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

$\displaystyle E=(1/2)mx'+mg \sin{\alpha}x+(k/2)x^2$

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.