- A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length 2l. The other end of the spring is attached to a fixed point \(A\). The particle \(P\) hangs in equilibrium vertically below \(A\), at the point \(E\) where \(A E = 6 l\). The particle \(P\) is then raised a vertical distance \(2 l\) and released from rest.
Air resistance is modelled as being negligible.
- Show that \(P\) moves with simple harmonic motion of period \(T\) where
$$T = 4 \pi \sqrt { \frac { l } { g } }$$
- Find, in terms of \(m , l\) and \(g\), the kinetic energy of \(P\) as it passes through \(E\)
- Find, in terms of \(T\), the exact time from the instant when \(P\) is released to the instant when \(P\) has moved a distance 31 .