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
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.