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Math Help - Cart on a ramp

  1. #1
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    Cart on a ramp

    B4.
    Consider a spring that is attached to a cart of mass m, which lies on a frictionless
    inclined plane. Let

    be the angle of the plane with the horizontal, and assume that
    the spring is an ideal spring with unstretched length
    l0 and constant k.

    (i)
    Establish the forces acting on the cart and compute the length l > l0 the spring
    stretches to hold the cart in equilibrium.

    (ii)
    Show that the mechanical energy of the cart is given by
    E=(1/2)mx'+mg \sin{\alpha}x+(k/2)x^2

    where
    x denotes the position measured up the slope along the plane, and chosen
    such that
    x = 0 when l = l0.

    (iii)
    Now assume that the cart is pushed up along the ramp to a distance x1 from
    the unstretched length
    l0, compressing the spring. What is the total mechanical
    energy of the cart in this case?

    (iv)
    When the cart is released from rest at position x1, what is its speed when the
    spring has returned to its unstretched length
    l0?

    (v)
    Show that the equation of motion for the cart is x''+(k/m)x=-gsin{\alpha}

    (vi)
    Compute the general solution of the equation and the period of oscillation of the

    cart.


    Last edited by mr fantastic; May 28th 2011 at 02:16 PM. Reason: Fixed latex tag.
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  2. #2
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    Quote Originally Posted by Duke View Post
    B4.
    Consider a spring that is attached to a cart of mass m, which lies on a frictionless

    inclined plane. Let
    be the angle of the plane with the horizontal, and assume that
    the spring is an ideal spring with unstretched length l0 and constant k.

    (i)
    Establish the forces acting on the cart and compute the length l > l0 the spring
    stretches to hold the cart in equilibrium.
    (ii)


    Show that the mechanical energy of the cart is given by
    E=(1/2)mx'+mg \sin{\alpha}x+(k/2)x^2
    where


    x denotes the position measured up the slope along the plane, and chosen
    such that x = 0 when l = l0.
    (iii)


    Now assume that the cart is pushed up along the ramp to a distance x1 from
    the unstretched length l0, compressing the spring. What is the total mechanical
    energy of the cart in this case?
    (iv)


    When the cart is released from rest at position x1, what is its speed when the
    spring has returned to its unstretched length l0?
    (v)


    Show that the equation of motion for the cart is x''+(k/m)x=-gsin{\alpha}
    (vi)


    Compute the general solution of the equation and the period of oscillation of the
    cart.

    Too many questions in one thread, certainly far too many without any effort shown. To get help here you will need to show all your work and say where you are stuck. MHF is not a homework service for providing solutions.

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