Goldstein_1_2_6_8_14_20

# Goldstein_1_2_6_8_14 - Homework 1 1 2 6 8 14 20 Michael Good 1 Show that for a single particle with constant mass the equation of motion implies

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Homework 1: # 1, 2, 6, 8, 14, 20 Michael Good August 22, 2004 1. Show that for a single particle with constant mass the equation of motion implies the follwing diﬀerential equation for the kinetic energy: dT dt = F · v while if the mass varies with time the corresponding equation is d ( mT ) dt = F · p . Answer: dT dt = d ( 1 2 mv 2 ) dt = m v · ˙ v = m a · v = F · v with time variable mass, d ( mT ) dt = d dt ( p 2 2 ) = p · ˙ p = F · p . 2. Prove that the magnitude R of the position vector for the center of mass from an arbitrary origin is given by the equation: M 2 R 2 = M X i m i r 2 i - 1 2 X i,j m i m j r 2 ij . Answer: M R = X m i r i M 2 R 2 = X i,j m i m j r i · r j Solving for r i · r j realize that r ij = r i - r j . Square r i - r j and you get 1

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r 2 ij = r 2 i - 2 r i · r j + r 2 j Plug in for r i · r j r i · r j = 1 2 ( r 2 i + r 2 j - r 2 ij ) M 2 R 2 = 1 2 X i,j m i m j r 2 i + 1 2 X i,j m i m j r 2 j - 1 2 X i,j m i m j r 2 ij M 2 R 2 = 1 2 M X i m i r 2 i + 1 2 M X j m j r 2 j - 1 2 X i,j m i m j r 2 ij M 2 R 2 = M X i m i r 2 i - 1 2 X i,j m i m j r 2 ij 6. A particle moves in the xy plane under the constraint that its velocity vector is always directed toward a point on the x axis whose abscissa is some given function of time f ( t ). Show that for f ( t ) diﬀerentiable, but otherwise arbitrary, the constraint is nonholonomic. Answer: The abscissa is the x-axis distance from the origin to the point on the x-axis that the velocity vector is aimed at. It has the distance f ( t ). I claim that the ratio of the velocity vector components must be equal to the ratio of the vector components of the vector that connects the particle to the point on the x-axis. The directions are the same. The velocity vector components are: v y = dy dt v x = dx dt The vector components of the vector that connects the particle to the point on the x-axis are: V y = y ( t ) V x = x ( t ) - f ( t ) For these to be the same, then v y v x = V y V x 2
dy dx = y ( t

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## This note was uploaded on 03/17/2012 for the course PHYS 202 taught by Professor Atkin during the Spring '12 term at Amity University.

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Goldstein_1_2_6_8_14 - Homework 1 1 2 6 8 14 20 Michael Good 1 Show that for a single particle with constant mass the equation of motion implies

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