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ENG102_SL3 - DATE DUE ENG102 Solution to homework set#3...

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DATE: October 9, 2011 DUE: October 14, 2011 ENG102, Solution to homework set #3. 1. Look at Problem 3/101 in the book. a. Solve the problem in the book and give the angle traveled by the mass m . The angle θ traveled is related to the distance s by s = . We start by calculating the force F acting on m . This consists of two compo- nents, the horizontal force necessary for keeping the mass in a circular orbit and a vertical necessary for keeping the mass from falling: F = q ( mg ) 2 + m ( v 2 /r ) 2 (1) The acceleration of the particle along its circular path is therefore a t = - μ k F/m (2) From our standard relationship vdv = ads , we then have the relationship be- tween distance s 1 and velocity v 1 , given an initial velocity v 0 : vdv = a t ds (3) vdv = - μ k q ( g ) 2 + ( v 2 /r ) 2 ds (4) - v q g 2 + ( v 2 /r ) 2 dv = μ k ds (5) Z v 1 v 0 - v q g 2 + ( v 2 /r ) 2 dv = μ k Z s 1 0 ds (6) Z v 2 0 /rg v 2 1 /rg 1 q 1 + ( v 2 /rg ) 2 d ( v 2 rg ) = 2 μ k r s 1 (7) The integral on the left hand side is known to be the inverse hyperbolic sine, sinh - 1 x = ln( x + 1 + x 2 ) , so: 2 μ k r s 1 = ln v 0 rg 2 + r v 0 rg 4 + 1 v 1 rg 2 + r v 1 rg 4 + 1 (8) The answer to the problem is found for
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