© 2001-2005 shannon w. helzer. all rights reserved. 1 unit 10 circular motion

© 2001-2005 Shannon W. Helzer. All Rights Reserved. 1

Unit 10Circular Motion


Setting the Standard – Angular/Circular Motion By now we should all be

very familiar with our standard.

However, our standard fails to consider the direction assigned to rotation.

As a result, we will now modify our standard to account for rotation.

A clockwise rotation would be considered to be negative.

A counterclockwise rotation would be considered to be positive.


Rotations

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2

t

As the Wheel Turns

Watch how the sine function (which demonstrates a wave) traces out as a wheel turns. The vertical axis represents horizontal position and the horizontal axis represents time. The amplitude of the sin wave is the radius of the wheel.

Rotations

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2

t


Circular Motion Circular motion is simply motion in which the path taken by a body forms a circular pattern. One common rotating body is a Compact Disc. This circular motion becomes more apparent when we watch a single point on the CD. Note how the rotation of this point traces out a sine wave. The closer to the center the point is, the lower the amplitude of the corresponding wave.

Circular Motion

-4

-3

-2

-1

0

1

2

3

4

0 1 2 3 4 5 6 7

Time

Po

siti

on


Circular Motion It is often desirable to convert circular motion to tangential motion or vice versa. For instance, we might wish to know how much linear distance a car travels in one complete

rotation of its tire. Consider the three points on the CD below located at 1.75 cm, 2.75 cm, and 3.50 cm from the

CD’s center. Note the difference in the linear distances traveled by these points in just one rotation (2

radians). What type of graph is the one below? There is a linear relationship between the angle of rotation and the linear distance traveled.

Distance v. Time

0

5

10

15

20

25

0 1 2 3 4 5 6

Angle of Rotation (Radians)

Dis

tan

ce (

cm)


Circular Motion The linear relationship between the linear distance and the angular distance is as follows

where s is the linear distance (arc length), r is the radius, and is the angle, in radians, through which the CD turned.

Suppose we turned the CD through an angle = 2.74 radians (157). How much linear distance would each point travel?

Distance v. Angle of Rotation

0

2.5

5

7.5

10

12.5

0 0.5 1 1.5 2 2.5 3

Angle of Rotation (Radians)

Dis

tan

ce (

cm)

s r


Circular Motion In addition to determining the arc length,

we are interested in knowing the angular displacement and the angular velocity of the point .

The angular displacement is found by subtracting the initial angular position from the final angular position.

Given our standard, what is the current angular position of the blue dot on the CD below?

If it rotates clockwise through an angle of 2.74 radians, then what is its final angular displacement?

The angular speed of a point on a rotating body may be determined using the equation below where is the angular speed and r is the distance of the point from the center of the rotating body.

2 1

2 1

t t

s r

11

s

r 2

2

s

r

2 1 2 11 1s s s s

t r r t

1v

r

v r


Angular Acceleration We may also wish to determine the

tangential acceleration of a body given the angular acceleration of the body.

We derive the equation that relates angular acceleration to tangential acceleration by following the same steps used in deriving the relationship between angular speed and tangential speed.

The dot on the CD below was initially rotating at an angular speed 1 = 2.5 RPM.

The CD speeds up for 4.0 s ending with an angular speed 2 = 6.2 RPM.

What was its angular and tangential accelerations during this time period?

2 1

2 1

t t

v r

11

v

r 2

2

v

r

2 1 2 11 1v v v v

t r r t

1a

r

a r


Circular Motion Equations We could follow the same derivation

procedure and derive the relationship between angular acceleration, , and the tangential acceleration.

Using these equations, our old kinematics equations are resurrected for circular motion.

Note: when using these equations, all calculations must be done in radians.

s r

v r

atvv 12

122

12

2 2 ddavv

2112 2

1attvdd

a r

2 1 t

2 22 1 2 12

22 1 1

1

2t t


Circular Motion Example WS 16 # 8 A door is opened through an angle of 79 in 1.50 s. The screw on the hinge is 3.50 cm from the hinge and the door knob is 85.0 cm

from the hinge. What is the tangential distance traveled by both the hinge and the knob? What was the average angular velocity of both the hinge and the knob? What was the average tangential speed of both the hinge and the knob?

2 1

t t

s r

v r


Circular Motion Example WS 16 # 9 Dr. Physics pulls a log through his orchard. The diameter of the front wheel is 0.40 m and that of the back wheel is 0.6 m. He accelerated to a constant speed of 2.14 m/s in 1.43 s. Afterwards, he pulled the log 400.00 m before beginning to come to a stop. What was the angular acceleration of the wheels while he was accelerating? What was the angular displacement of the wheels while he was accelerating? What was the final angular velocity of the wheels at the end of the acceleration? How many times did each wheel turn while traveling the 400.00 m?


Circular Motion Example WS 16 # 10 The blades (r = 0.4 m) in a box fan are initially rotating at 1500.0 RPM. How many radians per second is this value? How many meters per second is this value? Once the switch on the fan is turned to the “Off” position, the blades take 37.0 s

to come to a halt. What is the angular acceleration of the blades while they are stopping?

Tangential acceleration? What is the angular displacement of the blades while they are stopping?

Tangential distance?


Circular Motion Example WS 16 # 12 The large gear (d = 0.25 m) in a belt driven gear system rotates at 55 RPM. How fast is the small gear (d = 0.12 m) spinning at the same time? Would you expect the small gear to go faster or slower than the large gear? Explain. How does the tangential speed of the belt while moving around the Large gear

compare to the tangential speed of the belt while moving around the Small gear? Explain.

How fast is the small gear (d = 0.12 m) spinning at the same time (in m/s, rad/s, and RPM)?


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© 2001-2005 shannon w. helzer. all rights reserved. 1 unit 10 circular motion

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