, force and velocity Power P = W/t and as W = Fx then power = Fx/t
From which we get:
1. Suppose that a large locomotive must apply a force of 2250 kN to a train to keep them moving at a constant speed of 12 m s –1.
Draw and label a diagram
(a) What power must the locomotive generate?
(b) How much work is done by the locomotive’s engine in 30 minutes?
(c) How far has the locomotive travelled in this 30 minutes?
(d) What is the average frictional force that is opposing the motion of the locomotive?
P = F v
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Word Definition
You should keep a longer list of words and their meanings in your
own notes. You write an unfamiliar word during class and add its
meaning after class that day.
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STUDENT NAME……………………………….
STUDENT NUMBER………………
TEACHER…………………
PHYSICS A MOTION IN OUR WORLD
PRACTICAL MANUAL.
2020
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Contents
Information .................................................................................................................................. 74 About the Practical Manual ....................................................................................................... 74 Risk assessments ....................................................................................................................... 74 Pasco Datalogger Software ....................................................................................................... 74 Writing conclusions ................................................................................................................... 75
Practical 1: The pendulum ......................................................................................................... 76 Practical 2: Human motion ........................................................................................................ 82 Practical 3: Uniform motion and uniformly accelerated motion ............................................ 86 Practical 4: Motion of a basketball ............................................................................................ 91 Demonstration 1: Projectile motion ........................................................................................... 94
Practical 5: Video analysis of projectiles ................................................................................... 95 Appendix: How to use VideoPoint .......................................................................................... 100
Practical 6: Hooke’s law ........................................................................................................... 107 Demonstration 2: Simple Harmonic Motion .......................................................................... 110 Simulation 1: Simple Harmonic Motion ................................................................................. 113 Practical 7: Mass on a spring ................................................................................................... 114
Practical 8: One-dimensional collisions ................................................................................... 121
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Information
Your practical manual will be collected and marked at the end of the practical test and practical
exam. As part of this you are required to complete all sections of the practical manual unless
otherwise told by your teachers.
Your practical manual may be used when you do your practical exam (open book exam). For this
reason it is important that your work is complete and of a high standard.
About the Practical Manual
The practical manual has activities categorized in different ways:
Practical A full experiment where you are required to write an Aim, Theory, Method,
Results and Conclusion for the experiment.
Activity A simpler experiment where you only record results and some
analysis/conclusion.
Simulation An experiment done using a computer program.
Demonstration Something a teacher demonstrates because there is only one set of apparatus,
the experiment is dangerous or we don’t want to spend a long time on this
activity.
Risk assessments
As we will be using electrical equipment etc we need to do a risk assessment. The aim of
making a risk assessment is to identify the hazards connected with an activity, to assess the
seriousness of these hazards and to plan controls to reduce the risks connected with the
experiment to a minimum or at least to an acceptable level.
The hazard presented by an experiment is its potential to do harm and risk from an experiment
is the chance that it will cause harm in the circumstances of use.
Pasco Datalogger Software
Pasco datalogger software has been installed in two computers next to the printers in each
student access computer lab on Level 2 ( Room 2.09 and room 2.27 ).
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Writing conclusions
Your conclusion is a summary of the important things you have learnt in the experiment. As a
summary you cannot introduce anything new. If something is mentioned in the conclusion then it
should first be discussed in the results section. The conclusion should respond directly to the
Aim you have stated. It is the most important part of the experiment as it summarises concisely
and precisely what you have found out and what you think this means. You must be careful not
to just restate the results but to try and say why the results came out the way they did. It is not
necessary to start the conclusion with "It can be concluded` that..." – just start talking about what
you have observed and why this might be.
.
What can you conclude from your results?
This depends on several things, for example:
What results you actually get.
How accurately you know these values; that is, the error in the values you have measured and calculated.
Whether these measured quantities agree with known values found in textbooks.
It is important estimate greatest probable errors carefully whenever you make measurements and
calculate other data from them so that you can draw valid conclusions from the experimental
data.
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Practical 1: The pendulum
Materials (Use this list to assist you in writing a Method)
Clamp stand
Stopwatch
Metre ruler
50 g mass
String and scissors
Sticky tape
Introduction
In this experiment we are trying to find out how the period depends on the properties of an
oscillating system.
In this experiment you have to do accurate experimental measurements and try and work out
how the period varies with the length of the pendulum.
Some of the things you will do are:
measure five different lengths of the pendulum
set the pendulum swinging and devise a way to accurately measure the period
give an estimate of the accuracy of your measurements – you should revise errors in your course introduction study guide
draw a graph of period (T) as a function of length (L)
figure out and draw a graph involving T and L that will give you a straight line.
Aim
To determine how the period of a pendulum depends on its length.
Theory
You do not need to write any theory for this experiment.
Method
Write some text to introduce a diagram and then draw a labelled diagram of your experiment
and write down what you did to gather your results. You may describe the method in point form
or using paragraphs.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
Diagram ……...……………………………………………
……..…………………………………………….
……………………………………………………
……………………………………………..………
….…………………………………………………
……………………………………………………
……………………………………………………
……………………………………………………..
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Results
Record all of your measurements and calculated quantities in the table below.
Indicate and justify your estimates of uncertainty in all measured and calculated quantities.
Include errors for all your measurements.
L (m) Time for Ten Periods (s) T (s)
0.00
0.00
±
±
±
±
±
±
±
±
±
±
Greatest Probable Errors
Justify the values you have given.
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
……………………………………………………………………………………………………
On the next page draw a graph of T versus L, add a line or curve of best fit and suggest what
you think the shape of the graph is (i.e. linear, parabolic, etc).
On the page after that, plot a graph with some function of T and /or l on the axes which will
produce a straight line graph. Write the equation of that line (after calculating its gradient).
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Gradient = ….. -…….
----------------
….. -…….
= . So the equation of the line is:
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Discussion
Summarise what you have learnt from your graph.
……………………………………………………………………………………………………
………………………………………………………………………..…………………………..
Excel Table and Graphs
Follow the instructions in the course description “Drawing Graphs in Excel” and create a table
similar to that on the last page and two graphs, one as before of T versus L and one that gives a
linear relationship i.e. a straight line graph.
To get a linear relationship you will need to choose either a function of T (T2, T½ etc,) or a
function of L (L , L½ etc,) which you think will give you a straight line and calculate this in a
fourth column in an Excel table.
Stick your Excel table and two graphs in the space below.
Excel Table goes here
Excel Graph (T vs l) goes here
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Excel Graph (T2 vs l) goes here
Discussion
Summarise what you have learnt from your Excel table and two graphs.
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
Conclusion
Write a conclusion that explains what you discovered from your table and graphs in terms of the
pendulum’s motion.
It states as generally as is justified, what has been found out in THIS experiment.
The conclusion must be written in sentences and paragraphs!
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
………………………………………………………………………………………………..
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Practical 2: Human motion
Introduction
In this experiment you will be using the data logger to observe human motion as you walk
towards and away from a motion detector. The motion detector will be placed at the edge of a
table and you will walk towards and away from the detector.
Aim
Write your own Aim for this experiment.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
Method
Write some text to introduce a diagram and then draw a labelled diagram for your experiment
illustrating the apparatus used. Write text to describe what you will do. Refer to the Course
Description: “How to set up the Datalogger and Motion Detector.”
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
Diagram
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
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Results
Stick graphs below for at least two different types of motion.
Datalogger Graphs go here.
x-t, v-t, a-t
Constant velocity.
All three graphs should be printed
on one sheet of paper!
Discussion
Discuss the shape of your graphs. Compare their shape with what you might expect.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
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Datalogger Graphs go here.
x-t, v-t, a-t
Uniform acceleration.
All three graphs should be printed
on one sheet of paper!
Discussion
Discuss the shape of your graphs. Compare their shape with what you might expect.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
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Conclusion
Write a conclusion for both parts of the experiment.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
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Practical 3: Uniform motion and uniformly accelerated motion
Aim
To make quantitative and qualitative observations, drawing and analysing graphs of motion, of a
low-friction trolley as it travels along a horizontal smooth surface at a constant velocity and then
along a sloped track with constant acceleration.
Theory
Sketch graphs of displacement, velocity and acceleration versus time for (i) an object travelling
at constant velocity and (ii) a second set of graphs for an object undergoing constant
acceleration. Identify equations to go with each graph.
Constant velocity Uniform acceleration
Graphs Equation Graphs Equation
Method
Write some text to introduce a diagram and then draw a labelled diagram for your experiment
illustrating the apparatus used. Write text to describe what you will do. DO NOT describe the
same experiment three times BUT instead have a statement something like “The above steps
were repeated for ….”.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
Diagram
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……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
Results
A. Constant velocity Set up the frictionless track horizontal so that we can have constant velocity when the trolley is
pushed. Print graphs of distance against time, velocity against time and acceleration against
time from the data-logger program and stick them in below.
Graphs go here.
x-t, v-t, a-t
Constant velocity.
All three graphs should be printed
on one sheet of paper!
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Magnitude Units
v =
a =
Discussion
Summarise the shape of your graphs and include your measurements for velocity and
acceleration.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
B. Increasing velocity Set up the frictionless track with a slope so that we have an increasing velocity. Print graphs of
distance against time, velocity against time and acceleration against time and stick them in
below.
Datalogger Graphs go here.
x-t, v-t, a-t
Increasing velocity.
All three graphs should be printed
on one sheet of paper!
Magnitude Units
a =
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Discussion
Summarise the shape of your graphs and include your measurements for acceleration.
……………………………………………………………………………………………………
………………………………………………………………………………………………..…..
……………………………………………………………………………………………………
…………………………………………………………………………………………………....
C. Decreasing velocity Set up the track with a slope so that the car goes up the slope so that we have a decreasing
velocity. It may be more interesting to have the car go up the slope and then come back down
again. Print graphs of distance against time, velocity against time and acceleration against time
and stick them in below.
Datalogger Graphs go here.
x-t, v-t, a-t
Decreasing velocity.
All three graphs should be printed
on one sheet of paper!
Magnitude Units
a =
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Discussion
Summarise the shape of your graphs and include your measurements for acceleration.
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Conclusion
Write a conclusion for the different parts of the experiment.
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Practical 4: Motion of a basketball
Introduction
In this experiment use a motion detector and a datalogger to record the motion of a basket ball
falling vertically under gravity.
Aim
Write your own Aim for this experiment.
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Theory
Describe and explain the motion of the basketball for this situation.
Sketch graphs for y-t, v-t and a-t and write equations to describe this motion.
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Graphs Equations
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Method
Write some introductory text and then draw a diagram illustrating the arrangement of the
equipment (including datalogger and computer). Summarise the steps taken to record and
analyse the data.
Note: Mount a motion detector in the air pointing down. This may best be done by using two
retort stands on top of a table. Please do not put the motion detector on the ground as you may
break it if you drop a basketball onto it.
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Diagram ……...……………………………………………...
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Results
Datalogger Graphs go here.
x-t, v-t, a-t
All three graphs should be printed
on one sheet of paper!
Magnitude Units
a =
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Conclusion
Write a conclusion for the experiment. Include your measurement of the acceleration and
compare it to the known value due to gravity.
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Demonstration 1: Projectile motion
The projectile trolley Describe the demonstration you were shown of the projectile trolley. What information does this
give on the horizontal and vertical components of projectile motion?
Diagram
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Two falling balls Describe the demonstration you were shown with two balls, one projected horizontally and the
other dropped vertically. What information does this give on the horizontal and vertical
components of projectile motion?
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Diagram ……...……………………………………………...
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Practical 5: Video analysis of projectiles
Introduction
In this experiment you will examine the changes in the x (horizontal) and y (vertical)
components of displacement, velocity and acceleration of a projectile during its flight.
A ball (the projectile) is thrown as shown below and a digital camera is used to film its motion.
The camera is facing the wall and the procedure for using it is included on Study Smart. A metre
ruler is placed against the wall as shown in the picture for calibration.
The image sequence (movie) is loaded into the computer at the front of room 3.35 as explained
in the Appendix: How to use VideoPoint.
You then look at the instructions “using the digital camera (Finepix) to film the projectile” and
analysed frame by frame using the VideoPoint program. Try not to include too many extra
frames before and after the action.
Aim
Write your own Aim for this experiment.
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Method
Write some text to introduce a diagram and then draw a labelled diagram for your experiment.
Describe the method for this experiment.
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Diagram
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Results
Use the VideoPoint software to generate graphs of each of the x and y components for
displacement, velocity and acceleration of the ball against time.
VideoPoint Graphs go here.
Horizontal components
x-t, v-t, a-t
All three graphs should be printed
on one sheet of paper!
Magnitude Units
v =
a =
Discussion
Summarise the shape of your graphs and include your measurements for velocity and
acceleration.
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VideoPoint Graphs go here.
Vertical components
y-t, v-t, a-t
All three graphs should be printed
on one sheet of paper!
Magnitude Units
a =
Discussion
Summarise the shape of your graphs and include your measurements for acceleration.
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Conclusion
Write a conclusion for the experiment. State as generally as is justified, what has been found out
in THIS experiment.
Compare the vertical acceleration to the known value of acceleration due to gravity.
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Appendix: How to use VideoPoint
1. Open the VideoPoint software which is found in computer rooms 3.09, 3.10 and 3.35. Go to Start-
Programs-VideoPoint-VideoPoint 2.5.
2. This will bring up a new window. Click the “Open Movie” button.
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3. The movie is located at :
G:\StudentResources\Science\USFP\Physics\MotioninOurWorld\Ross’sProjectileVideo
\Projectile.AVI
4. This will bring up a new window. Make sure the “number of features or objects to be located” is 1. Then click “OK”.
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5. Now you need to collect the data. Move the cursor over the movie window area. The cursor
should look like this . The bottom right of the movie window should have the italicized
text “Point S1”. This is the first video point to be located by you in the frame that currently
appears in the movie window.
6. The movie will automatically advance to the next frame. Continue clicking on the location
of the ball in each frame until the last frame of the movie.
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7. You have now collected data for this movie. However, the data is currently in pixels. You
now need to scale the movie. This involves marking the position of each end of a 1m ruler
that was placed in the movie.
To scale the movie:
Click on the scale icon in the left hand side of the screen. A dialog box will appear.
Enter 1m into the “Known Length” box and click “Continue”.
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You then need to click on one end of the 1m ruler.
Then click on the other end of the 1m ruler. The points “Scale 1A” and “Scale 1B” should
appear at either end of the ruler with a straight line draw between these points.
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8. Once you have collected your data, you can start producing graphs.
In this experiment, you will need to generate six graphs.
These are:
Position-time graph (x – component)
Velocity-time graph (x – component)
Acceleration-time graph (x – component)
Position-time graph (y – component)
Velocity-time graph (y – component)
Acceleration-time graph (y – component)
To generate a graph, click on the graph icon in the left hand side of the screen. A dialog
box will appear.
Select which of the two components (x or y) you want to plot and the variable (position,
velocity or acceleration) that you want to plot. Then click “OK”.
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The graph will appear in the screen, as shown below.
Repeat this process to generate the other five graphs needed for the experiment.
NOTE. The scale chosen by videopoint for some graphs is too sensitive, resulting in a large
scatter of points.
To fix this, click on the top number of the Y scale and enter a much larger number.
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Practical 6: Hooke’s law
You are provided with a spring, a mass carrier, five 50 g masses, a hook, a metre ruler and a
clamp stand.
Aim
The Aim of this experiment is to determine the relationship between the Force applied and its
extension.
Method
The equipment is set up as shown in the diagram below.
metre ruler clamp
stand
mass’s stick
spring
The mass carrier and the masses each have a mass of 50 g.
Each mass is placed gently on to the spring and the extension (the change in the length) is measured.
A graph of Force versus extension is plotted.
Remove the masses from the spring as soon as you complete your measurements.
Results
(a) Record your results in the table below. Show all measurements taken.
Mass (Kg) Force (N) Extension (m)
0.00 0.00 0.00
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(b) Plot a graph of force against extension.
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(c) Is the origin an experimental point? Explain your answer.
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(d) Measure the slope of the graph and state its units.
Magnitude Units
slope =
(e) Write a complete equation relating the force and extension of the spring.
Conclusion
Write a conclusion for the experiment. Summarising the purpose of the experiment (see Aim and
Method) and then summarise your Results noting the relationship between Force and extension
and quoting the equation you have identified.
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Demonstration 2: Simple Harmonic Motion A mass on a spring pulled down and released is an example of simple harmonic motion (SHM).
The mass will freely oscillate up and down which we describe as SHM.
We investigate simple harmonic motion with the following apparatus.
(Instructions: Set motion sensor on narrow beam setting to 50 Hz frequency. Drag position/velocity/acceleration
to one graph. Use small spring on large clamp stand with 200 g slotted mass. Use sticky tape to tape mass
to spring and spring to hook on clamp stand. Put you eye above spring and line up mass with
motion sensor. Take care to ensure that the mass can not fall and damage the motion sensor.)
We use the data-logger and motion sensor to record position-time, velocity-time and
acceleration-time graphs for the motions of a mass oscillating vertically on a stretched spring.
If necessary adjust the time axes of the three graphs to have the same scale and origins aligned
so that the phase relationship between the three quantities can be observed.
Draw graphs on the following page of displacement, velocity and acceleration against time for
the SHM motion. Make t = 0 the point where the displacement is zero, and then make the
displacement positive.
mass which is pulled down and
then released.
spring
motion sensor
connected to
data logger
Make sure the mass
and motion sensor are
at least 30 cm away
form each other
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displacement
positive
negative
time
positive
negative
time
velocity
positive
negative
time
acceleration
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Results
We can see from the displacement/time graph and the acceleration/time graph that they are
directly proportional to each other but are exactly out of phase with each other, i.e. as one goes
up the other goes down. This leads to the following force/displacement graph.
a = - constant x s
hence F = - constant x s
When we look at the mass we find that it accelerates towards its rest position, but as it
approaches the rest position its acceleration decreases. Once it has reached the rest position, the
acceleration is zero, but is does have an upwards velocity. This means it continues past the rest
position. As it moves past the rest position it begins to decelerate. This deceleration increases as
it moves away from the rest position. Finally it stops and then accelerates down towards the rest
position in the same way as before.
The greatest displacement the object ever has from its position of rest is called the amplitude of
the motion.
The velocity of the object is zero at its maximum displacement. The velocity is maximum when
the displacement is zero.
As the acceleration varies then SHM is an example of non-uniformly accelerated motion. (We
can not use the equations of motion here.)
force
displacement
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Simulation 1: Simple Harmonic Motion
A web site with a simulation of SHM is found at:
http://www.surendranath.org/GPA/Oscillations/SHM/SHM.html
Run this simulation and summarise what you discover.
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Practical 7: Mass on a spring
Introduction
In this experiment we will investigate the motion of a mass undergoing Simple Harmonic
Motion:
You will:
Observe the phase relationship between displacement, velocity and acceleration.
Observe the relationship between the period of oscillation and the mass.
Compare the direct measurement of the period to that calculated from the mass and spring constant.
Compare the direct measurement of the spring constant as the ratio of the weight to extension (as you did in Hooke’s law) with the value found from a graph of T2 against m.
Note:
Use a datalogger and motion sensor to record position-time, velocity-time and acceleration- time graphs for the motions of different masses oscillating vertically on a stretched spring.
The masses you need to use for your experiment will depend on the spring you use. eg 50, 100, 150, 200 and 250 g
Aim
Write your own Aim for this experiment.
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Method
Write some text to introduce a diagram and then draw a labelled diagram for your experiment.
Write up your own description of how the experiment was done by your group being careful to
identify possible sources of error. You should include enough information about the equipment
and experimental arrangement to allow another student at the same level in Physics to do the
experiment just as you have done it. Include information about using the Datalogger.
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Diagram
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Results In the space below stick position-time, velocity-time and acceleration-time graphs recorded for
one mass. Your graph should show 10 periods (10T) of oscillation.
Datalogger Graphs go here.
y-t, v-t, a-t
All three graphs should be printed
on one sheet of paper!
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Identify the phase relationship between the y-t and v-t graph. Express your answer as a function of period T.
Identify the phase relationship between the y-t and a-t graph. Express your answer as a function of period T.
Using the displacement-time graph measure the time for 10T and hence calculate the period of oscillation for the mass (label this as T1).
Measure the maximum amplitude from the displacement-time graph. Measure the maximum velocity from the velocity-time graph.
Remembering that x = A sin ωt and v = A ω cos ωt calculate the angular frequency ω by
calculating the ratio.
Then calculate the period and call this T2 (T2 = 2π/ω).
Enter these values for T1 and T2 below and estimate an error/uncertainty for both.
Phase relationship y-t and v-t =
Phase relationship y-t and a-t =
=
T1 measured from s/t graph = ±
T2 calculated from 2π/ω = ±
Discussion 1
Summarise your results so far including (i) discussing the phase relationship between
displacement, velocity and acceleration and (ii) comparing the two values for period.
Note anything else distinguishing about the graphs.
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Repeat the experiment for a number of different masses and record your results in the table
below. DO NOT print any more graphs – do all your analysis on the computer.
(OPTIONAL………………………………)
k found from the ratio of force /
extension
T is found from the y-t
graph
maximum
velocity
from the v-
t graph
maximum
amplitude
from the
y-t graph
is
found
from the
relations
hip
v =
ωAcosω
t
T2 is
found
from
2π/ω
M1 (kg)
F
(N)
Ext
(m)
k
(Nm-1)
T
(s)
M2 (kg)
vmax A max T2
k average =
NOTE. M1 does not include carrier which was on at x = 0. M2 includes carrier which must
be accelerated by the spring.
Using Excel, draw a graph of period (T1) versus mass and determine the shape of this graph.
Excel Graph goes here.
T versus m
Shape of T-m graph
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From this information work out the quantities required to produce a linear graph and draw this graph. Include a Trendline (line of best fit) and an equation.
From the linear graph determine the spring constant k from the Trendline or the equation.
Compare this value to the average value of that found from the ratio of force / extension.
Excel Graph goes here.
Linear graph
Magnitude units
Slope of graph =
k graph =
k average (table) =
Discussion 2
Summarise your results so far including (i) shape of T-m graph, (ii) quality of linear graph, (iii)
comparison of two values for k.
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Conclusion
Write a conclusion for the experiment (in paragraphs). Your Conclusion should include:
The values you have found for k by the two methods (with uncertainties).
The level of support (or otherwise) that this provides for the validity of the equation under
investigation.
OPTIONAL.
Summarise the graphs y-t, v-t and a-t (i) comparing the phase relationship between displacement, velocity and acceleration and (ii) compare the two values for period.
Summarise (i) the shape of the T-m graph, (ii) the quality of the linear graph and (iii) comparison of two values for k.
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Practical 8: One-dimensional collisions
NB This experiment needs to be done over two days if students are to
gather data for a number of elastic and inelastic collisions (eight in total).
It may be practical for different groups to do different collisions and
share their results with the whole class.
Materials
Low friction track and low friction cars.
Datalogger and two motion detectors – one placed at either end of the low friction track. Use f = 20 Hz.
Introduction
In this experiment you will investigate the conservation of momentum and kinetic energy in real
one-dimensional collisions between low-friction cars. You will study a number of collisions
chosen from the following:
one moving car colliding with a stationary car of equal mass elastically / inelastically
two cars of equal mass travelling in opposite directions colliding elastically / inelastically
one moving car colliding with a stationary car of different mass elastically / inelastically
two cars of unequal mass travelling in opposite directions colliding elastically / inelastically
Cars with magnets will be used for elastic collisions. Cars with blu-tac or Velcro will be used for
inelastic collisions.
Label the cars A and B and measure their masses before continuing.
Aim
Write your own Aim for this experiment.
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Theory
Draw a diagram illustrating a one dimensional collision between two masses (eg circles) approaching each other and then bouncing apart in opposite directions after the collision.
Label the masses m1 and m2 and identify initial and final velocities u1, u2, v1 and v2.
Summarise the conservation laws for momentum and kinetic energy as they apply to two bodies experiencing a one-dimensional collision.
These equations should be written in two ways (i) in terms of p’s and KE’s and (ii) in terms
of m1, m2, u1, u2, v1 and v2.
Explain the conditions under which each is obeyed and how these are approximated in the actual apparatus used in this experiment.
Identify in terms of elastic and inelastic collisions.
Diagram
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Method
Write some text to introduce a diagram and then draw a labelled diagram for your experiment.
Write up your own description of how the experiment was done being careful to identify reasons
why each of the conservation laws might not work and note other possible sources of error.
You should include enough information about the equipment and experimental arrangement to
allow another student at the same level in Physics to do the experiment just as you have done it.
Include information about using the Datalogger.
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Diagram
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Page 124 USFP Physics 2020 – Motion in our World Work Book
10/1/2020 You are learning from a Taylors College Work Book
Results
Print and stick below a set of graphs for one of your collisions. There should be two velocity-
time graphs. Indicate on your graph the car i.e. whether it is A or B and use a sketch to show the
direction of motion of each car before and after the collision.
Design and construct a table in Excel similar to the one below suitable for recording:
masses
measured initial and final velocities
calculated initial and final momenta
calculated initial and final kinetic energies
calculated percentage Ek lost in each case. Transfer the data from your first experiment to the Table.
Print your Excel table and stick it in the space below.
Collision* m1 u1 v1 m2 u2 v2 pi pf Agree? i f %KElost
1
2
3
4
5
6
7
8
*Type of Collision
1. one moving car colliding with a stationary car of equal mass elastically 2. one moving car colliding with a stationary car of equal mass inelastically 3. two cars of equal mass travelling in opposite directions colliding elastically 4. two cars of equal mass travelling in opposite directions colliding inelastically 5. one moving car colliding with a stationary car of different mass elastically 6. one moving car colliding with a stationary car of different mass inelastically 7. two cars of unequal mass travelling in opposite directions colliding elastically 8. two cars of unequal mass travelling in opposite directions colliding inelastically
NOTE: If short of time and/or motion sensors do only 2 and 6 above.
Work Book USFP Physics 2020 – Motion in our World Page 125
10/1/2020 You are learning from a Taylors College Work Book
Discussion
Compare the initial and final momentum for the different collisions. Was momentum conserved
for the elastic/inelastic collisions? Were there particular experiments which appeared to work
better than others? If momentum was not conserved identify reasons for this.
Compare the initial and final Kinetic energy for the different collisions. Was Kinetic energy
conserved for the elastic/inelastic collisions? Do you expect Kinetic energy to be conserved for
this collision? If Kinetic energy was not conserved identify reasons for this.
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Conclusion
Write a conclusion for the experiment (in paragraphs). Your Conclusion should include:
Identify the elastic collisions and discuss whether momentum and Kinetic energy were conserved for these collisions. Compare the results to what you might expect theoretically.
Identify the inelastic collisions and discuss whether momentum and Kinetic energy were conserved for these collisions. Compare the results to what you might expect theoretically.
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Page 126 USFP Physics 2020 – Motion in our World Work Book
10/1/2020 You are learning from a Taylors College Work Book
Motion in our World
t
s v
av
t
v a
av
F ma t
uv a
atuv yauv 2 22
2
2
1 atuty W = mg
2
= = t T
f2 =
v = r 2
c a r
Fc = mac = mω 2r y = A sin ωt
2
c
v a
r
v = ω A cos ωt a = - ω2 A sin ωt
a = - ω2 y F kx
k
m T 2 2
l T
g
F t = I = mv - mu = p W = F. x
EK = ½ m v 2 – ½ m u 2 Q = FF x
EP = m g h EE = ½ k x 2
Work Book USFP Physics 2020 – Motion in our World Page 127
10/1/2020 You are learning from a Taylors College Work Book
Page 128 USFP Physics 2020 – Motion in our World Work Book
10/1/2020 You are learning from a Taylors College Work Book