1 problem found
A particle \(P\) of mass \(m\) moves on a smooth fixed straight horizontal rail and is attached to a fixed peg \(Q\) by a light elastic string of natural length \(a\) and modulus \(\lambda\). The peg \(Q\) is a distance \(a\) from the rail. Initially \(P\) is at rest with \(PQ=a\). An impulse imparts to \(P\) a speed \(v\) along the rail. Let \(x\) be the displacement at time \(t\) of \(P\) from its initial position. Obtain the equation \[ \dot x^2 = v^2 - k^2 \left( \sqrt{x^2+a^2} -a\right)^{\!2} \] where \( k^2 = \lambda/(ma)\), \(k>0\) and the dot denotes differentiation with respect to \(t\). Find, in terms of \(k\), \(a\) and \(v\), the greatest value, \(x_0\), attained by \(x\). Find also the acceleration of \(P\) at \(x=x_0\). Obtain, in the form of an integral, an expression for the period of the motion. Show that in the case \(v\ll ka\) (that is, \(v\) is much less than \(ka\)), this is approximately \[ \sqrt {\frac {32a}{kv}} \int_0^1 \frac 1 {\sqrt{1-u^4}} \, \d u \, . \]