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.

Example: A total eclipse of the sun

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

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

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

Centripetal and Tangential Accelerations in Rotational Kinematics

Exam Example 22: Throwing a Discus (example 9.4)

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

Rotation about an Axis Shifted from a Center of Mass Position

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)

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

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