1. Rotation Kinematics

2. Need for angular quantities
In rotation the axis of rotation is physically located on the object.
All points on a rotating system have different velocities.
All points on a radial line (along a radius) have the same instantaneous direction, but different speeds.
All points the same distance from the axis have the same speeds, but have different directions.

3. Need for angular quantities
Examine four points.
The time required to move the distances shown is the same for all four points.
However, points more distant from the axis must travel farther.
These points must have greater speed.
Therefore, linear distance and tangential speed cannot be used to describe the motion of all the points simultaneously.
However, there is a quantity that all points on a rotating system share . A B C D
All points on a rotating system move through the same angle in a time t .

4. θ Δθ θ0 Need for angular quantities During rotation, objects move
from in initial angle θ0
to a final angle θ
Experiencing and angular displacement Δθ .
All angular quantities are in radians. θ Δθ θ0

5. Rotational kinematics variables
Equation
Positive
Negative
Zero

6. Rotational kinematics variables
Equation
Positive
Negative
Zero
Initial angular position θ0
rad
ccw cw 0o
Final angular position θ
Change in angular position Δθ
Stationary

7. Rotational kinematics variables
Equation
Positive
Negative
Zero
Initial angular position θ0
rad
ccw cw 0o
Final angular position θ
Change in angular position Δθ
Stationary
Angular velocity ω
rad/s
Initial angular velocity ω0
Final angular velocity
Change in angular velocity Δω
Uniform Circular

8. Rotational kinematics variables
Equation
Positive
Negative
Zero
Initial angular position θ0
rad
ccw cw 0o
Final angular position θ
Change in angular position Δθ
Stationary
Angular velocity ω
rad/s
Initial angular velocity ω0
Final angular velocity
Change in angular velocity Δω
Uniform Circular
Angular Acceleration α
rad/s2
ccw & faster
cw & slower
ccw & slower
cw & faster

9. Rotational kinematic Equations

10. Rotational kinematic Equations
In terms of Δθ
When ω0 = 0
Strategy
Use if problem states uniform circular motion (constant speed)
Not relevant
Time is not mentioned, or the variables to solve time are missing
Time and angular displacement (or angular position) are mentioned
Time and angular velocity are mentioned

11. conversions θ The angle of the rotation (rad)
ω The angular velocity (rad/s)
α The angular acceleration (rad/s2)
conversions
The number of cycles (revolutions, rotations, orbits, etc.) is tied to the angular displacement. To convert between cycles and Δθ …
rpm’s (revolutions per minute) is tied to angular velocity.
Converting angular quantities into tangential quantities.

12. Example 1 angular displacement tangential displacement
An object rotates with a constant angular speed. It completes 10 rev in 2.0 s. Determine each of the following for a point P, that is 4.0 m from the axis.
angular displacement
tangential displacement
angular velocity
(D) tangential velocity

13. Example 1 (E) angular acceleration (F) tangential acceleration
An object rotates with a constant angular speed. It completes 10 rev in 2.0 s. Determine the object’s
(E) angular acceleration
(F) tangential acceleration
(G) radial acceleration

14. Example 2 Δθ = ω0 = 0 rad/s ω = α = 6.0 rad/s2 t = 12.6 rad/s
An object initially at rest rotates with an angular acceleration of 6.0 rad/s2. How many revolutions has the object completed when it reaches the point where it is spinning at 120 rpm?
Δθ =
ω0 = rad/s
ω =
α = rad/s2
t =
12.6 rad/s
120 rpm’s can be converted into angular velocity.
Angular displacement can be converted into revolutions.

15. Example 3 (A) angular displacement Δθ = 230 rad ω0 = 20 rad/s ω =
An object with an initial angular velocity of 20 rad/s accelerates at 10 rad/s2 for 5.0 s. Determine the object’s
(A) angular displacement
Δθ =
ω0 = 20 rad/s
ω =
α = 10 rad/s2
t = s
230 rad
(B) radial acceleration of a point 3.0 m from the axis
Need tangential velocity, but have radial variables.
Now you can solve for radial accel.

16. Example 4 (A) final angular velocity (B) angular acceleration Δθ =
An object initially at rest rotates with a uniform angular acceleration. After 10 seconds a point 3.0 m from the axis has a tangential speed of 20 m/s. Determine the object’s
(A) final angular velocity
(B) angular acceleration
Δθ =
ω0 = 0 rad/s
ω =
α =
t = 10 s
5.0 rad/s
0.5 rad/s2
(C) radial acceleration
(D) angular displacement