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Physics 311

Problem Set 12

Announcements:

• Note – this homework is due on Wednesday, December 6th. The last day of classes.

• FINAL, Physics 311 in Phys 123 on Tuesday December 12th, 1-4pm.

Due: We 12/6

Rigid Body Motion

Problem 12.1) [10 pts] Euler’s Equations and conserved quantities

Consider an object that is rotating freely, subject to zero torque.

(a) Use Euler’s equation [M&T Eq.(11.120)] to prove that the magnitude of the angular momentum L is constant. (Hint: multiply the ith equation by Li = Iiωi and add).

(b) In a similar way, use Euler’s equation prove that the kinetic energy of rotation is constant.

Problem 12.2) [5 pts] Angular Momentum

Denote the angular momentum with respect to the origin of the space fixed coordinates S by L' and the angular momentum with respect to the origin of the body coordinates B by L. Assume the origins of S and B do not coincide (r'α = R + rα) and the origin of B is at the center of mass, such that the vector R between origins is R = r'cm. Show that the angular momentum L' = P α r 0α × ps,α can be written as a sum of L'cm and L, where L'cm is the angular momentum with respect to the origin of S of a point mass M = P α mα located at the center of mass, and the following holds: