OCR M2 (Mechanics 2) 2011 June

1
\includegraphics[max width=\textwidth, alt={}, center]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-2_314_931_242_607} A sledge with its load has mass 70 kg . It moves down a slope and the resistance to the motion of the sledge is 90 N . The speed of the sledge is controlled by the constant tension in a light rope, which is attached to the sledge and parallel to the slope (see diagram). While travelling 20 m down the slope, the speed of the sledge decreases from \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it descends a vertical distance of 3 m .

1. Calculate the change in energy of the sledge and its load.
2. Calculate the tension in the rope.

2 A car of mass 1250 kg travels along a straight road inclined at \(2 ^ { \circ }\) to the horizontal. The resistance to the motion of the car is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car and \(k\) is a constant. The car travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope and the engine of the car works at a constant rate of 21 kW .

1. Calculate the value of \(k\).
2. Calculate the constant speed of the car on a horizontal road.

3 A uniform lamina \(A B C D E\) consists of a square \(A C D E\) and an equilateral triangle \(A B C\) which are joined along their common edge \(A C\) to form a pentagon whose sides are each 8 cm in length.

1. Calculate the distance of the centre of mass of the lamina from \(A C\).
2. The lamina is freely suspended from \(A\) and hangs in equilibrium. Calculate the angle that \(A C\) makes with the vertical.

4 Two small spheres \(A\) and \(B\) are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before they collide directly. The direction of motion of \(B\) is reversed in the collision. The speeds of \(A\) and \(B\) after the collision are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(2.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. (a) Show that the direction of motion of \(A\) is unchanged by the collision.
   (b) Calculate the coefficient of restitution between \(A\) and \(B\). The mass of \(B\) is 0.2 kg .
2. Find the mass of \(A\).
   \(B\) continues to move at \(2.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes a vertical wall at right angles. The wall exerts an impulse of magnitude 0.68 N s on \(B\).
3. Calculate the coefficient of restitution between \(B\) and the wall.

5 A particle is projected with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\) from a point \(O\) and moves freely under gravity. The horizontal and vertically upwards displacements of the particle from \(O\) at any subsequent time \(t \mathrm {~s}\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the particle.
2. Calculate the values of \(x\) when \(y = 0.6\).
3. Find the direction of motion of the particle when \(y = 0.6\) and the particle is rising.

6 \begin{figure}[h] \end{figure} A container is constructed from a hollow cylindrical shell and a hollow cone which are joined along their circumferences. The cylindrical shell has radius 0.2 m , and the cone has semi-vertical angle \(30 ^ { \circ }\). Two identical small spheres \(P\) and \(Q\) move independently in horizontal circles on the smooth inner surface of the container (see Fig. 1). Each sphere has mass 0.3 kg .

1. \(P\) moves in a circle of radius 0.12 m and is in contact with only the conical part of the container. Calculate the angular speed of \(P\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-3_278_209_1845_1009} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(Q\) moves with speed \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is in contact with both the cylindrical and conical surfaces of the container (see Fig. 2). Calculate the magnitude of the force which the cylindrical shell exerts on the sphere.
3. Calculate the difference between the mechanical energy of \(P\) and of \(Q\). \section*{[Question 7 is printed overleaf.]}

7 \begin{figure}[h] \end{figure} A uniform solid cone of height 0.8 m and semi-vertical angle \(60 ^ { \circ }\) lies with its curved surface on a horizontal plane. The point \(P\) on the circumference of the base is in contact with the plane. \(V\) is the vertex of the cone and \(P Q\) is a diameter of its base. The weight of the cone is 550 N . A force of magnitude \(F \mathrm {~N}\) and line of action \(P Q\) is applied to the base of the cone (see Fig. 1). The cone topples about \(V\) without sliding.

1. Calculate the least possible value of \(F\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-4_528_1143_1302_500} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The force of magnitude \(F \mathrm {~N}\) is removed and an increasing force of magnitude \(T \mathrm {~N}\) acting upwards in the vertical plane of symmetry of the cone and perpendicular to \(P Q\) is applied to the cone at \(Q\) (see Fig. 2). The coefficient of friction between the cone and the horizontal plane is \(\mu\).
2. Given that the cone slides before it topples about \(P\), calculate the greatest possible value for \(\mu\).