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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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    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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    Why are you using the natural length as l/2

    Are you treating the the string as 2 strings?
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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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    (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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    Thanks all, got it now.
 
 
 
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Updated: April 13, 2006
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