Two particles, \(A\) of mass \(m\) and \(B\) of mass \(M\), are fixed to the ends of a light inextensible string \(AB\) of length \(r\) and lie on a smooth horizontal plane. The origin of coordinates and the \(x\)- and \(y\)-axes are in the plane.
Initially, \(A\) is at \((0,\,0)\) and \(B\) is at \((r,\,0)\). \(B\) is at rest and \(A\) is given an instantaneous velocity of magnitude \(u\) in the positive \(y\) direction.
At a time \(t\) after this, \(A\) has position \((x,\,y)\) and \(B\) has position \((X,\,Y)\). You may assume that, in the subsequent motion, the string remains taut.
- Explain by means of a diagram why
\[X = x + r\cos\theta\]
\[Y = y - r\sin\theta\]
where \(\theta\) is the angle clockwise from the positive \(x\)-axis of the vector \(\overrightarrow{AB}\).
- Find expressions for \(\dot{X}\), \(\dot{Y}\), \(\ddot{X}\) and \(\ddot{Y}\) in terms of \(\ddot{x}\), \(\ddot{y}\), \(\dot{x}\), \(\dot{y}\), \(r\), \(\ddot{\theta}\), \(\dot{\theta}\) and \(\theta\), as appropriate.
Assume that the tension \(T\) in the string is the only force acting on either particle.
- Show that
\[\ddot{x}\sin\theta + \ddot{y}\cos\theta = 0\]
\[\ddot{X}\sin\theta + \ddot{Y}\cos\theta = 0\]
and hence that \(\theta = \dfrac{ut}{r}\).
- Show that
\[m\ddot{x} + M\ddot{X} = 0\]
\[m\ddot{y} + M\ddot{Y} = 0\]
and find \(my + MY\) in terms of \(t\) and \(m, M, u, r\) as appropriate.
- Show that
\[y = \frac{1}{m+M}\left(mut + Mr\sin\!\left(\frac{ut}{r}\right)\right).\]
- Show that, if \(M > m\), then the \(y\) component of the velocity of particle \(A\) will be negative at some time in the subsequent motion.