# midterm - Use your solution for an ellipse to determine a...

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Physics 821 Spring Quarter 2007 Dr. Herbst MIDTERM EXAMINATION TUESDAY, MAY 8 8:30 – 10:18 AM 100 POINTS 1) (40 points) The figure below shows a hoop with mass m and radius a rolling on the inside surface of a fixed hoop with center O and radius b>a under the influence of gravity. a) (20 pts) Find the Lagrangian equation of motion for the angle θ . If there is a constraint, discuss whether it is holonomic or non-holonomic. b) (10 pts) Obtain an expression for the time derivative of θ if the hoop starts from rest at an arbitrary angle. c) (10 pts) Determine what, if any, are the constants of the motion. 2) (30 points) a) (20 pts) A bound particle of mass m moves under the attractive potential V = -k/r . Determine the apsidal distances (minimum and maximum r ) in terms of the total energy E and the angular momentum l . Under what circumstances are there two valid solutions?

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Unformatted text preview: Use your solution for an ellipse to determine a formula for the energy in terms of the semi-major axis a . b) (10 pts) Consider scattering by a central force V(r). Determine an expression for the time as a function of radial distance r assuming that t=0 pertains to the distance of closest approach, 3) (30 points) a) (20 pts) Find the principal moments of inertia and the principal axes for the following system: I xx = a I yy = a I zz = b I xy = c ( c < a ) I xz = I yz = 0 b) (10 pts) Express the unit vectors in the principal axis system in terms of the unit vectors in the initial axis system by a rotation matrix with specific angles. Some Basic Equations d dt " L ˙ q # L q = Q k = " # \$ % & f q k h = ˙ q j j " L ˙ q j \$ L dh dt = " L t l = mr 2 ˙ " V = V + l 2 2 mr 2 l = mvb T = m 2 ˙ r 2 + r 2 ˙ 2 ( ) = \$ 2 %...
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## This note was uploaded on 06/13/2011 for the course PHYSICS 821 taught by Professor Eric during the Spring '07 term at Ohio State.

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midterm - Use your solution for an ellipse to determine a...

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