Rolling, torque, and angular momentum Chapter 11 Rolling, torque, and angular momentum

ds/dt = d(θR)/dt = R dθ/dt vcom = ds/dt = ωR Smooth rolling Smooth rolling – object is rolling without slipping or bouncing on the surface Center of mass is moving at speed vcom Point P (point of momentary contact between two surfaces) is moving at speed vcom s = θR ds/dt = d(θR)/dt = R dθ/dt vcom = ds/dt = ωR

Rolling: translation and rotation combined Rotation – all points on the wheel move with the same angular speed ω Translation – all point on the wheel move with the same linear speed vcom

Rolling: pure rotation Rolling can be viewed as a pure rotation around the axis P moving with the linear speed vcom The speed of the top of the rolling wheel will be vtop = (ω)(2R) = 2(ωR) = 2vcom

Chapter 11 Problem 2

fs opposes tendency to slide Friction and rolling Smooth rolling is an idealized mathematical description of a complicated process In a uniform smooth rolling, P is at rest, so there’s no tendency to slide and hence no friction force In case of an accelerated smooth rolling acom = α R fs opposes tendency to slide

Rolling down a ramp Fnet,x = M acom,x fs – M g sin θ = M acom,x R fs = Icom α α = – acom,x / R fs = – Icom acom,x / R2

Torque revisited Using vector product, we can redefine torque (vector) as:

Angular momentum Angular momentum of a particle of mass m and velocity with respect to the origin O is defined as SI unit: kg*m2/s

Newton’s Second Law in angular form

Angular momentum of a system of particles

Chapter 11 Problem 33

Angular momentum of a rigid body A rigid body (a collection of elementary masses Δmi) rotates about a fixed axis with constant angular speed ω Δmi is described by

Angular momentum of a rigid body

Conservation of angular momentum From the Newton’s Second Law If the net torque acting on a system is zero, then If no net external torque acts on a system of particles, the total angular momentum of the system is conserved (constant) This rule applies independently to all components

Conservation of angular momentum

Conservation of angular momentum

More corresponding relations for translational and rotational motion (Table 11-1)

Chapter 11 Problem 51

Answers to the even-numbered problems Chapter 11: Problem 4 (a) 8.0º; (b) more

Answers to the even-numbered problems Chapter 11: Problem 18 (a) (6.0 N · m)ˆj + (8.0 N · m) ˆk; (b) (− 22 N · m)ˆi

Answers to the even-numbered problems Chapter 11: Problem 26 (a) (6.0 × 102 kg · m2/s) ˆk; (b) (7.2 × 102 kg · m2/s)ˆk

Answers to the even-numbered problems Chapter 11: Problem 32 (a) 0; (b) (−8.0N · m/s)tˆk; (c) − 2.0/√t ˆk in newton·meters for t in seconds; (d) 8.0 t−3 ˆk in newton·meters for t in seconds

Answers to the even-numbered problems Chapter 11: Problem 42 (a) 750 rev/min; (b) 450 rev/min; (c) clockwise