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2021 Paper 2 Q10
D: 1500.0 B: 1500.0
non-inertial frame constrained motion energy method bead on wire constant acceleration parabolic constraint lagrangian mechanics

A train moves westwards on a straight horizontal track with constant acceleration \(a\), where \(a > 0\). Axes are chosen as follows: the origin is fixed in the train; the \(x\)-axis is in the direction of the track with the positive \(x\)-axis pointing to the East; and the positive \(y\)-axis points vertically upwards. A smooth wire is fixed in the train. It lies in the \(x\)--\(y\) plane and is bent in the shape given by \(ky = x^2\), where \(k\) is a positive constant. A small bead is threaded onto the wire. Initially, the bead is held at the origin. It is then released.

  1. Explain why the bead cannot remain stationary relative to the train at the origin.
  2. Show that, in the subsequent motion, the coordinates \((x, y)\) of the bead satisfy \[ \dot{x}(\ddot{x} - a) + \dot{y}(\ddot{y} + g) = 0 \] and deduce that \(\tfrac{1}{2}(\dot{x}^2 + \dot{y}^2) - ax + gy\) is constant during the motion.
  3. Find an expression for the maximum vertical displacement, \(b\), of the bead from its initial position in terms of \(a\), \(k\) and \(g\).
  4. Find the value of \(x\) for which the speed of the bead relative to the train is greatest and give this maximum speed in terms of \(a\), \(k\) and \(g\).

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1997 Paper 3 Q11
D: 1700.0 B: 1500.0
moments of inertia rigid body compound pendulum angular momentum constraint motion separation condition clapper and bell Lagrangian mechanics

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A heavy symmetrical bell and clapper can both swung freely in a vertical plane about a point \(O\) on a horizontal beam at the apex of the bell. The mass of the bell is \(M\) and its moment of inertia about the beam is \(Mk^{2}\). Its centre of mass, \(G\), is a distance \(h\) from \(O\). The clapper may be regarded as a small heavy ball on a light rod of length \(l\). Initially the bell is held with its axis vertical and its mouth above the beam. The clapper ball rests against the side of the bell, with the rod making an angle \(\beta\) with the axis. The bell is then released. Show that, at the moment when the clapper and bell separate, the clapper rod makes an angle \(\alpha\) with the upwards vertical, where \[ \cot\alpha=\cot\beta-\frac{k^{2}}{hl}\mathrm{cosec}\beta. \]

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