1. Kinematics of Rotation of Rigid Bodies
Angular displacement Δθ = θ – θ0
Δθ > 0 if rotation is counterclockwise
Δθ < 0 if rotation is clockwise z
Angle of rotation s s’
Angle of rotation θ is
a dimensionless quantity,
but it is a vector r’ r
If θ is in radians, then
s=rθ and s’=r’θ with the same θ.

2. Example:
A total eclipse
of the sun

3. Angular Velocity and Accelerationz z
Average angular velocity
Instantaneous angular velocity
ωz>0
ωz<0
Instantaneous angular speed is a scalar
Average angular acceleration
Instantaneous angular acceleration

4. Rotational motion Quantity Linear motion θ Displacement x ω0z
Kinematic Equations of Linear and Angular Motion
with Constant Acceleration
(1)
(3)
(4)
(2)
Rotational motion
Quantity
Linear motion θ
Displacement x
ω0z
Initial velocity
v0x ωz
Final velocity vx αz
Acceleration ax t
Time

6. Ice-skating stunt “crack-the-whip”
Relations between Angular and Tangential Kinematic Quantities
Ice-skating stunt
“crack-the-whip”

7. Centripetal and Tangential Accelerations in Rotational Kinematics

8. Exam Example 22: Throwing a Discus (example 9.4)

9. Rotational Kinetic Energy and Moment of Inertia
Kinetic energy of one particle
Rotational kinetic energy is the kinetic energy of the
entire rigid body rotating with the angular speed ω ω
Definition of the moment of inertia
Total mechanical energy h
Translational
kinetic energy
Rotational
kinetic energy
Potential
energy
0cm
Parallel-Axis Theorem
Proof: =0

10. Rotation about an Axis Shifted from a Center of Mass Position

11. Solution: (a) Work-energy theorem Wnc = - μk m1g Δy , since FN1 = m1g,
Exam Example 23: Blocks descending over a massive pulley (problem 9.83) ω
Data: m1, m2, μk, I, R, Δy, v0y=0 m1 R
Find: (a) vy; (b) t, ay; (c) ω,α; (d) T1, T2
Solution: (a) Work-energy theorem
Wnc= ΔK + ΔU, ΔU = - m2gΔy,
Wnc = - μk m1g Δy , since FN1 = m1g,
ΔK=K=(m1+m2)vy2/2 + Iω2/2 = (m1+m2+I/R2)vy2/2 since vy = Rω x m2 ay Δy y
(b) Kinematics with constant acceleration: t = 2Δy/vy , ay = vy2/(2Δy)
(c) ω = vy/R , α = ay/R = vy2/(2ΔyR)
(d) Newton’s second law for each block:
T1x + fkx = m1ay → T1x= m1 (ay + μk g) ,
T2y + m2g = m2ay → T2y = - m2 (g – ay)

12. Moments of Inertia of Various Bodies
Scaling law and order of magnitude I ~ ML2

13. Moment of Inertia Calculations
Cylinder rotating about axis of symmetry
The total mass of the cylinder is
Result: