13 A particle P of mass \(m\) is fixed to one end of a light spring of natural length \(a\) and modulus of elasticity man \({ } ^ { 2 }\), where \(n > 0\). The other end of the spring is attached to the ceiling of a lift. The lift is at rest and P is hanging vertically in equilibrium.
- Find, in terms of \(g\) and \(n\), the extension in the spring.
At time \(t = 0\) the lift begins to accelerate upwards from rest. At time \(t\), the upward displacement of the lift from its initial position is \(y\) and the extension of the spring is \(x\).
- Express, in terms of \(g , n , x\) and \(y\), the upward displacement of P from its initial position at time \(t\).
- Given that \(\ddot { y } = k t\), where \(k\) is a positive constant, express the upward acceleration of \(P\) in terms of \(\ddot { x } , k\) and \(t\).
- Show that \(x\) satisfies the differential equation
$$\ddot { \mathrm { x } } + \mathrm { n } ^ { 2 } \mathrm { x } = \mathrm { kt } + \mathrm { g } .$$
- Verify that \(\mathrm { x } = \frac { 1 } { \mathrm { n } ^ { 3 } } ( \mathrm { k } n \mathrm { t } + \mathrm { gn } - \mathrm { k } \sin ( \mathrm { nt } ) )\).
- By considering \(\dot { x }\) comment on the motion of P relative to the ceiling of the lift for all times after the lift begins to move.