OCR M3 (Mechanics 3) 2009 June

A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \text{ m s}^{-1}\) and \(6 \text{ m s}^{-1}\) respectively. The speed of the sphere immediately after it bounces is \(5 \text{ m s}^{-1}\).

1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \text{ m s}^{-1}\), and hence find the coefficient of restitution between the surface and the sphere. [3]
2. State the direction of the impulse on the sphere and find its magnitude. [3]

\includegraphics{figure_2} Two uniform rods, \(AB\) and \(BC\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is \(1.5\) m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is \(3\) m (see diagram). The weight of \(AB\) is \(80\) N and the frictional force acting on \(AB\) at \(A\) is \(14\) N.

1. Write down the horizontal component of the force acting on \(AB\) at \(B\) and show that the vertical component of this force is \(33\) N upwards. [4]
2. Given that the force acting on \(BC\) at \(C\) has magnitude \(50\) N, find the weight of \(BC\). [4]

\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(4\) kg and \(2\) kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \text{ m s}^{-1}\). The spheres are moving in opposite directions, each at \(60°\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.

1. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres. [8]
2. Find the coefficient of restitution between the spheres. [2]

A motor-cycle, whose mass including the rider is \(120\) kg, is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \text{ m s}^{-1}\) and when it has travelled a distance of \(x\) m beyond \(A\) its speed is \(v \text{ m s}^{-1}\). The engine develops a constant power of \(8\) kW and resistances are modelled by a force of \(0.25v^2\) N opposing the motion.

1. Show that \(\frac{480v^2}{v^3 - 32000} \frac{dv}{dx} = -1\). [5]
2. Find the speed of the motor-cycle when it has travelled \(500\) m beyond \(A\). [6]

\includegraphics{figure_5} Each of two identical strings has natural length \(1.5\) m and modulus of elasticity \(18\) N. One end of one of the strings is attached to \(A\) and one end of the other string is attached to \(B\), where \(A\) and \(B\) are fixed points which are \(3\) m apart and at the same horizontal level. \(M\) is the mid-point of \(AB\). A particle \(P\) of mass \(m\) kg is attached to the other end of each of the strings. \(P\) is held at rest at the point \(0.8\) m vertically above \(M\), and then released. The lowest point reached by \(P\) in the subsequent motion is \(2\) m below \(M\) (see diagram).

1. Find the maximum tension in each of the strings during \(P\)'s motion. [3]
2. By considering energy,
   1. show that the value of \(m\) is \(0.42\), correct to 2 significant figures, [5]
   2. find the speed of \(P\) at \(M\). [3]

\includegraphics{figure_6} A particle \(P\) of mass \(m\) kg is attached to one end of a light inextensible string of length \(L\) m. The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(OP\), at time \(t\) s, is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is \(0.05\).

1. Show that \(\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin \theta\). [2]
2. Hence show that the motion of \(P\) is approximately simple harmonic. [2]
3. Given that the period of the approximate simple harmonic motion is \(\frac{4}{3}\pi\) s, find the value of \(L\). [2]
4. Find the value of \(\theta\) when \(t = 0.7\) s, and the value of \(t\) when \(\theta\) next takes this value. [4]
5. Find the speed of \(P\) when \(t = 0.7\) s. [3]

\includegraphics{figure_7} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(OP\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).

1. Find \(v^2\) in terms of \(u\), \(a\), \(g\) and \(\theta\) and show that \(R = \frac{mu^2}{a} + mg(3\cos\theta - 2)\). [7]
2. Given that \(P\) just reaches the highest point of the circle, find \(u^2\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v^2\) is \(ag\). [4]
3. Given instead that \(P\) oscillates between \(\theta = \pm\frac{1}{5}\pi\) radians, find \(u^2\) in terms of \(a\) and \(g\). [2]