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Physics 325 Homework 9

# Physics 325 Homework 9 - i That the angular momentum of the...

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Physics 325 Homework 9 – Fall 2010 Due: Thursday 4 November at 11:15 am 1) [15 points] A hoop of mass m and radius R rolls without slipping down an inclined plane of mass M , that makes an angle θ with the horizontal. Find the Lagrange equations and the integrals of the motion if the plane can slide without friction along a horizontal surface. [ Hint : Recall that the kinetic energy of rotation can be written as T rot = 1 2 I ! 2 , with I the moment of inertia. For a hoop, I = mR 2 .] 2) [15 points] An object of mass m is free to move in three dimensions and is subject to a force of the form: ! F r , , z ( ) = " k r 2 ˆ e r . At t =0, the mass has a position ! r = r 0 , 0 ,0 ( ) and velocity ! " r = " r 0 , " 0 ,0 ( ) . a) Using cylindrical coordinates, write the Lagrangian for this system. b) Derive the differential equations for the motion of the object. From these differential equations show the following:
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Unformatted text preview: i) That the angular momentum of the mass is conserved. ii) That the motion of the object is restricted to the z =0 plane. 3) [15 points] A simple pendulum of length b and bob mass m is attached to a massless support moving vertically upwards with constant acceleration α . Determine (a) the equations of motion and (b) the period for small oscillations. 4) [15 points] A particle is free to slide along a smooth cycloidal trough whose surface is given by the parametric equations: x = a 4 2 " sin2 ( ) y = " a 4 1 " cos2 ( ) where ! " ! # and a is a constant. a) Find the Lagrangian function and the equation of motion of the particle. b) Show that the particle undergoes simple harmonic motion without any small angle approximation. Total 60 points...
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