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

2. 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

3. 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

4. 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

5. Chapter 11
Problem 2

6. 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

7. 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

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

9. 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

10. Newton’s Second Law in angular form

11. Angular momentum of a system of particles

12. Chapter 11
Problem 33

13. 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

14. Angular momentum of a rigid body

15. 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

16. Conservation of angular momentum

17. Conservation of angular momentum

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

19. Chapter 11
Problem 51

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

21. 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

22. 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

23. 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

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