Chapter12-page22

Chapter12-page22 - n coupled differential equations that we...

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Physics 235 Chapter 12 - 22 - The potential energy of the system is the potential energy associated with the tension in the string. We assume that the displacements from the equilibrium positions are small. We ignore the gravitational forces acting on the masses (and the associated gravitational potential energy). In order to calculate the force acting on mass j we calculate the vertical components due to the tension in the left and right section of the string: F j = ! " q j ! q j ! 1 d 2 ! q j ! q j ! 1 ( ) 2 # $ % % % ( ( ( ! q j ! q j + 1 d 2 ! q j ! q j + 1 ( ) 2 # $ % % % ( ( ( ) d q j ! 1 ! 2 q j + q j + 1 ( ) In the last step we have made the assumption that the vertical displacement is small compared to the distance d . Since the force on mass j depends not only on the position of mass j but also on the position of masses j - 1 and j + 1. We can use the force on the n masses to obtain
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Unformatted text preview: n coupled differential equations that we can try to solve. Consider the following trial function: q j t ( ) = a j e i ! t Substituting this function into our differential equation we obtain F j = m !! q j = ! m 2 a j e i t = # d a j ! 1 e i t ! 2 a j e i t + a j + 1 e i t ( ) or d a j " 1 " 2 d " m 2 $ % & ( ) a j + d a j + 1 = The amplitudes a can be complex. Based on the type of motion we expect the system to carry out, we can try to parameterize the amplitude dependence on j in the following way: a j = ae i j " ( ) where a is now a real number. Taking this expression for a j and substituting it into the previous equation we obtain ae ! i d e ! i $ ! 2 d ! m % 2 & ( ) * + + d e i & ( ) * + =...
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This note was uploaded on 01/08/2012 for the course PHY 235 taught by Professor Morgan during the Spring '09 term at SUNY Stony Brook.

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