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# M4 damped harmonic Motion watch

1. A light elastic string of natural length l and modulus of 4mln2 has a particle of mass m attached to its midpoint. One end of the string is attaced to a fixed point A on a smooth horizontal table and the other end to a fixed point B on the same table where AB = 2l. The particle moves along the line AB and is subject to a resistance of mangitude 2mnv, where v is the speed of the particle. The midpoint of AB is O.

a) Show that, so long as both parts of the string remain taut, the displacement x of P from O at time t satisfies the differntial equation

2. A ----<-L------|------R->---- B

The tensions in the left and right parts of the string are

L = (4mln^2)(l + x - l/2)/(l/2) = 8mn^2 (l/2 + x)
R = (4mln^2)(l - x - l/2)/(l/2) = 8mn^2 (l/2 - x)

m d^2x/dt^2
= R - L - 2mn dx/dt
= -16mn^2 x - 2mn dx/dt

d^2x/dt^2 + 2n dx/dt + 16n^2x = 0
3. Why are you using the natural length as l/2

Are you treating the the string as 2 strings?
4. (Original post by Bengaltiger)
A light elastic string of natural length l and modulus of 4mln2 has a particle of mass m attached to its midpoint. One end of the string is attaced to a fixed point A on a smooth horizontal table and the other end to a fixed point B on the same table where AB = 2l. The particle moves along the line AB and is subject to a resistance of mangitude 2mnv, where v is the speed of the particle. The midpoint of AB is O.

a) Show that, so long as both parts of the string remain taut, the displacement x of P from O at time t satisfies the differntial equation

Woohoo my first ever tex post!
Edit: rats, beaten to it.
5. (Original post by Bengaltiger)
Why are you using the natural length as l/2

Are you treating the the string as 2 strings?
yeah.
6. Thanks all, got it now.

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