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2009 Paper 1 Q10
D: 1500.0 B: 1500.0
mechanics motion on a slope wedge pulley connected particles Newton's second law optimisation trigonometric identity smooth surface

A triangular wedge is fixed to a horizontal surface. The base angles of the wedge are \(\alpha\) and \(\frac\pi 2-\alpha\). Two particles, of masses \(M\) and \(m\), lie on different faces of the wedge, and are connected by a light inextensible string which passes over a smooth pulley at the apex of the wedge, as shown in the diagram. The contacts between the particles and the wedge are smooth.

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  1. Show that if \(\tan \alpha> \dfrac m M \) the particle of mass \(M\) will slide down the face of the wedge.
  2. Given that \(\tan \alpha = \dfrac{2m}M\), show that the magnitude of the acceleration of the particles is \[ \frac{g\sin\alpha}{\tan\alpha +2} \] and that this is maximised at \(4m^3=M^3\,\).

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1998 Paper 2 Q10
D: 1600.0 B: 1500.0
mechanics wedge motion on a slope conservation of momentum energy conservation Newton's laws differential equation smooth surface particle on inclined plane

A wedge of mass \(M\) rests on a smooth horizontal surface. The face of the wedge is a smooth plane inclined at an angle \(\alpha\) to the horizontal. A particle of mass \(m\) slides down the face of the wedge, starting from rest. At a later time \(t\), the speed \(V\) of the wedge, the speed \(v\) of the particle and the angle \(\beta\) of the velocity of the particle below the horizontal are as shown in the diagram.

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\noindent Let \(y\) be the vertical distance descended by the particle. Derive the following results, stating in (ii) and (iii) the mechanical principles you use:
  1. \(V\sin\alpha=v\sin(\beta-\alpha)\);
  2. \(\tan\beta=(1+m/M)\tan\alpha\);
  3. \(2gy=v^2(M+m\cos^2\beta)/M\).
Write down a differential equation for \(y\) and hence show that $$y={gMt^2\sin^2\beta \over 2\,(M+m\cos^2\beta)}.$$

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1994 Paper 1 Q11
D: 1500.0 B: 1469.5
mechanics motion on a slope inclined plane pulley Newton's second law energy conservation kinematics light inextensible string inequality

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The diagram shows a small railway wagon \(A\) of mass \(m\) standing at the bottom of a smooth railway track of length \(d\) inclined at an angle \(\theta\) to the horizontal. A light inextensible string, also of length \(d\), is connected to the wagon and passes over a light frictionless pulley at the top of the incline. On the other end of the string is a ball \(B\) of mass \(M\) which hangs freely. The system is initially at rest and is then released.
  1. Find the condition which \(m,M\) and \(\theta\) must satisfy to ensure that the ball will fall to the ground. Assuming that this condition is satisfied, show that the velocity \(v\) of the ball when it hits the ground satisfies \[ v^{2}=\frac{2g(M-m\sin\theta)d\sin\theta}{M+m}. \]
  2. Find the condition which \(m,M\) and \(\theta\) must satisfy if the wagon is not to collide with the pulley at the top of the incline.

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