1. Rotational Kinematics

2. The need for a new set of variables
We have talked about things in linear motion and in purely rotational movement, but many object both spin and move linearly
Rolling balls
Planets in orbit
Tennis balls or baseballs or volleyballs after they have been hit
Most rotational kinematics variables will be Greek letters

3. Radians
So far we have talked about everything in degrees, but it now makes sense to switch to radians because a radian relates an angle (rotation) to a distance on the circle (linear)
A radian is defined as the measure of a central angle  that makes an arc length s equal in length to the radius r of the circle.
If we call the arc length (the linear movement) x, then x = rƟ
360o=2π radian so 1 revolution = 2π and T would be the time it takes to go 2π

4. Angular Velocity Variable is ω (omega)
Angular velocity is the change in angular position per time (ω= ΔƟ/t)
To get the “connection” equation for velocity, divide each side of the position “connection” equation by t, you get v = rω

5. Angular Acceleration Variable is α (alpha)
Angular acceleration is the change in angular velocity per time (Δω/t)
If you divide each side of the velocity connection equation by time, you get a = rα

6. A demo…..

7. Rotational Kinematics Equations
Using these equations and relationships we can write the rotational kinematics equations

8. Centripetal Acceleration

9. Here are all the formulas
Quantity
Linear
Rotational
Connection
Displacement
Velocity
Acceleration
1st kinematic
2nd kinematic
3rd kinematic
Centripetal acceleration Δx ΔƟ
Δx=rΔƟ
v (m/s)
ω (rad/s)
v=rω
a (m/s2)
α (rad/s2)
a=rα
v=rω

10. Examples
A knight swings a mace of radius 1m in two complete revolutions. What is the translational displacement of the mace?

11. Examples
A compact disc player is designed to vary the disc’s rotational velocity so that the point being read by the laser moves at a linear velocity of 1.25 m/s . What is the CD’s rotational velocity in rev/s when the laser is reading information on an inner portion of the disc at a radius of 0.03 m?

12. Examples
A carpenter cuts a piece of wood with a high powered circular saw. The saw blade accelerates from rest with an angular acceleration of 14 rad/s2 to a maximum speed of 15,000 rpms. What is the maximum speed of the saw in radians per second?

13. Examples
How long does it take the saw to reach its maximum speed?

14. Examples
How many complete rotations does the saw make while accelerating to its maximum speed?

15. Examples
A safety mechanism will bring the saw blade to rest in 0.3 seconds should the carpenter’s hand come off the saw controls. What angular acceleration does this require? How many complete revolutions will the saw blade make in this time?

16. The Rotor
An amusement park ride called the Rotor spins with an angular speed of 4 radians/s. It has a radius of 3.5 m. What is the minimum coefficient of friction so the riders don’t slip? ff mg N +y +x