7. \section*{Figure 4} A light elastic spring has natural length \(l\) and modulus of elasticity \(4 m g\). One end of the spring is attached to a point \(A\) on a plane that is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The other end of the spring is attached to a particle \(P\) of mass \(m\). The plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle \(P\) is held at a point \(B\) on the plane where \(B\) is below \(A\) and \(A B = l\), with the spring lying along a line of greatest slope of the plane, as shown in Figure 4. At time \(t = 0\), the particle is projected up the plane towards \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g l )\). At time \(t\), the compression of the spring is \(x\).

1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \omega ^ { 2 } x = - g , \text { where } \omega = \sqrt { \left( \frac { g } { l } \right) }$$
2. Find \(x\) in terms of \(l , \omega\) and \(t\).
3. Find the distance that \(P\) travels up the plane before first coming to rest.