Goldstein_21_7_12

Goldstein_21_7_12 - Homework 1: # 1.21, 2.7, 2.12 Michael...

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Homework 1: # 1.21, 2.7, 2.12 Michael Good Sept 3, 2004 1.21. Two mass points of mass m 1 and m 2 are connected by a string passing through a hole in a smooth table so that m 1 rests on the table surface and m 2 hangs suspended. Assuming m 2 moves only in a vertical line, what are the generalized coordinates for the system? Write the Lagrange equations for the system and, if possible, discuss the physical significance any of them might have. Reduce the problem to a single second-order differential equation and obtain a first integral of the equation. What is its physical significance? (Consider the motion only until m 1 reaches the hole.) Answer: The generalized coordinates for the system are θ , the angle m 1 moves round on the table, and r the length of the string from the hole to m 1 . The whole motion of the system can be described by just these coordinates. To write the Lagrangian, we will want the kinetic and potential energies. T = 1 2 m 2 ˙ r 2 + 1 2 m 1 r 2 + r 2 ˙ θ 2 ) V = - m 2 g ( R - r ) The kinetic energy is just the addition of both masses, while V is obtained so that V = - mgR when r = 0 and so that V = 0 when r = R . L = T - V = 1 2 ( m 2 + m 1 r 2 + 1 2 m 1 r 2 ˙ θ 2 + m 2 g ( R - r ) To find the Lagrangian equations or equations of motion, solve for each component: ∂L ∂θ = 0 ∂L ˙ θ = m 1 r 2 ˙ θ d dt ∂L ˙ θ = d dt ( m 1 r 2 ˙ θ ) = m 1 r 2 ¨ θ + 2 m 1 r ˙ r ˙ θ 1
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Thus m 1 r ( r ¨ θ + 2 ˙ θ ˙ r ) = 0 and ∂L ∂r = - m 2 g + m 1 r ˙ θ 2 ∂L ˙ r = ( m 2 + m 1 r d dt ∂L ˙ r = ( m 2 + m 1 r Thus m 2 g - m 1 r ˙ θ 2 + ( m 2 + m 1 r = 0 Therefore our equations of motion are: d dt ( m 1 r 2 ˙ θ ) = m 1 r ( r ¨ θ + 2 ˙ θ ˙ r ) = 0 m 2 g - m 1 r ˙ θ 2 + ( m 2 + m 1 r = 0 See that m 1 r 2 ˙ θ is constant, because d dt ( m 1 r 2 ˙ θ ) = 0. It is angular momentum. Now the Lagrangian can be put in terms of angular momentum and will lend the problem to interpretation. We have ˙ θ = l/m 1 r 2 , where l is angular momentum. The equation of motion m 2 g - m 1 r ˙ θ 2 + ( m 2 + m 1 r = 0 becomes ( m 1 + m 2 r - l 2 m 1 r 3 + m 2 g = 0 The problem has been reduced to a single non-linear second-order differential equation. The next step is a nice one to notice. If you take the first integral you get 1 2 ( m 1 + m 2 r 2 + l 2 2 m 1 r 2 + m 2 gr + C = 0 To see this, check by assuming that C = - m 2 gR : d dt ( 1 2 ( m 1 + m 2 r 2 + l 2 2 m 1 r 2 - m 2 g ( R - r )) = ( m 1 + m 2 r ¨ r - l 2 m 1 r 3 ˙ r + m 2 g ˙ r = 0 ( m 1 + m 2 r - l 2 m 1 r 3 + m 2 g = 0 Because this term is T plus V , this is the total energy, and because its time derivative is constant, energy is conserved.
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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_21_7_12 - Homework 1: # 1.21, 2.7, 2.12 Michael...

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