1. Rotational Motion
WHS
Lee Wignall

2. Important Terms and Concepts
Radians/degrees: how to convert from one to another
Rotational motion: what is it?
Arc length
Angular displacement (in radians)
Angular velocity (ω: omega)
Angular acceleration (α: alpha)
Rotational motion equations vs. linear motion equations
Tangential velocity/acceleration
Centripetal acceleration
Causes of rotational motion (gravity/torque)
Conservation of Angular Momentum

3. Basic Quantities Converting from degrees to radians:
So far we’ve done everything in degrees, but now we will make a change to radians for rotational motion.
Why? Because things usually “turn” more than just one revolution (360 degrees).
Converting from degrees to radians:
2π = θr so, θr = θd π
360 Θd

4. Arc Length
Arc length (m)
Radius (m)
angle (radians)

5. ω = Δθ Δt Angular Velocity Linear equation for velocity:
change in distance
change in time
Angular velocity:
change in angle
“omega”: radians/s
“angular displacement”
ω = Δθ Δt

6. ω = Δθ Δt Angular Velocity Angular velocity: change in angle
change in time
“angular displacement”
“omega”
ω = Δθ Δt
COUNTER CLOCKWISE = positive direction (+)
CLOCKWISE = negative direction (-)

7. α = Δω Δt Angular Acceleration Linear equation for acceleration:
change in velocity
change in time
Angular acceleration:
change in angular velocity
“alpha”: radians/s2
α = Δω Δt

8. Angular vs. Linear Motion Equations

9. Homework Angular Displacement: page 247 (1-4)
Angular Speed: page 248 (1-4)
Angular Acceleration: page 250 (1-3)
Angular Kinematics: page 252 (1-5)

10. Sample Rotational Motion Problem #1:
A wheel is slowly turning at 1.0 rad/s when an outside force causes the angular velocity to increase to 4.0 rad/s. If it takes 5 full revolutions of the wheel to reach it’s final angular velocity, what was the angular acceleration provided by the force?

11. Sample Rotational Motion Problem #2:
A helicopter blade accelerates at 35.5 rad/s2. If the initial angular velocity of the blade was 5 rad/s, how long will it take the blades to complete 1500 turns?
What will be its angular velocity at that point?
If it takes an angular velocity of 2000 rad/s to achieve liftoff, how many seconds will that take?

12. Tangential Velocity and Acceleration
Radius (m)
vt = rω
Angular velocity (rad/s)
Tangential velocity (m/s)
at = rα
Angular acceleration (rad/s2)
Tangential acceleration (m/s2)

13. Homework Finish Today’s Work: Tangential speed page 255 (1-4)
Angular Displacement: page 247 (1-4)
Angular Speed: page 248 (1-4)
Angular Acceleration: page 250 (1-3)
Angular Kinematics: page 252 (1-5)
Finish
Today’s Work:
Tangential speed page 255 (1-4)
Tangential acceleration: page 256 (1-3)

14. A mysterious planet has a gravitational pull of 15 m/s2 but it’s spinning so fast that a 50 kg. rock at the equator no longer falls downward. At what latitudes (north and south) will the rock have a downward acceleration of 10 m/s2?

15. A 200 gram ball on a 0. 75 meter string has an angular velocity of 1
A 200 gram ball on a 0.75 meter string has an angular velocity of 1.5 rad/s. The person spinning it supplies a force that provides an acceleration of 2.0 rad/s2. The string will break when the tension reaches 150 N. How many seconds until the string breaks? At what angle will the ball go flying off at? Assume you the ball begins at 0 degrees and spins counter-clockwise.

16. Conservation of Angular Momentum
The velocity is the TANGENTIAL VELOCITY
So, the unit for angular momentum L is:
“In any closed system, the total amount of angular momentum is conserved.”

17. Formation of Galaxies
Horsehead Nebula

18. Formation of Galaxies
Crab Nebula

19. Formation of Galaxies

20. Formation of Galaxies
The Earth and tossed pizza dough do the same thing.

25. Figure Skaters
I is the “moment of inertia” which is also called the radial mass. It’s a measure of how much mass is how far away from the axis of rotation.

26. Figure Skaters
Instead of thinking about moments of inertia and angular velocity, we can think about this spinning skater in terms of radius and tangential velocity. It’s easier.

27. Figure Skaters
So, in order to conserve angular momentum L, if we DECREASE the radial distance for the overall mass, then we will experience an INCREASE in the tangential velocity, meaning we spin faster.