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2. Chapter 10 More on Angular Momentum and Torque

3. Main Points of Chapter 10 Any moving object can have angular momentum around a fixed point Angular momentum and torque can be expressed as vector products Angular momentum is conserved The work-energy theorem is valid for rotational motion Precession and nutation

4. 10-1 Generalization of Angular Momentum We can apply the definition of angular momentum: (9-37) to a single object moving in a straight line:

5. At point A (the distance of closest approach), the object has the same angular momentum around O as it would if it were in orbit around O. 10-1 Generalization of Angular Momentum

6. (10-4) 10-1 Generalization of Angular Momentum We generalize the expression for angular momentum: and find that it is constant for an object moving at constant velocity. r increases while θ decreases:

7. The Vector Product 10-1 Generalization of Angular Momentum For any two vectors, the vector product has magnitude: (B1-1) Its direction is perpendicular to the A-B plane; a right-hand rule gives the sign:

8. 10-2 Generalization of Torque (10-5) We can immediately write an analogous expression for the torque: and verify that it has the proper relation to angular momentum (which it does): (10-6)

9. 10-3 The Dynamics of Rotation (10-11) (10-12) This is the rotational version of Newton’s second law: Substituting for the angular momentum:

10. (10-13) (10-14) 10-3 The Dynamics of Rotation Torque and angular momentum can be measured from any reference point P Then can be related to τ and L about the center of mass:

11. (10-15) 10-3 The Dynamics of Rotation The second equation is the time derivative of the first:

12. 10-3 The Dynamics of Rotation Angular Impulse (10-21) Defined analogously to linear impulse: Same angular impulse can be delivered by large torque for short time, or smaller torque for longer time

13. 10-4 Conservation of Angular Momentum If there is no net external torque on a system, angular momentum is conserved: (10-23) For central forces, angular momentum around origin is conserved Motion is in a plane

14. 10-4 Conservation of Angular Momentum This leads to Kepler’s second law: The radius vector of a planetary orbit sweeps out equal areas in equal times.

15. Nonrigid Objects 10-4 Conservation of Angular Momentum Can change rotational inertia through internal forces Examples: rotating skater, collapsing interstellar dust cloud Angular momentum must remain constant Decreased rotational inertia means increased angular speed, and vice versa

16. 10-4 Conservation of Angular Momentum Example: Rapidly spinning neutron star inside supernova remnant

17. 10-5 Work and Energy in Angular Motion (10-29) Rotational kinetic energy: Instantaneous power: (10-32)

18. (10-33) (10-35) (10-38) 10-5 Work and Energy in Angular Motion Infinitesimal work done by torque: leading to the work-energy theorem in its usual form: The kinetic energy has both translational and rotational contributions:

19. 10-6 Collecting Parallels between Rotational and Linear Motion

20. 10-8 Precession Nutation Interplay of gravity, angular momentum, and torque can lead to perturbations of precession, called nutation. Precession The axis of rotation can itself rotate.

21. 10-8 Precession Torque on a Spinning Top A force perpendicular to the axis of rotation can cause the axis itself to rotate If there is no such force, the axis will not rotate – this leads to the stability of gyroscopes

22. Precession Wp = Torque causing prescession / Angurlar momentum of the spinning object Wp = T / L = mgl / I.w m – mass of object l.– moment of inertia l – Torque arm of the rate of precession

23. Summary of Chapter 10 Angular momentum can be defined for any moving object; for an object moving at constant velocity, it is constant Vector product A x B has magnitude AB sin θ, is perpendicular to AB plane Rotational quantities are analogous to linear quantities Angular momentum is conserved in the absence of external torques

24. Summary of Chapter 10, cont. Nonrigid objects can change rotational inertia through internal forces Angular momentum, Iω, remains constant Total kinetic energy of object in both rotation and translation has contributions from each The axis of rotation can itself experience a torque, which tends to change its direction