physics assignment
Applied Physics
Module 0 Part 2: Mathematical Tools for Physics
MEASURING DIRECTION & POSITION
RECTANGULAR COORDINATES USE X,Y POINTS TO INDICATE DISPLACEMENTS AND DIRECTIONS.
POLAR COORDINATES USE MAGNITUDES (LENGTHS) AND ANGULAR DIRECTION. THE ANGULAR DIRECTION MAY BE EXPRESSED IN DEGREES OR RADIANS.
DIRECTIONS CAN ALSO BE INDICATED IN GEOGRAPHIC TERMS SUCH AS NORTH, SOUTH, EAST AND WEST.
OFTEN, GEOGRAPHIC MEASURES AND ANGULAR MEASURES ARE COMBINED TO INDICATE DIRECTION.
Up = + Down = - Right = + Left = +
y
x
+
+
-
-
Quadrant IQuadrant II
Quadrant III Quadrant IV
0 o
90 o
180 o
270 o
360 o
Rectangular Coordinates
Physics Mathematics Direction Indication:
RADIANS = ARC LENGTH / RADIUS LENGTH
CIRCUMFERENCE OF A CIRCLE = 2 x RADIUS
RADIANS IN A CIRCLE = 2 R / R
1 CIRCLE = 2 RADIANS = 360O
1 RADIAN = 360O / 2 = 57.3O
y
x
+
+
-
-
Quadrant IQuadrant II
Quadrant III Quadrant IV
0 radians radians
3/2 radians
2 radians
/2 radians
NOTICE THAT THESE DIRECTIONS ARE NOT PRECISE !
TRIGNOMETRY
TRIGNOMETRIC RELATIONSHIPS ARE BASES ON THE RIGHT TRIANGLE (A TRIANGLE CONTAINING A 900 ANGLE). THE MOST FUNDAMENTAL CONCEPT IS THE PYTHAGOREAN THEOREM (A2 + B2 = C2) WHERE A AND B ARE THE SHORTER SIDES (THE LEGS) OF THE TRIANGLE AND C IS THE LONGEST SIDE CALLED THE HYPOTENUSE.
RATIOS OF THE SIDES OF THE RIGHT TRIANGLE ARE GIVEN NAMES SUCH AS SINE, COSINE AND TANGENT. DEPENDING ON THE ANGLE BETWEEN A LEG (ONE OF THE SHORTER SIDES) AND THE HYPOTENUSE (THE LONGEST SIDE), THE RATIO OF SIDES FOR A PARTICULAR ANGLE ALWAYS HAS THE SAME VALUE NO MATTER WHAT SIZE THE TRIANGLE.
A RIGHT TRIANGLE
B
C
900
900
A
900+ + = 1800
A
B
C
Sin = A / C
Cos = B / C
Tan = A / B
A RIGHT TRIANGLE
C
A1 sin
C
B1 cos
B
A1 tan
90 o
y
x
+
+
-
- 0 radians radians
3/2 radians
2 radians
Quadrant III
Quadrant IV
Quadrant I
Quadrant II
Sin Cos Tan
/2 radians
90 o
0 o 180 o
270 o
360 o
+ + +
+ _ _
_ _ +
_ + _
m67 50tan
o h
m0.80m6750tan o
h
a
o
h
h tan
On sunny day , a tall building casts a shadow that is 67 m long. The angle between the sun’s rays and the ground is 50 degree. Determine the height of the building
mh a
62
mh a
62
0 50
0 50
0 h
0 h
A scalar quantity is one that can be described by a single number: temperature, speed, mass
A vector quantity deals fundamentally with both magnitude and direction: velocity, force, displacement
By convention, the length of a vector arrow is proportional to the magnitude of the vector.
8 lb 4 lb
Arrows are used to represent vectors. The direction of the arrow gives the direction of the vector.
To add one vector to another.
The tail of one vector, in this case A, is moved to the head of the other vector B. The vector sum ( C is the vector that extends from the tail of one vector to the head of the other. The sum of the vectors is called the resultant .
+x
+y
FinishStart
Tail to head
A
C
B
6 m 3 m
The sum of the vectors is called the resultant and equals: 9 m
Example: Add 2 vectors. Both vectors have the same direction
2.00 m
6.00 m
Example: Add 2 vectors. Vectors are perpendicular to each other.
6.32 m 222 m 00.6m 00.2 C
m32.6m 00.6m 00.2 22 C
00.600.2tan
4.1800.600.2tan 1
Start Finish
Tail to head
Finish
Start Tail to head
A
B
C +x
+y
When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed.
A
B
BA
A
B
BA
BAC
BCA
A
B
B C
Tail to head
Tail to head
Start
Start
Finish
Finish
Addition and subtraction of vectors: Component method
Vector in the rectangular coordinate system of Figure . The vector A can be expressed as the sum of two vectors along the x and y axes, , where A x and A y are called the components of A. In two dimensions. The vector components of A are tow perpendicular vectors A x and A y . The direction of A x is parallel to the x axis, and that of A y is parallel to the y axis. The magnitudes of the components are obtained from the definitions of the sine and cosine of an angle: cos θ = A x / A and sin θ = A y / A, or
yx AAA
A
It is often easier to work with the scalar components rather than the vector components.
. of
componentsscalar theare and
A
yx AA
)ˆcaret a ( 1. magnitude with rsunit vecto are ˆ and ˆ yyx
yxA ˆˆ yx
AA
x A
y A
A
x
y
A
A tan
x
y
o
Component method of vector addition, A + B = C. These resultant components form the two sides of a right angle with a hypotenuse of the magnitude of C; thus, the magnitude of the resultant is
The direction of the resultant C is calculated from the tangent , tan θ = C x / C y . To solve for the angle θ, use θ = tan −1 ( C y / C x ). The procedure can be summarized as follows:
1. Sketch the vectors on a coordinate system. 2. Find the x and y components of all the vectors, with the appropriate signs. 3. Sum the components in both the x and y directions. 4. Find the magnitude of the resultant vector from the Pythagorean theorem. 5. Find the direction of the resultant vector using the tangent function.
To add vectors numerically, first find the components of all the vectors. The signs of the components are the same as the signs of the cosine and sine in the given quadrant. Then, sum the components in the x direction, and sum the components in the y direction. As shown in Figure , the sum of the x components and the sum of the y components of the given vectors ( A and B) comprise the x and y components of the resultant vector ( C).
yx CCC
x A
y A
x B
y B
x C
y C
x
y BAC
yxA ˆˆ yx
AA
yxB ˆˆ yx
BB
yxyxyxC ˆˆˆˆˆˆ yyxxyxyx
BABABBAA
xxx BAC
yyy BAC
Example
A displacement vector has a magnitude of 110 m and points at an angle of 55.0 degrees relative to the x axis. Find the x and y components of this vector.
rysin
m 900.55sinm 110sin ry
rxcos
m 7.620.55cosm 110cos rx
yxr ˆm 90ˆm 7.62
C
0 55
x
y
0 90
In science, we often encounter very large and very small numbers. Using scientific numbers makes working with these numbers easier
Scientific numbers use powers of 10