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    A plane curve is defined by the parametric equations
    x=a(theta+sin theta), y=a(1-cos theta), where -pi<=theta<=pi.
    If alpha is the angle between the tangent to this curve and the x axis, and if s is the distance along the curve from the point where theta =0 in the direction of increasing theta, prove that

    alpha =1/2 theta,
    s=4a sin alpha

    Write down the tangential and normal components of acceleration for a particle moving along a smooth wire in the shape of this curve. If the wire is placed with the y axis vertically upwards and the particle slides along the wire under the influence of uniform gravity g, show that the position s(t) of the particle satisfies the differential equation

    (d^2s)/(dt^2) + (g/4a)s =0

    Please see the attachments.
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Updated: February 1, 2006

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