Problem Set 2 Solution

So one t is a stable equili ium position v m0 v t d m0

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Unformatted text preview: m1m2 m1v12 Gm2 2 2! r1 2! d 3 2 # $ v1 # $" # # 2 d r1 d( e 1 % m2 ) v1 For E3 ! 0 , the particle is either at the stable mquilibrium point x = 0Gor beyond x ! " x4 . , (m1 % m2 ) The E4 , the particle comes in from "# to "the and returns. of forces acting on 2nd star. For result will be the same if we consider x5 equilibrium 76 CHAPTER 2 2-50. can see that in this problem x = a and x = –a are unstable equilibrium positions, and x = 0 2-44. So one t is a stable equili' ium position. v ' m0 v T d & m0 v br Ft T & m0 # F$-d # # F t $ θ (t ) # a) v 2) 2 2 ( ( 4Uvx ) T dt v v F 2t 2 4U 0 k 0 2 ( 1* 2 ( *2 m0 % 2 c) Around the) origin, F " #1 * 20 2 ) # kx1% ! & $ & + + c, c c a c , m1 m m2 m a 2 mg m2 g g t d) To escape to infinity from x = 0, tthe 1particle & ' c 2 needs toF 2e2 at least to the peak of the potential, t $ x(t) # - v(t)dt # ( m0 2 % 2 * m0 ) 2 F+ c 2U 0 From the figure, the forces acting on n 0 masses give the equations,of motion mvmi the & U max & U 0 % vmin & m 2 m1 !!1 ! m1 g $ T (1) x b) e) From energy conservation, we havex ! m g $ 2T cos % v m2 !!2 2 (2) 2U 0 ' dx x( &v& *1 # 2 + , dt m a2 & b $ 1 ' $ d2 We note that, in the ideal case, becausexthe initialxvelocity is the escape velocity found in d), (3) 2! ideally x is always smaller or equal to a, then fro4 the above expression, m 2 2 2 min mv where x2 is related to x1 by the U 0 x & mv ) relation a2 2 2 2 % m This gives ( and cos % ! d (& b $ x1 ' 2 * . At equilibrium, !!1 ! x 2 ! 0 and T ! 't 1 g . ' 8U 0 ( as the equilibrium x !! ) + a * exp * t #1 x 2 , ma 2 + + -, values for the coordinates dx ma a)x c) From a) we find m ln & % x(t) & t& 2U 0 . ' 8U 0 avm4 m d #x ' x2 ( ' 8U 0 ( ( 0 1 0 *1 # 2 + x10 t!#b $ (4) * exp * t ma 2 + ) 1+ , a , -2 2 4 m12 $ m2 , v F 1* 2 x c m2 d x20 ! (5) 2 2 NEWTONF IAN MECHANICS—SINGLE PARTICLE 75 4 m1 $ m2 Now if # 10 , then m0 We recognize that our expression x10 is identical to...
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This note was uploaded on 12/14/2013 for the course PHYS 301 taught by Professor Argryes during the Fall '13 term at University of Cincinnati.

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