Bengaltiger
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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

\frac {d^2x}{dt^2} + 2n\frac {dx}{dt} + 16n^2x = 0

Thanks in advance.
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Jonny W
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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
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Bengaltiger
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Why are you using the natural length as l/2

Are you treating the the string as 2 strings?
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Christophicus
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(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

\frac {d^2x}{dt^2} + 2n\frac {dx}{dt} + 16n^2x = 0

Thanks in advance.
ma = T1 - T2 - 2mnv
ma = 4mln^2\frac{(l-x-l/2)}{(l/2)} - 4mln^2\frac{(l+x-l/2)}{(l/2)} - 2mnv
=&gt; a = 4ln^2\frac{(l/2-x)}{(l/2)} - 4ln^2\frac{(l/2+x)}{(l/2)} - 2nv
=&gt; \frac{d^2x}{dt^2} = 4ln^2\frac{(-2x)}{(l/2)} - 2n\frac{dx}{dt}
=&gt; \frac{d^2x}{dt^2} + 2n\frac{dx}{dt} + 16n^2x = 0

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

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