only for prof. goodwill

Physics 225 Fall 2014

Problem Set #1

NOTE: Show ALL work and ALL answers on a piece of separate loose leaf paper. Not on this sheet.

1) Convert the following using your unit conversion sheet. For

your answers, use correct scientific notation when appropriate. a) 75 miles/hour to meters/second b) 40 kilometers/hour to yards/minute c) your age (in years only) into seconds d) 1.8 m3 to in3 e) 235 fm to meters

2) Do the following calculus operations …

a) y = x3 + 4x – 7, find dx dy

b) x = 3tt 4 1 2 +− , find

dt dx

c) fb a bS

3 −= , find the derivative of “S” with respect

to “b” d) v = 4t4 – 3t2 + 6, integrate “v” with respect to “t”,

showing the proper notation. 3) When lightning and thunder

occur in movies they are typically shown to happen at the same time. (In case you don’t know … thunder is the sound that lightning makes.) Let’s say you are standing 3 miles from a lightning strike. a) Given that the average speed of sound is 342 m/s, calculate the time it takes for you to hear the thunder. b) Given that the average speed of light is 300,000,000 m/s, calculate the time it takes for you to see the lightning. c) So, what is wrong with the way lightning is presented in movies? In what situation will you hear the thunder and see the lightning at the same time?

4) You are late for class so you are driving at a constant

velocity of 90 mi/hr north on the 57. After driving for 6 minutes you are at Mile Marker #13. Assuming that the mile markers are increasing in number as you drive, at what mile marker were you located at the beginning of the 6 minutes?

5) You are preparing for your next ½ marathon by running around Grant Blvd. in Corona. This road is a perfect circle with a 1 mile diameter. You begin your run at the westernmost point of the road. It takes you 4.35 min to get to the easternmost point. (a) Calculate your average speed for this part of your run. (b) Calculate your average velocity for this part of your run. You get a cramp in your leg and walk back to your starting point at 6 ft/s. (c) What is your average speed for your round trip? (d) What is your average velocity for the round trip?

6) State the type of VELOCITY for graphs (a) and (b). Tell

whether; Increasing, Decreasing, Constant, or Zero; and Positive or Negative.

7) Draw the velocity graphs for the following position graphs.

t

(A) (B) x x

t

t

x

x

t(A) (B)

1) a) 33.5 m/s b) 729 yd/min d) 1.1 x 105 in2 e) 2.35 x 10–13 m 2) a) 3x2 + 4

b) 1t 2 1 −

c) f a b3

db dS 2 −=

2) d) ct6tt 5 4

35 ++−

3) a) 14.1 s b), c) not given 4) #4 5) a) 9.68 m/s b) 6.16 m/s c) 3.08 m/s

Physics 225 Fall 2014

Problem Set #2

1) The velocity of an object is given by v(t) = 9t2 – 4t + 6.

At t = 1 s the position of the object is at 10 m. Calculate the acceleration equation and position equation for the object.

2) Derive the equation, v2 = v02 + 2aΔx, by combining the

other two main motion equations given in class. 3) Mitch is riding a stolen recumbent bicycle at a constant

velocity of 40 m/s. At some point in time, he applies the brakes which make the recumbent slow to a stop over a period of 5.0 s. (a) What is the acceleration of the bicycle? (b) How far does the bicycle travel before coming to a stop?

4) A Cessna Jet (call number 90-RC) has a lift-off velocity of

120 km/hr. The runway that 90-RC takes off from is 240 m long. (a) What is the minimum acceleration that the jet has to have in order to take off before it reaches the end of the runway? (b) How long does it take for 90-RC to take off?

5) A train is traveling at 100 m/s. The engineer applies the

brakes because he sees a recumbent bicycle on the tracks ahead. The brakes cause an acceleration of -2.2 m/s2. The engineer applies the brakes when the train is 960 m from the bicycle. How long does it take the train to reach the bicycle?

6) An object has a position equation of

4t6t4t 3 2

)t(x 23 ++−= . (a) At what times does the object

have a velocity of zero? (b) What is the location of the object when its acceleration is zero?

7) While chasing Mitch for destroying his bicycle, Skid does the following in his Pinto: 1. He starts from rest and accelerates at 0.6 m/s2 for 15 s. 2. Travels at a constant velocity for 2 minutes. 3. Slows to a stop with an acceleration of –0.75 m/s2. What is the total distance that Skid travels?

8) A juggling bag is thrown straight up into the air and is

caught 2 s later. (a) What is the initial velocity of the ball? (b) Calculate the maximum height that it reaches.

9) A coyote with and anvil falls off of a high cliff. Luckily the

coyote has a parachute. The chute opens and they both fall at a constant velocity of 10 m/s. When they are 50 m above the ground, the coyote drops the anvil. (a) How long does it take the anvil to hit the ground? (b) What is the velocity of the anvil right before it hits the ground?

10) Mitch is skydiving. He jumps out of a plane and reaches a

terminal velocity (constant velocity) of 20 m/s. When he is 1000 m above the ground, Skid, standing directly below Mitch, fires a rocket launcher at Mitch. The rocket has an initial velocity of 125 m/s. (a) How much time does Mitch have to figure out how to survive this predicament? (b) Assuming Mitch does not have his wits about him, at what location above the ground will the rocket intersect with him?

1) a(t) = 18t – 4 7) 1202 m

x(t) = 3t3 –2t2 + 6t +3 8) a) 9.8 m/s 3) a) - 8 m/s2 b) 4.9 m

b) 100 m 9) a) 2.33 s 4) a) 2.32 m/s2 b) -32.9 m/s b) 14.4 s 10) a) 10.9 s 5) 10.9 s b) 782 m 6) a) 1 s, 3 s

b) 5.33 m

Physics 225 Fall 2014

Problem Set #3

Note: Bold face type implies vectors. 1) Given three vectors; F1 = 35, F2 = 40, F3 = 80 (see

the diagram for the angles), find the resultant vector of all three using the component method.

2) Repeat Problem #1 using unit vectors. Make sure

you write out each vector using unit vectors first and show ALL of your work.

3) Using the vectors F1 and F2 from Problem #1, find

the following, leaving your answers in unit vector notation. a) F1 – F2 b) 2F2 – F1 c) F1 + F2 d) F2F1 e) F2 (F1 + F2) / F1

4) For the vector A = – 8j + 4k, find its magnitude and

direction. Also state in what plane the vector resides.

5) Given the vector: A = 3i + 4j, find any three vectors B that also lie in the xy plane and have the property that A = B but A ≠ B. Write out these vectors with unit vector notation.

6) Given two vectors; S = 2.5, R = 4.2, (see the

diagram for the angles), find the dot product using (a) the magnitudes and (b) unit vector notation.

7) Calculate the angle between the two vectors given by:

b = 3i – 2j – 8k , c = i – 5j + 4k. 8) Determine if the following vectors are perpendicular

to each other. a) P = 3i – 2j + k , Q = 4i + 9j + 6k. b) D = i + 2j – 5k , F = 2i – 7j + k.

9) Determine the vector product of each set of vectors.

a) P = 3i – 2j + k , Q = 4i + 9j + 6k. b) D = i + 2j – 5k , F = 2i – 7j + k.

1) 78.4, 17.1° 6) – 9.09 3) a) – 39.4i – 9.5j 7) 109.5° b) 56.3i + 45.8j 8) a) Perp.

c) 75 b) Not Perp. d) – 900i + 1072j 9) a) – 21i – 14j + 35k e) – 6.4i + 72.1j b) – 33i – 11j – 11k

4) 8.94, 26.6°, yz plane

25°

30°

F1

F2

F3

40°

x

y

65°

S

R

35°

Physics 225 Fall 2014

Problem Set #4

1) Mitch is out in the backwoods scaring up some dinner with

his bow and arrows. He shoots an arrow horizontally 1.25 m above the ground. He completely misses the rabbit he was shooting at and the arrow plunges into the ground 150 m away. Calculate the initial velocity of the arrow.

2) Sideshow Bob is launched from a cannon at an angle of 30°

with a velocity of 50 m/s. (a) What is Bob’s velocity at the highest point in its trajectory? (b) How long does it take to get to this point? (c) How far does Bob travel horizontally right before he hits the ground? (d) What is the y-component of Bob’s velocity when its velocity vector is pointing downward at an angle of 45° below the horizontal?

7) A baseball is thrown with an initial speed of 25 m/s at an angle of 40° with the horizontal. At the same moment that the ball is launched, a player is 45 m away from the thrower. At what speed and which direction should the player run in order to catch the ball at the same level at which it was released?

3) Skid is going to attempt a field goal by kicking the football

from a point 40 yards out from the goal. The ball must clear the crossbar that is 3 m off of the ground. Skid kicks the ball with a velocity of 20 m/s at an angle of 45° (a) Does the ball clear the crossbar? (b) Is the ball rising or falling as it approaches the crossbar? Show your work for all.

8

4) A stone is thrown upward from the top of a 20 m high cliff

at an angle of 25°. The stone has an initial velocity of 15 m/s. Make the BASE of the cliff the origin. (a) Write out the position, velocity, and acceleration vectors for the stone. (b) What is the magnitude of the velocity of the stone 3 s after it is launched?

5) A baseball is hit at angle into the air. When the ball is at a height of 9.1 m, it has a velocity according to v = (7.6 m/s)i + (6.1 m/s)j. What is the magnitude and direction of the initial velocity of the baseball?

6) A projectile is launched with a speed of vo at an angle of θo

with respect to the horizontal. Derive an expression for the maximum height it reaches above its starting point in terms of g, θo, vo.

) Calculate the angle of launch for an object so that

maximum height of the object is equal to its horizontal range.

9) Skid wants to fly his plane from Champaign, Il to Chicago,

Il. Champaign is directly south of Chicago. Relative to the air, Skid’s plane can fly at 100 mph. The wind is blowing due east at 40 mph. (a) In what direction should Skid fly in order to travel directly to Chicago? (b) What will his speed be relative to the ground?

1) 297 m/s 6) g2

sinv H

22 o θ=

2) a) 43.3 m/s b) 2.55 s 7) 5.53 /s, away c) 221 m 8) 76° 4) b) 26.8 m/s 9) a) 23.6° west of north 5) 16.6 m/s, 62.6°

Physics 225 Fall 2014

Problem Set #5

1) Greezy-X has a 150 N bird- feeder in his backyard. It is suspended by two ropes tied to a third as shown in the diagram. Calculate the tension in each of the three ropes.

2) A block of mass 25 kg is being

dragged across a frictionless surface by a rope with an unknown tension. See diagram. The block starts from rest and travels 30 m in 6 s. Calculate the tension in the rope.

3) A mass, m1 = 5 kg, resting on

a frictionless horizontal table is connected to a cable that passes over a pulley which is then attached to a second hanging mass, m2 = 10 kg. See diagram. Calculate the tension in the cable.

4) Two blocks are fastened to the

ceiling of an elevator. See diagram. Each block has a mass of 10 kg. The elevator accelerates upward at 2 m/s2. Find the tension in each rope.

5) A mass, m1 = 5 kg, resting on a

frictionless ramp is connected to a cable that passes over a pulley which is then attached to a second hanging mass, m2 = 10 kg. See diagram. How long will it take the boxes to move 3 m if they are released from rest?

6) Skid has fuzzy dice hanging from the

rearview mirror of his Pinto. See diagram. When Skid’s car accelerates to the right the string holding up one of the dice makes an angle of 4° with the vertical. Calculate the acceleration of the Pinto.

7) A block is at the base of a 5 m long ramp of angle 35°. If the block is initially given a velocity of 12 m/s up the ramp, how far from the ramp will the block land?

60° 30°

8) Three planets, Hctim of mass 4.88 x 1024 kg, Yzeerg of mass

3.18 x 1023 kg, and Yrral of mass 5.98 x 1024 kg in the solar system Sraggeb are in harmonic convergence (i.e. they are perfectly aligned). See diagram. At the exact time that they are aligned, calculate the net force acting on planet Yzeerg.

9) Two blocks are in contact on a frictionless surface as shown.

Block #1 has a mass of 5 kg and Block #2 has a mass of 2 kg. A force of 14 N is applied to the right on Block #1. (a) Calculate the contact force between two blocks. (b) If the force is applied to the left on Block #2, will the contact force be the same or different. Explain your answer.

10) An object of 4 kg has two forces acting on it: F1 = 3i – 2j

+ 18k, F2 = 13i – 10j + 10 k. If the object is started from rest and the forces are applied over a period of 3 s, find using complete unit vector notation (a) the acceleration of the object and (b) the velocity of the object at the end of the 3 s interval.

1) 130 N, 75 N, 150 N 7) 11.44 m 2) 51.0 N 8) 9.96 x 1016 N 3) 32.7 N 9) a) 4 N 4) 236 N, 118 N 10) a) a = 4i – 3j + 7k 5) 1.13 s b) v = 12i – 9j + 21k 6) 0.685 m/s2

35°

4°

5.0 x 1010 m 3.0 x 1010 m

Hctim Yzeerg Yrral

F

2 1

35°

rope

Physics 225 Fall 2014

Problem Set #6

1) Greezy works in a factory because he does not make enough

money from The Charming Beggars. He is pushing a very large crate of 50 kg across the floor at a constant velocity with a force of 200 N. He is pushing on the crate upwards at an angle of 20°. See diagram. (a) What is the coefficient of kinetic friction between the ground and the crate? (b) Greezy increases his force to 400 N. What is the acceleration of the crate?

2) Two masses, m1 = 10 kg and m2 = 5 kg, are connected by a

string that passes over a pulley as shown below. Mass #1 is on a frictional surface. The system is released from rest. After 1.2 s, mass #2 drops 1.0 m. Calculate the coefficient of kinetic friction between the surface and mass #1.

3) A large block of mass 25 kg is being

accelerated along a frictionless surface by a horizontal force F. A smaller block of mass 4 kg is in contact with the front surface of the large block and will slide downward unless F is sufficiently large. The coefficient of static friction between the blocks is 0.71. Calculate the smallest magnitude that F can have in order to keep the block from sliding downward.

4) Skid of mass 45 kg is in orbit around the planet Hctim in

the solar system Sraggeb. Skid is traveling at 4.05 x 104 m/s. Hctim has a mass of 1.9 x 1027 kg and a radius of 7.14 x 107 m. Calculate Skid’s altitude above the surface of the planet.

5) An air puck of mass 0.25 kg is traveling in a circle of radius 1 m. The puck is attached to a cable that passes through a hole in a frictionless table. The other end of the cable is attached to a hanging weight of 1 kg. The air puck is moving in a circular path on the table so that the hanging weight will remain motionless. See diagram below. (a) Find the tension in the cable, (b) Name the force that is causing the air puck to move in a circle. (c) What is the velocity of the puck?

20° 6) At Magic Mountain there is a ride in which people stand up

against the inside wall of a large cylinder of radius 3 m. The cylinder is then spun with a velocity of 15 m/s at which point the floor drops away leaving the riders suspended against the wall. What is the minimum coefficient of static friction between the rider’s clothing and the wall in order to keep the rider from sliding down?

7) A mass is hung from a ceiling by a

cable that is 2 m in length. The mass is swung around so that it travels in a horizontal circle. While swinging the cable makes an angle of 30° with the vertical. Find the velocity of the mass in order to maintain the angle.

F

30°

8) A horizontal spring with a mass of 8 kg is pulled out to a

distance of 15 cm with an external force of 200 N. If the mass is released (i.e. no more external force) from this point, what will be its acceleration when it is 10 cm from equilibrium? [Do this one for Problem Set #7] [IOW, skip it for this week.]

1) a) 0.445 b) 4.38 m/s2 5) a) 9.8 N c) 6.26 m/s 2) 0.288 6) 0.131 3) 400 N 7) 2.38 m/s 4) 5.9 x 106 m 8) 16.7 m/s2

Physics 225 Fall 2014

Problem Set #7

1) A box of mass 30 kg is being dragged from rest up a

frictional ramp by a rope with tension 500 N. See diagram. The ramp has coefficients of static and kinetic friction of 0.5 and 0.3, respectively. (a) Calculate the net work done on the object if it moves 5m up the ramp. (b) At the end of 5 m, what is the box’s velocity? (Do not use motion equations.)

2) Do problem 1a) over using unit vector notation. Make your

x-axis and y-axis horizontal and vertical, respectively. 3) An object moves from a point {3, - 2, 1} to another point

{- 4, 1, 2} while a force F = (3x + 1)i – 2j + 6z2k acts on it. Calculate the work done by the force.

4) Skid is dragged kicking and screaming to Mike’s Physics

class. A force of 100 N is applied to Skid over a distance 50 m. If it takes 5 minutes to do this, what is the power required?

NOTE: You must use energy methods for the next six

problems in order to get full credit. 5) A projectile is launched at an angle of 60° above the

horizontal with a velocity of 40 m/s. Calculate the maximum height of the projectile while it is flying through the air.

6) A block of mass 0.25 kg is placed on a vertical spring of

constant 5 x 103 N/m. The block is pushed down in order to compresses the spring 0.1 m. The block is released from this position. It is forced upwards and eventually leaves the spring and continues traveling up. Calculate the maximum height above the point of release that the block will achieve.

7) A block is at the base of a ramp of angle 35°. The ramp is

5 m long and has coefficients of static and kinetic friction of 0.6 and 0.2, respectively. If the block is initially given a velocity of 12 m/s up the ramp, what will be its velocity right before it hits the ground?

8) An 80 N box is pushed up a 30° ramp by a 100 N force that is applied parallel to the ramp surface. The ramp has coefficients of static and kinetic friction of 0.8 and 0.22, respectively. The box is moved 20 m up the ramp. Calculate the change in kinetic energy of the box while it is moved up the ramp.

9) Two masses, m1 = 10 kg and m2 = 20 kg, are connected by

a string that passes over a pulley as shown below. The system is released from rest. (a) If the surface of the table is frictionless, calculate the velocity of the blocks after they have moved 2 m.

Ramp Rope

(b) Now, the surface has coefficients of static and kinetic friction of 0.9 and 0.6, respectively. Calculate the velocity of the blocks after they have moved 2 m. (You must use energy methods for this problem.)

Skid hangin’

out

25° Box

35°

10) You and your posse are riding a roller coaster with no

friction. The first tall peak is 140 m high which is followed by a valley that is 10 m high with a radius of curvature of 12 m. The next peak is 60 m high with a radius of curvature 25 m. (a) If the car just barely starts from rest at the first tall peak, what is the velocity of the car at the bottom of the valley? (b) How hard is the car seat pushing up on you? (You have a mass of 35 kg.) (You don’t have to use energy methods for this one.) (c) What would be the required height of the first tall peak in order for the car, at the 60 m high peak, just barely not to fly of the track?

1) a) 1378 J b) 9.59 m/s 7) 11.3 m/s 3) 11.5 J 8) 895 J 4) 16.7 W 9) a) 5.11 m/s b) 4.28 m/s 5) 61.2 m 10) a) 50.5 m/s, b) 7781 N 6) 10.2 m c) 72.5 m

Physics 225 Fall 2014

Problem Set #8

1) The potential energy function of a certain force acting on an

object is U(x) = 3x3 – 5x + 9. Calculate the force on the object when the object is at x = 2 m.

2) A uniform sheet of wood 6 m on

a side has a square piece 2 m on a side cut out of it. The cut-out is centered vertically on the right side of the sheet. The center of the sheet is located at the origin. See diagram. Find the center of mass of the sheet.

3) A cubical box has been

constructed from metal plates of uniform density and negligible thickness. The box is open at the top. See diagram. The sides of the cube are 40 cm. Find the x, y, and z coordinates of the center of mass of the box.

4) Three cricket balls have masses of 4 kg, 6 kg, and 8 kg,

respectively. The ball A has a velocity of 12 m/s while the other balls are initially at rest. The ball A collides with the ball B and rebounds in the opposite direction with a velocity of 7 m/s. (a) Calculate the momentum of ball A before the first collision. (b) Calculate the velocity of ball B after the first collision. Ball B now hits ball C (which has gum on it) and sticks to it. (c) What is the velocity of the balls after this second collision?

5) A 3 kg sphere moving at 30 m/s makes a perfectly inelastic

collision with a second sphere that is at rest. The spheres have a velocity of 10 m/s after the collision. (a) What is the mass of the second sphere? (b) How much kinetic energy is lost in the collision?

6) Skid of weight 440 N is stuck in the middle of a circular

frozen pond of radius 5 m. He cannot move because the pond is absolutely frictionless. He happens to have his 2.6 kg physics textbook so he looks for a solution to his problem. Unable to find a solution he throws his physics textbook in a fit a rage. Skid throws it at a velocity of 4 m/s directly away from him. After throwing the book, how long does it take Skid to reach the edge of the pond?

7) A 4 g object moving to the right at 20 cm/s makes an elastic

head-on collision with a 10 g object moving to the left at 5 cm/s. Calculate the velocity of each ball after the collision.

8) Skid is driving in his car of 2400 kg at 80 m/s towards

Mitch. Mitch is driving in his car of 1600 kg at 60 m/s towards Skid. Calculate the velocity of the center of mass of the Skid/Mitch system.

9) Greezy-X of 50 kg is standing at the end of a flatboat of

180 kg and is 6 m from the shore. See diagram. Greezy then walks to the other end of the 4 m long boat. Assume there is no friction between the boat and the water. Calculate how far Greezy is from shore. HINT: Think about what the center of mass is doing before, during, and after he walks the across the boat.

10) In a game of pool, the cue ball with 5 m/s makes an elastic

collision with the 8 ball which is initially at rest. After the collision, the 8 ball moves off at an angle of 30° with respect to the original direction of the cue ball. Assume that each ball has the same mass. (a) Find the direction of motion of the cue ball after the impact. (b) Calculate the speed of each ball after the collision.

11) A steel ball of mass 3 kg

strikes a solid wall at 10 m/s at an angle of 60°. It bounces off the wall with the same speed and angle. See diagram. If the ball is in contact with the wall for 0.2 s, what is the average force exerted on the ball by the wall? Hint: Think velocity components.

1) 31 N 7) v1’ = -15.7 cm/s, 2) (- 0.25, 0) v2’ = 9.29 cm/s 3) XCM = 20 cm, YCM = 20 cm 8) 24 m/s to the right ZCM = 16 cm 9) 2.87 m 4) a) 48 kg⋅m/s, b) 12.7 m/s, 10) a) 60°, c) 5.44 m/s b) vc = 2.5 m/s, 5) a) 6 kg b) 900 J v8 = 4.33 m/s 6) 21.7 s 11) - 260 N

Z

6 m

Y X

Sticky Gum

60° A B C

60°

Physics 225 Fall 2014

Problem Set #9

1) A bullet of mass 30 g collides completely inelastically with a ballistic pendulum of mass 2.5 kg. The pendulum swings up to a maximum height of 15 cm. Calculate the velocity of the bullet right before it collides with the pendulum.

2) Mitch is sitting at rest on a wheeled chair in a long hallway.

Mitch and the chair have a combined weight of 540 N. Skid throws a 15 kg medicine ball horizontally at Mitch with a velocity of 6 m/s. Mitch catches the medicine ball. If the coefficient of static friction between the chair wheels and the floor is 0.05, how far will Mitch travel after catching the ball?

3) A block of mass 1.3 kg is resting at the base of a frictionless

ramp. A bullet of mass 50 g is traveling parallel to the ramp surface at 250 m/s. It collides with the block, enters it, and exits the other side at 100 m/s. How far up the ramp will the block travel?

4) Ball A of mass 1.2 kg collides elastically with Ball B of mass

1.8 kg. Ball A is traveling at 12 m/s initially. Ball B is at rest initially. The surface is frictionless. After the collision ball B rolls to the right and compresses the massless, springed platform. The spring has a constant of 500 N/m. How far is the spring compressed?

5) You are swinging a ball of 2 kg on a string of 85 cm above your head. The string is making an angle of 75° with the vertical. The ball is moving in a plane that is 1.5 m from the ground. If you release the string from your hand, how far from you will the ball land?

6) An object is rotating at a constant angular speed of

33 rev/min. (a) What is its angular speed in rad/s? (b) Through what angle, in radians, does it rotate in 1.5 s?

7) A machine rotates at an angular velocity of 1.6 rad/s. Its

velocity is then increased to 6.2 rad/s at an angular acceleration of 0.7 rad/s2. (a) How many revolutions does it make during its acceleration? (b) How long does the acceleration take?

8) A dentist’s drill starts from rest and after 3.2 s it has an

angular velocity of 2.51 x 104 rev/min. (a) Find the angular acceleration of the drill. (b) Calculate the angle through which the drill rotates. 40°

9) Mike is riding his unicycle of wheel diameter 20 in. Mike rides for 30 m with an initial angular velocity of 0.75 rad/s and an acceleration of 1.2 rad/s2. (a) How long does it take to ride the 30 m? (b) What is the unicycle’s linear velocity at 30 m?

A B 1) 145 m/s 7) a) 4.08 rev 2) 1.68 m b) 6.57 s 3) 2.65 m 8) a) 821 rad/s2 4) 0.576 m b) 4205 rads 5) 3.03 m 9) a) 13.4 s 6) a) 3.46 rad/s b) 4.28 m/s b) 5.19 rad

Physics 225 Fall 2014

Problem Set #10

1) Derive the equation for the moment of inertia of a one

dimensional stick of mass M and length L about an axis perpendicular to the stick through one end.

2) What is the moment of inertia about

the y-axis for the system as shown. The stick has a length of 2L and a mass of M. The solid sphere has a diameter of L and a mass of 2M.

3) A merry-go-round of mass 150 kg and radius 1.5 m is set in

motion by a rope that has been wrapped around it. Calculate the constant force that you would have to apply to the rope in order for the merry-go-round to start from rest and achieve and angular velocity of 0.5 rev/s in 2 s.

4) A mass of 8 kg is resting on a

horizontal table with a coefficient of kinetic friction of 0.35. The mass is connected to a cable that passes over a pulley of mass 3.5 kg and radius 21 cm. The other end of the cable is attached to a second hanging mass of 15 kg. The system is released from rest. (a) Calculate the acceleration of the system. (b) How many times does the pulley rotate in 2 s?

5) A mass of 5 kg is resting on a

frictionless ramp. It is connected to a cable that passes over a pulley of radius 2 cm which is then attached to a second hanging mass. The system is moving at a constant velocity. (a) Calculate the mass of the hanging block. (b) If the blocks move 1.5 m in 3 seconds what is the angular velocity of the pulley?

6) A plank of wood 2 m long and mass 30 kg is supported by

three ropes. A box of 700 N is resting on the plank. See diagram. Find the tension in each rope.

7) A beam has three forces acting on it as shown below. Sum

the torques around (a) Point A and (b) Point B to find the net torque.

8) Mitch of 700 N walks out along a beam that is 6 m long

and weighs 200 N. The box at the end weighs 80 N. (a) Draw a Free-Body diagram for the beam. (b) When Mitch is at x = 1 m, calculate the tension in the wire and the components of the reaction force at the hinge. (c) The wire can handle a maximum tension of 900 N. How far out can Mitch walk before the wire will break?

1) 2ML 3 1

5) a) 2.87 kg, b) 25 rad/s

2) 2ML 30 421 6) a) 29.6 Nm, b) 35.6 Nm

3) 177 N 7) T1 = 501 N, T2 = 672 N, T3 = 385 N 4) a) 4.83 m/s2 8) b) T = 343 N, Rx = 172 N, Ry = 683 N b) 7.32 times c) 5.14 m

35°

y-axis 25 N

30°

B A 20° 45°

2 m 10 N

4 m 30 N

Mitch

60°

x

T1 T2

T3 40°

0.5 m

2 m

Physics 225 Fall 2014

Problem Set #11

1) A 15 m ladder of 500 N rests against a frictionless wall.

The ladder makes an angle of 60° with the floor. There is friction between the ground and the ladder. (a) Calculate the horizontal and vertical forces acting on the base of the ladder when Skid of 800 N is standing on the ladder 4 m from the bottom. (b) If the ladder is on the verge of slipping when Skid is 9 m up the ladder, calculate the coefficient of static friction between the ladder and the ground.

2) A spherical Yo-Yo is released from rest 1.2 m above the

ground. The Yo-Yo unwinds as it falls. What is the Yo-Yo’s linear velocity right before it hits the ground? (Use energy methods.)

3) A mass of 8 kg is resting on a horizontal table with a

coefficient of kinetic friction of 0.35. The mass is connected to a cable that passes over a pulley of mass 3.5 kg and radius 21 cm. The other end of the cable is attached to a second hanging mass of 15 kg. The system is released from rest. Calculate the angular velocity of the pulley after the masses have moved 1.5 m. (Use energy methods.)

4) A force vector F = 2.0i – 3.0k acts on a pebble with

position vector r = 0.5j – 2.0k, relative to the origin. What is the resulting torque acting on the pebble about the origin?

5) A 4.8 m diameter merry-go-round is rotating freely with an

angular velocity of 0.8 rad/s. Its total moment of inertia is 1950 kg m2. Skid, Mitch, Larry, and Greezy all jump on the edge of the merry-go-round at the same time. They each have a mass of 65 kg. What is the angular velocity of the merry-go-round now?

6) Disc #1 of mass 5 kg and radius 2.6 m is attached to an axle that is rotating at 20 rad/s. Disc #2 of mass 2 kg and radius 1.7 m is dropped from rest. It slides down the axle and sticks to Disc #1. What is the angular velocity of the two discs?

Disc #2

Disc #1 Before After 7) Skid has landing on an asteroid of radius 500 km and

gravitational acceleration 3.0 m/s2. (a) What will Skid’s escape velocity need to be in order to take off in his ship Beggar I? (b) What will Beggar I’s maximum height from the surface be if Skid takes off with a velocity of 1000 m/s?

8) Two identical neutron stars of mass 1030 kg and radius 105

m are separated by a distance of 1010 m. If released from rest, what are the stars’ velocities when their separation is one-half of their original?

1) a) FS = 267 N, Fn1 = 1300 N 5) 0.452 rad/s b) 0.324 6) 17.1 rad/s 2) 4.1 m/s 7) a) 1732 m/s 3) 18.1 rad/s b) 250 km 4) – 1.5i – 4j – k 8) 82 km/s

Physics 225 Fall 2014

Problem Set #12

1) The Martian satellite Phobos travels in an approximately

circular orbit of radius 9.4 x 106 m with a period of 7 hours and 39 min. Calculate the mass of Mars from this information.

2) The mean distance of Mars from the Sun is 1.52 times that

of Earth from the Sun. From Kepler’s law of periods, calculate the number of years required for Mars to make one revolution about the Sun.

3) A box on a spring starts 6 cm from equilibrium. The box

starts from rest and has a mass of 2 kg and a spring constant of 1200 N/m. (a) What is the period of the oscillation?, (b) Write out the position, velocity and acceleration equations for the box’s motion.

4) An object oscillates with an amplitude of 6 cm on a

horizontal surface. The spring has a constant of 2 kN/m. It has a maximum speed of 2.2 m/s. Find the (a) mass of the object, (b) the frequency of motion, (c) and the period of motion.

5) A 3 kg object oscillating on a spring of force constant

2 kN/m has total energy 0.9 J. (a) What is the amplitude of motion? (b) What is the maximum speed?

6) Skid of 2 kg is oscillating on a spring. Skid has a period of

s and has an amplitude of 40 cm. Find (a) the angular frequency, (b) the total energy of the system (c) At what location will Skid’s velocity be 0.6 m/s?

3

7) A simple pendulum on Earth has a period of 2 s. What

is its period on the moon? (The moon has a mass of 7.36 x 1022 kg and a radius of 1.74 x 106 m.)

8) A pendulum consists of a disc attached to

a rod. The disc has a mass of 500 g and radius 10 cm. The rod has a mass of 270 g and a length of 500 mm. Calculate the period of oscillation.

1) 6.5 x 1023 kg 6) a) 2.09 Hz 2) 1.88 years b) 0.699 J 3) a) 0.257 s c) 27.9 cm 4) a) 1.49 kg b) 5.84 Hz 7) 4.92 s c) 0.171 s 8) 1.5 s 5) a) 3 cm b) 77.5 cm/s

Physics 225 Fall 2014

The Last Problem Set

1) A sound wave from your speakers playing The Charming

Beggars has a frequency of 262 Hz and travels with a velocity of 330 m/s. How far apart are the wave crests (compressions)?

2) Greezy-X is in his fishing boat on the Kankakee River. He

notices wave crests that pass the bow of his anchored boat every 3 s. He measures the distance between the crests to be 7.5 m. What is the velocity of the waves?

3) A wave traveling in the positive x direction has a frequency

of 25 Hz. See the diagram for other information. Find the (a) amplitude, (b) wavelength, (c) period, (d) velocity of the wave, (e) write out the wave equation for the wave, and (f) calculate the maximum transverse speed of the wave.

4) Two waves, one of amplitude 0.3 m and the other of amplitude 0.4 m are traveling towards each other. (a) What is the largest resultant amplitude of these two waves when they meet? (b) What is the smallest resultant amplitude of these two waves when they meet? (c) Under what conditions will the answers to (a) and (b) occur?

5) While tuning his guitar, Mitch hears one beat every 0.25 s

when comparing two strings. One of the strings is sounding a true “C” note which is 262 Hz. What could the possible frequencies be of the other string?

6) Transverse waves are traveling at a velocity of 30 m/s on a

rope that has a tension of 9 N. If the tension on the string is increased to 22 N what will be the new velocity of the waves?

7) The 3rd and 5th harmonics that sound when a guitar string is

struck are 1152 Hz and 1920 Hz. The string is 65 cm long and is under 300 N of tension. (a) What is the fundamental frequency for this string? (b) Find the mass per unit length of the string.

18 cm

8) A string of 3 g/m is fixed at both ends. A standing wave of

frequency 2000 Hz is set up in which the nodes are 8 cm apart. (a) What is the wavelength of the standing wave? (b) Calculate the velocity of the waves. (c) What is the tension of the string?

10 cm

9) A rope of length 8 m and mass 40 g is fixed at both ends.

The tension of the rope is 49 N. (a) What are the positions of the nodes and the antinodes for the 3rd harmonic? (b) Calculate the frequency for this harmonic.

1) 1.26 m 6) 46.9 m/s 2) 2.5 m/s 7) a) 384 Hz 3) a) 0.09 m b) 1.2 g/m

b) 0.20 m 8) a) 0.16 m c) 0.04 s b) 320 m/s d) 5 m/s c) 307 N

f) 14.14 m/s 9) a) Nodes: 0, 2.67 m, 5.33 m 5) 266 Hz, 258 Hz 8.0 m b) 18.6 Hz