1. Chapter 11 Rolling, Torque, and angular Momentum

2. REGAN PHY342102 11: Rolling, Torque and Angular Momentum Rolling: Rolling motion (such as a bicycle wheel on the ground) is a combination of translational and rotational motion. COM motion. P O P O R  S

3. REGAN PHY342103 The kinetic energy of rolling. A rolling object has two types of kinetic energy, a rotational kinetic energy due to the rotation about the centre of mass of the body and translational kinetic energy due to the translation of its centre of mass. COM motion. P O P O R  S

4. REGAN PHY342104 Rolling Down a Ramp If a wheel rolls at a constant speed, it has no tendency to slide. However, if this wheel is acted upon by a net force (such as gravity) this has the effect of speeding up (or slowing down) the rotation, causing an acceleration of the centre of mass of the system, a com along the direction of travel. It also causes the wheel to rotate faster. These accelerations tend to make the wheel SLIDE at the point, P, that it touches the ground. If the wheel does not slide, it is because the FRICTIONAL FORCE between the wheel and the slide opposes the motion. Note that if the wheel does not slide, the force is the STATIC FRICTIONAL FORCE ( f s ).  R P

5. REGAN PHY342105  R P Rolling down a ramp (cont.) For a uniform body of mass, M and radius, R, rolling smoothly (i.e. not sliding) down a ramp tilted at angle,  (which we define as the x-axis in this problem), the translational acceleration down the ramp can be calculated, from

6. REGAN PHY342106 R0R0 R  T Mg

7. REGAN PHY342107 Example 1: A uniform ball of mass M=6 kg and radius R rolls smoothly from rest down a ramp inclined at 30 o to the horizontal. (a) If the ball descends a vertical height of 1.2m to reach the bottom of the ramp, what is the speed of the ball at the bottom ? 1.2m

8. REGAN PHY342108 Example 1 (cont): (b) A uniform ball, hoop and disk, all of mass M=6 kg and radius R roll smoothly from rest down a ramp inclined at 30 o to the horizontal. Which of the three objects reaches the bottom of the slope first ? 1.2m

9. REGAN PHY342109 Torque was defined previously for a rotating rigid body as  =rFsin . More generally, torque can be defined for a particle moving along ANY PATH relative to a fixed point. i.e. the path need not be circular. z x y O   F F redrawn at origin r x F =  z x y O  

10. REGAN PHY3421010 Angular Momentum z x y O   p p redrawn at origin r x p = l z x y O  l p p

11. REGAN PHY3421011 Newton’s 2 nd Law in Angular Form.

12. REGAN PHY3421012 The net external torque,  net acting on a system is equal to the rate of change of the total angular momentum of the system ( L ) with time.

13. REGAN PHY3421013 z x y   mm

14. REGAN PHY3421014 Conservation of Angular Momentum

15. REGAN PHY3421015 Example1: Pulsars (Rotating Neutron Stars) Vela supernova remnant, pulsar period ~0.7 secs Crab nebula, SN remnant observed by chinese in 11th century Pulsars have similar periodicities ~0.1-1s. beforeafter! SN1987A

16. REGAN PHY3421016 Rotational period of crab nebula (supernova remnant) =1.337secs optical x-ray PULSAR = PULSAting Radio Star (neutron-star) Lighthouse effect Star quakes

17. REGAN PHY3421017 TRANSLATIONALROTATIONAL