OCR M2 (Mechanics 2) Specimen

1
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_236_949_269_603} A barge \(B\) is pulled along a canal by a horse \(H\), which is on the tow-path. The barge and the horse move in parallel straight lines and the tow-rope makes a constant angle of \(15 ^ { \circ }\) with the direction of motion (see diagram). The tow-rope remains taut and horizontal, and has a constant tension of 500 N .

1. Find the work done on the barge by the tow-rope, as the barge travels a distance of 400 m . The barge moves at a constant speed and takes 10 minutes to travel the 400 m .
2. Find the power applied to the barge.

2 A uniform circular cylinder, of radius 6 cm and height 15 cm , is in equilibrium on a fixed inclined plane with one of its ends in contact with the plane.

1. Given that the cylinder is on the point of toppling, find the angle the plane makes with the horizontal. The cylinder is now placed on a horizontal board with one of its ends in contact with the board. The board is then tilted so that the angle it makes with the horizontal gradually increases.
2. Given that the coefficient of friction between the cylinder and the board is \(\frac { 3 } { 4 }\), determine whether or not the cylinder will slide before it topples, justifying your answer.

3
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_389_698_1706_694} A uniform lamina \(A B C D\) has the shape of a square of side \(a\) adjoining a right-angled isosceles triangle whose equal sides are also of length \(a\). The weight of the lamina is \(W\). The lamina rests, in a vertical plane, on smooth supports at \(A\) and \(D\), with \(A D\) horizontal (see diagram).

1. Show that the centre of mass of the lamina is at a horizontal distance of \(\frac { 11 } { 9 } a\) from \(A\).
2. Find, in terms of \(W\), the magnitudes of the forces on the supports at \(A\) and \(D\).

4
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-3_563_707_274_721} A rigid body \(A B C\) consists of two uniform rods \(A B\) and \(B C\), rigidly joined at \(B\). The lengths of \(A B\) and \(B C\) are 13 cm and 20 cm respectively, and their weights are 13 N and 20 N respectively. The distance of \(B\) from \(A C\) is 12 cm . The body hangs in equilibrium, with \(A C\) horizontal, from two vertical strings attached at \(A\) and \(C\). Find the tension in each string.

5 A cyclist and his machine have a combined mass of 80 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 4 m above the level of \(A\).

1. Find the gain in kinetic energy and the gain in gravitational potential energy of the cyclist and his machine. During the ascent the resistance to motion is constant and has magnitude 70 N .
2. Given that the work done by the cyclist in ascending the hill is 8000 J , find the distance \(A B\). At \(B\) the cyclist is working at 720 watts and starts to move in a straight line along horizontal ground. The resistance to motion has the same magnitude of 70 N as before.
3. Find the acceleration with which the cyclist starts to move horizontally.

6 An athlete 'puts the shot' with an initial speed of \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(11 ^ { \circ }\) above the horizontal. At the instant of release the shot is 1.53 m above the horizontal ground. By treating the shot as a particle and ignoring air resistance, find

1. the maximum height, above the ground, reached by the shot,
2. the horizontal distance the shot has travelled when it hits the ground.

7
\includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-4_314_757_285_708} A ball of mass 0.08 kg is attached by two strings to a fixed vertical post. The strings have lengths 2.5 m and 2.4 m , as shown in the diagram. The ball moves in a horizontal circle, of radius 2.4 m , with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Each string is taut and the lower string is horizontal. The modelling assumptions made are that both strings are light and inextensible, and that there is no air resistance.

1. Find the tension in each string when \(v = 10.5\).
2. Find the least value of \(v\) for which the lower string is taut.

8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.24 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a horizontal table, with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the magnitude of the impulse exerted by \(A\) on \(B\).
2. Show that \(m \leqslant 0.08\). It is given that \(m = 0.06\).
3. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
4. Find the speeds of \(A\) and \(B\) immediately after the collision.