Dynamics of Rotational Motion

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

2. Newton’s Second Law for Rotation about a Fixed Axis Only Ftan contributes to the torque τz . • One particle moving on a circle: Ftan=matan and atan= rαz rFtan= mr2 αzτz = I αz τzI (ii) Rigid body (composed of many particles m1, m2, …) Example 10.3: a1x,T1,T2? a1x Pulley: T2R-T1R=Iαz T2-T1=(I/R2)a1x y Glider: T1=m1a1x a1x=a2y=Rαz a2y Object: m2g-T2=m2a1x Only external torques (forces) count ! X

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

4. Rigid-Body Rotation about a Moving Axis 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. Energy: Proof: General Work-Energy Theorem: E – E0 = Wnc , E = K + U Rolling without slipping: vcm= Rω, ax = R αz

5. Rolling Friction Rolling Motion 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 Icm=MR2/2 and rolls down with ay=Rαz (examples 10.4, 10.6; problems 10.20, 10.75) Find: (a) ay, (b) vcm, (c)T Mg-T=May τz=TR=Icmαz ay=2g/3 , T=Mg/3 y ay

7. Exam Example 25: Race of Rolling Bodies(examples 10.5, 10.7; problem 10.22, problem 10.29) y Data: Icm=cMR2, h, β Find: v, a, t, and min μs preventing from slipping FN x = h / sinβ β x Solution 1: Conservation of Energy Solution 2: Dynamics (Newton’s 2nd law) and rolling kinematics a=Rαz fs v2=2ax

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

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: For a body rotating around a symmetry axis: I1ω1z = I2ω2z Example: Angular acceleration due to sudden decrease of the moment of inertia ω0 < ωf 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) Ball: m Board: I=(1/3)ML2 L h=L sinα Mg

11. Gyroscopes and Precession Precession is a circular motion of the axis due to spin motion of the flywheel about axis. Period of earth’s precession is 26,000 years. Dynamics of precession: Precession angular speed: Circular motion of the center of mass requires a centripetal force Fc = M Ω2 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.

12. Analogy between Rotational and Translational Motions