1. Dynamics of Rotational Motion The main problem of dynamics: How a net force affects (i) translational (linear) motion ( Newtons’ 2 nd law) (ii) rotational motion ??? (iii) combination of translational and rotational motions ??? m αzαz Axis of rotation Lever arm l is the distance between the line of action and the axis of rotation, measured on a line that is to both. Definition of torque: A torque applied to a door Units: [ τ ] = newton·meter = N·m τ z > 0 if the force acts counterclockwise τ z < 0 if the force acts clockwise

2. Newton’s Second Law for Rotation about a Fixed Axis (i)One particle moving on a circle: F tan =ma tan and a tan = r α z rF tan = mr 2 α z τ z = I α z Only F tan contributes to the torque τ z. τ z I (ii) Rigid body (composed of many particles m 1, m 2, …) Only external torques (forces) count ! Example 10.3: a 1x,T 1,T 2 ? a 1x a 2y y Pulley: T 2 R-T 1 R=I α z a 1x =a 2y =R α z X T 2 -T 1 =(I/R 2 )a 1x Glider: T 1 =m 1 a 1x Object: m 2 g-T 2 =m 2 a 1x

3. Work-Energy Theorem and Power in Rotational Motion Rotational work: Work-Energy Theorem for Rigid-Body Rotation: Power for rotational work or energy change: Proof:

4. Rigid-Body Rotation about a Moving Axis Proof: General Theorem: Motion of a rigid body is always a combination of translation of the center of mass and rotation about the center of mass. Rolling without slipping: v cm = Rω, a x = R α z Energy: General Work-Energy Theorem: E – E 0 = W nc, E = K + U

5. Rolling Motion Rolling Friction Sliding and deformation of a tire also cause rolling friction.

6. Combined Translation and Rotation: Dynamics Note: The last equation is valid only if the axis through the center of mass is an axis of symmetry and does not change direction. Exam Example 24: Yo-Yo has I cm =MR 2 /2 and rolls down with a y =R α z (examples 10.4, 10.6; problems 10.20, 10.75) Find: (a) a y, (b) v cm, (c) T Mg-T=Ma y τ z =TR=I cm α z a y =2g/3, T=Mg/3 ayay y

7. Exam Example 25: Race of Rolling Bodies (examples 10.5, 10.7; problem 10.22, problem 10.29) β Data: I cm =cMR 2, h, β Find: v, a, t, and min μ s preventing from slipping y x Solution 1: Conservation of EnergySolution 2: Dynamics (Newton’s 2 nd law) and rolling kinematics a=R α z x = h / sin β v 2 =2ax fsfs FNFN

8. Angular Momentum (i) One particle: (ii) Any System of Particles: It is Newton’s 2 nd law for arbitrary rotation. Note: Only external torques count since (iii) Rigid body rotating around a symmetry axis: (nonrigid or rigid bodies) Unbalanced wheel: torque of friction in bearings. Impulse-Momentum Theorem for Rotation

9. Principle of Conservation of Angular Momentum Total angular momentum of a system is constant (conserved), if the net external torque acting on the system is zero: Example: Angular acceleration due to sudden decrease of the moment of inertia For a body rotating around a symmetry axis: I 1 ω 1z = I 2 ω 2z ω 0 < ω f Origin of Principles of Conservation There are only three general principles of conservation (of energy, momentum, and angular momentum) and they are consequences of the symmetry of space-time (homogeneity of time and space and isotropy of space).

10. Hinged board (faster than free fall) h=L sinα Mg m Ball: Board: I=(1/3)ML 2

11. Gyroscopes and Precession Dynamics of precession: Precession is a circular motion of the axis due to spin motion of the flywheel about axis. Precession angular speed: Circular motion of the center of mass requires a centripetal force F c = M Ω 2 r supplied by the pivot. Nutation is an up-and-down wobble of flywheel axis that’s superimposed on the precession motion if Ω ≥ ω. Period of earth’s precession is 26,000 years.

12. Analogy between Rotational and Translational Motions Physical ConceptRotationalTranslational Displacementθs Velocityωv Acceleration α a Cause of accelerationTorque τ Force F InertiaMoment of inertia I = Σmr 2 Mass m Newton’s second lawΣ τ = I α ΣF = ma Work τ θτ θFs Kinetic Energy(1/2) Iω 2 (1/2) mv 2 MomentumL = I ωp = mv