OCR M2 (Mechanics 2) 2010 January

1 Find the average power exerted by a climber of mass 75 kg when climbing a vertical distance of 40 m in 2 minutes.

2 A small sphere of mass 0.2 kg is dropped from rest at a height of 3 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.8 m above the ground.

1. Calculate the magnitude of the impulse which the ground exerts on the sphere.
2. Calculate the coefficient of restitution between the sphere and the ground.

3 \begin{figure}[h] \end{figure} A uniform conical shell has mass 0.2 kg , height 0.3 m and base diameter 0.8 m . A uniform hollow cylinder has mass 0.3 kg , length 0.7 m and diameter 0.8 m . The conical shell is attached to the cylinder, with the circumference of its base coinciding with one end of the cylinder (see Fig. 1).

1. Show that the distance of the centre of mass of the combined object from the vertex of the conical shell is 0.47 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-2_497_572_1836_788} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The combined object is freely suspended from its vertex and is held with its axis horizontal. This is achieved by means of a wire attached to a point on the circumference of the base of the conical shell. The wire makes an angle of \(80 ^ { \circ }\) with the slant edge of the conical shell (see Fig. 2).
2. Calculate the tension in the wire.

4 A car of mass 700 kg is moving along a horizontal road against a constant resistance to motion of 400 N . At an instant when the car is travelling at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the driving force of the car at this instant.
2. Find the power at this instant. The maximum steady speed of the car on a horizontal road is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the maximum power of the car. The car now moves at maximum power against the same resistance up a slope of constant angle \(\theta ^ { \circ }\) to the horizontal. The maximum steady speed up the slope is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Find \(\theta\).

5 Two spheres of the same radius with masses 2 kg and 3 kg are moving directly towards each other on a smooth horizontal plane with speeds \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The spheres collide and the kinetic energy lost is 81 J . Calculate the speed and direction of motion of each sphere after the collision.

6
\includegraphics[max width=\textwidth, alt={}, center]{8e1225a2-cb98-4b71-a4af-0150f093f852-3_698_1047_1297_550} A particle \(P\) is projected with speed \(V _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 1 }\) from a point \(O\) on horizontal ground. When \(P\) is vertically above a point \(A\) on the ground its height is 250 m and its velocity components are \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically upwards (see diagram).

1. Show that \(V _ { 1 } = 86.0\) and \(\theta _ { 1 } = 62.3 ^ { \circ }\), correct to 3 significant figures. At the instant when \(P\) is vertically above \(A\), a second particle \(Q\) is projected from \(O\) with speed \(V _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta _ { 2 } . P\) and \(Q\) hit the ground at the same time and at the same place.
2. Calculate the total time of flight of \(P\) and the total time of flight of \(Q\).
3. Calculate the range of the particles and hence calculate \(V _ { 2 }\) and \(\theta _ { 2 }\).

7 \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.2 kg is moving on the smooth inner surface of a fixed hollow hemisphere which has centre \(O\) and radius \(5 \mathrm {~m} . P\) moves with constant angular speed \(\omega\) in a horizontal circle at a vertical distance of 3 m below the level of \(O\) (see Fig.1).

1. Calculate the magnitude of the force exerted by the hemisphere on \(P\).
2. Calculate \(\omega\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8e1225a2-cb98-4b71-a4af-0150f093f852-4_592_773_1231_687} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string is now attached to \(P\). The string passes through a small smooth hole at the lowest point of the hemisphere and a particle of mass 0.1 kg hangs in equilibrium at the end of the string. \(P\) moves in the same horizontal circle as before (see Fig. 2).
3. Calculate the new angular speed of \(P\).