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F_09s - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.01 Physics I Fall Term 2009 Final Exam (Analytic Part) Name Section Table and Group Answers without work shown will not be given any credit. Comments on how you are approaching the problem and what you are doing at any given point will be helpful to the grader in awarding partial credit, as will carefully drawn diagrams when appropriate. Problem 1 (30 points) Problem 2 (30 points) Problem 3 (30 points) Problem 4 (30 points) Problem 5 (30 points) Total (150 points) 1
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Problem 1: Central Field Motion (30 points) An object of mass M moves under the influence of an attractive central force ~ F = - Ar 4 ˆ r where ˆ r is a unit vector in the radial direction. a) (10 points) If the object is in a circular orbit of radius r 0 , find its speed v 0 as a function of M , A and r 0 . Solution Use ~ F = m~a - Ar 4 0 ˆ r = - Mv 2 0 /r 0 ˆ r v 2 0 = ( A/M ) r 5 0 v 0 = q ( A/M ) r 5 0 b) (8 points) For the object in the circular orbit described in a) find the total energy E and the magnitude of the angular momentum L about the origin as functions of M , A and r 0 . Assume that the potential energy is zero at the origin. Solution First find the potential energy at r = r 0 . The rest follows. U ( r 0 ) = - Z r 0 0 F ( r ) dr = A Z r 0 0 r 4 dr = ( A/ 5) r 5 0 E = U + K = U ( r 0 ) + (1 / 2) Mv 2 0 = ( A/ 5) r 5 0 + ( A/ 2) r 5 0 = (7 / 10) Ar 5 0 ~ L = ~ r × ~ p = Mr 0 v 0 ˆ z when the orbit is circular L | ~ L | = Mr 0 q ( A/M ) r 5 0 = q AMr 7 0 Continued on next page. 2
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A second object of the same mass is in an identical circular orbit in the same plane, but orbiting in the opposite direction. They collide and stick together. c) (4 points) Describe in words the motion of the resulting composite object. Solution Immediately after the collision, the composite object is at rest. The force on the object is radially inward. Thus the subsequent motion is along a straight line through the origin. The motion will be periodic but not simple harmonic motion. d) (8 points) For the motion of that composite object give the total energy E , the angular momentum L about the origin, the maximum speed v max and the maximum distance from the origin r max as functions of M , A and r 0 .
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