ps8 - Show that the long diagonal of the cube is a...

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PHY 504 Problem Set #8 due 29 October 2010 1. Problem 4.15. 2. A rotation about the z -axis by φ is followed by a rotation about the x ' axis by θ . What is the axis and angle of the resulting rotation? 3. A child riding on a platform rotating at ω rad/s drops a ball from a height h , while standing a distance r from the center of the platform. How far from her feet does the ball hit the platform? Solve the problem in both the lab frame and the child’s frame, using the notation of Chapter 4. 4. A bullet is shot directly up into the air in Lexington (latitude 38 o N), with an initial velocity of 1000 m/s. Neglecting air resistance, calculate how far it lands from its initial position. Are centrifugal or Coriolis effects dominant? Justify any approximations you make. 5. A rigid body consists of eight equal point masses m situated at the corners of a cube and connected by massless rods of length a . (a) Calculate the inertia tensor about a corner of the cube.
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Unformatted text preview: Show that the long diagonal of the cube is a principal axis. (b) If the cube is hanging by a string attached to a corner, and a torque N z is applied about the vertical axis, what will be its angular acceleration? (c) Suppose the body hangs by one of its corners in a uniform gravitational field and swings in a vertical plane. Write down a Lagrangian in terms of the swinging angle and the moment of inertia. What is the frequency of the pendulum? 6. A thin rod of mass m and length l has one end on a table and is balanced vertically. The top of the rod is given a small angular displacement and allowed to fall. (a) Obtain an equation of motion for the angle made by the rod with the table, if the lower end of the rod is not allowed to slide. With what speed does the moving end of the rod hit the table? (b) How does (a) change if it can slide without friction?...
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This note was uploaded on 11/16/2010 for the course PHY 504 taught by Professor Staff during the Fall '08 term at Kentucky.

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