1. Rotational Kinematics
Mark Lesmeister
Dawson High School AP Physics 1
Objectives:
Express the motion of an object using narrative, mathematical, and graphical representations.
Relate angular and tangential quantities for a rigid object rotating about a fixed axis.
Solve problems involving constant angular velocity or constant angular acceleration for a rotating object.
Rotate a student on a spinning chair, or spinning platform, with his or her hands out, and have the student bring his or her hands in.
Review how we progressed through kinematics (describing motion) through dynamics, explaining motion, and then used energy and momentum for certain situations. We will do the same here.

2. Acknowledgements © Mark Lesmeister/Pearland ISD
Selected questions © 2010 Pearson Education Inc.
Selected graphics and questions from Cutnell and Johnson, Physics 9e: Instructor’s Resource Site, © 2015 John Wiley and Sons.

3. Discovery Lab: Rotational Motion
Go to this page on your laptop or computer:
Click on Run Now (or Download if it is your computer and you want to have a copy of the app.)
Experiment with different angular velocities on the intro tab.
Essential Knowledge 3.A.1: An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration.
Enduring Understanding 3.F. “… The rate of change of the rotational motion is most simply expressed by defining the rotational kinematic quantities of angular displacement, angular velocity, and angular acceleration, analogous to the corresponding linear quantities,

4. Consider the motion of a rigid body about a fixed axis
, or an axis that is moving parallel to itself like a rolling ball.

5. Angular Position
On your Ladybug app, click on the handle on the turntable with your mouse and move it to various positions on the circle, and observe the effect on angle.
If one segment of the turntable rotates 50 degrees, how far would the other segments rotate?
Move the bug to various positions, both along a radius and along a circle.
Does the angle change when you move along a radius? q
Since each part of the rotating object keeps the same relative position, one number is sufficient to state the location of every point in the object.

6. Angular Position
All the points in the object maintain the same relative position.
When the object rotates, every point rotates through the same angle.
The angle θ gives the angular position of every point in the object. q
Like position, theta must be measured relative to a reference line.

7. Angular Displacement
When the object rotates, each point undergoes the same angular displacement Δθ.
If we measure the angular displacement in radians, the distance traveled by a point a distance r from the center is Dq
Our rotation is like one dimensional motion. We will just use + and -.
This is how radian measure is defined.

8. Angular Displacement Example
Two people ride on a carousel. One rides on a horse located 5 meters from the center. The other rides on a swan located 3 meters from the center.
When the carousel goes around ¼ of a revolution, how far does each person travel?
The angular displacement = pi/2, so s = r pi/2. The horse goes 7.8 m and the swan goes 4.7 m.

9. Over the course of a day (twenty-four hours), what is the angular displacement of the second hand of a wrist watch in radians?
a) rad
b) rad
c) rad
d) rad
e) rad

10. Angular velocity
Set your Ladybug app to display radians instead of degrees.
Change the angular velocity to some reasonable number of radians /s. (6.3 rad= 1 rev)
Move the bug to various distances from the center, and observe the angular velocity (w) and tangential velocity v.
What is the relationship between w and v? Dq
We say radians per second. But radians are dimensionless. Otherwise, s = r Theta would give radian meters for units.
Like linear motion, we use + and -.

11. Angular velocity
The angular velocity is the rate of change of angular position.
The dimensions of the angular velocity are T-1.
The magnitude of the angular velocity is the angular speed.
Counterclockwise rotation corresponds to a positive velocity. Dq

12. Angular and Tangential velocities
Since s = r Dq, vt = rw Dq

13. Angular velocity practice
What is the angular speed, in radians/second, of a motor that spinning with rpm? Dq

14. 8. 2. 1. The planet Mercury takes only 88 Earth days to orbit the Sun
The planet Mercury takes only 88 Earth days to orbit the Sun. The orbit is nearly circular, so for this exercise, assume that it is. What is the angular velocity, in radians per second, of Mercury in its orbit around the Sun?
a) 8.3 × 107 rad/s
b) 2.0 × 10 5 rad/s
c) 7.3 × 10 4 rad/s
d) 7.1 × 10 2 rad/s
e) This cannot be determined without knowing the radius of the orbit.

15. 8.2.2. Complete the following statement: For a wheel that turns with constant angular speed,
a) each point on its rim moves with constant acceleration.
b) the wheel turns through “equal angles in equal times.”
c) each point on the rim moves at a constant velocity.
d) the angular displacement of a point on the rim is constant.
e) all points on the wheel are moving at a constant velocity.

16. Angular acceleration
Select q, w, a under Show Graphs in the Ladybug app.
Start with an angular velocity of 1 rad/s in the velocity block.
Observer what happens to the motion when you change the angular acceleration to 1 rad/s/s.cceleration refers to the increase or decrease in rotational speed of the particle. Dq
We say radians per second per second.

17. Angular acceleration
The angular acceleration is the rate of change of angular velocity.
The dimensions of angular acceleration are T-2.
This acceleration refers to the increase or decrease in rotational speed of the particle. Dq

18. Angular and Tangential Quantities
Each angular quantity has a corresponding tangential quantity.
The arc length s is the distance travelled along the circle, in m .
The tangential velocity is the speed of the particle in the direction tangent to the circle, in m/s.
The tangential acceleration is the acceleration of the particle tangent to the circle, in m/s2.
These formulas relate what the object as a whole is doing to what each part is doing.

19. Describing Rotational Motion Practice
406/1.7

20. Rotational motion of a rigid body is analogous to linear motion.
Straight line motion
Rotation about a fixed axis
Linear position x.
Linear displacement Dx
Linear velocity v
Linear acceleration a
Angular position q.
Angular displacement Dq
Angular velocity ω
Angular acceleration α
Remind students not to confuse this analogy with finding the tangential quantities. These are only mention to help them remember the formulas.

21. Equations of motion for constant acceleration
Straight line motion
Rotation about a fixed axis
The same methods we used to derive the equations of straight line motion could be used to derive the equations for rotation about a fixed axis.

22. Problem solving with rotational motion.
Convert tangential velocities, speeds etc. to their angular counterparts by dividing by the radius.
Solve for the angular motion by using the angular equations of motion just as we did with linear motion.
Convert quantities to their tangential counterparts if needed.

23. Angular kinematics examples
Toast falling off a table usually starts to fall when it makes an angle of 30 degrees with the horizontal, and falls with an angular speed of where l is the length of one side. On what side of the bread will the toast land if it falls from a table 0.5 m high? If it falls from a table 1.0 m high? Assume l = 0.10 m, and ignore air resistance.
When a turntable rotating at 33 rev/min is turned off, it comes to rest in 26 s. Assuming constant angular acceleration, find the angular acceleration and the angular displacement. If the turntable is 0.20 m in radius, how far would an ant riding on the outside edge have moved in that time?
The formula is valid in terms of radian measure.

24. Rotational Kinematics Practice
406/1.7

25. The propeller of an airplane is at rest when the pilot starts the engine; and its angular acceleration is a constant value. Two seconds later, the propeller is rotating at 10 rad/s. Through how many revolutions has the propeller rotated through during the first two seconds?
a) 300
b) 50
c) 20
d) 10
e) 5

26. A ball is spinning about an axis that passes through its center with a constant angular acceleration of  rad/s2. During a time interval from t1 to t2, the angular displacement of the ball is  radians. At time t2, the angular velocity of the ball is 2 rad/s. What is the ball’s angular velocity at time t1?
a) rad/s
b) rad/s
c) rad/s
d) rad/s
e) zero rad/s

27. The Earth, which has an equatorial radius of 6380 km, makes one revolution on its axis every hours. What is the tangential speed of Nairobi, Kenya, a city near the equator?
a) m/s
b) m/s
c) 148 m/s
d) 232 m/s
e) 465 m/s

28. The original Ferris wheel had a radius of 38 m and completed a full revolution (2 radians) every two minutes when operating at its maximum speed. If the wheel were uniformly slowed from its maximum speed to a stop in 35 seconds, what would be the magnitude of the instantaneous tangential speed at the outer rim of the wheel 15 seconds after it begins its deceleration?
a) m/s
b) m/s
c) m/s
d) m/s
e) m/s

29. A long, thin rod of length 4L rotates counterclockwise with constant angular acceleration around an axis that is perpendicular to the rod and passes through a pivot point that is a length L from one end as shown. What is the ratio of the tangential acceleration at a point on the end closest to the pivot point to that at a point on the end farthest from the pivot point?
a) 4
b) 3
c) 1/2
d) 1/3
e) 1/4

30. A long, thin rod of length 4L rotates counterclockwise with constant angular acceleration around an axis that is perpendicular to the rod and passes through a pivot point that is a length L from one end as shown. What is the ratio of the tangential speed (at any instant) at a point on the end closest to the pivot point to that at a point on the end farthest from the pivot point?
a) 1/4
b) 1/3
c) 1/2
d) 3
e) 4

31. Rotation and Centripetal Acceleration
If an object is rotating about a fixed axis, even at a constant speed, every point in that object is undergoing a centripetal acceleration as well.

32. Centripetal Acceleration in Angular Form

33. at aC Total acceleration
The total acceleration is the vector sum of angular acceleration, which points to the center of the circle, and tangential acceleration, which is tangent to the circle. at aC

34. Speed and Period
Since the speed we are considering is constant, we can calculate it by dividing the distance for one revolution, i.e. the circumference, by the time for one revolution, which is called the period.

35. Rolling Objects

36. An airplane starts from rest at the end of a runway and begins accelerating. The tires of the plane are rotating with an angular velocity that is uniformly increasing with time. On one of the tires, Point A is located on the part of the tire in contact with the runway surface and point B is located halfway between Point A and the axis of rotation. Which one of the following statements is true concerning this situation?
a) Both points have the same tangential acceleration.
b) Both points have the same centripetal acceleration.
c) Both points have the same instantaneous angular velocity.
d) The angular velocity at point A is greater than that of point B.
e) Each second, point A turns through a greater angle than point B.

37. A wheel starts from rest and rotates with a constant angular acceleration. What is the ratio of the instantaneous tangential acceleration at point A located a distance 2r to that at point B located at r, where the radius of the wheel is R = 2r?
a) 0.25
b) 0.50
c) 1.0
d) 2.0
e) 4.0

38. 8. 6. 1. The wheels of a bicycle have a radius of r meters
The wheels of a bicycle have a radius of r meters. The bicycle is traveling along a level road at a constant speed v m/s. Which one of the following expressions may be used to determine the angular speed, in rev/min, of the wheels? a) b) c) d) e)

39. Josh is painting yellow stripes on a road using a paint roller. To roll the paint roller along the road, Josh applies a force of 15 N at an angle of 45 with respect to the road. The mass of the roller is 2.5 kg; and its radius is 4.0 cm. Ignoring the mass of the handle of the roller, what is the magnitude of the tangential acceleration of the roller?
a) 4.2 m/s2
b) 6.0 m/s2
c) 15 m/s2
d) 110 m/s2
e) 150 m/s2

40. 8. 6. 5. A bicycle wheel of radius 0
A bicycle wheel of radius 0.70 m is turning at an angular speed of 6.3 rad/s as it rolls on a horizontal surface without slipping. What is the linear speed of the wheel?
a) 1.4 m/s
b) 28 m/s
c) m/s
d) 4.4 m/s
e) 9.1 m/s