- A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The particle \(P\) is at rest at the point \(B\) on the table, where \(A B = l\).
At time \(t = 0 , P\) is projected along the table with speed \(U\) in the direction \(A B\).
At time \(t\)
- the elastic string has not gone slack
- \(B P = x\)
- the speed of \(P\) is \(v\)
- Show that
$$v ^ { 2 } = U ^ { 2 } - \frac { 2 g x ^ { 2 } } { l }$$
By differentiating this equation with respect to \(x\), prove that, before the elastic string goes slack, \(P\) moves with simple harmonic motion with period \(\pi \sqrt { \frac { 2 l } { g } }\)
Given that \(U = \sqrt { \frac { g l } { 2 } }\)find, in terms of \(l\) and \(g\), the exact total time, from the instant it is projected from \(B\), that it takes \(P\) to travel a total distance of \(\frac { 3 } { 4 } l\) along the table.