OCR M2 (Mechanics 2) 2009 June

1 A boy on a sledge slides down a straight track of length 180 m which descends a vertical distance of 40 m . The combined mass of the boy and the sledge is 75 kg . The initial speed is \(3 \mathrm {~ms} ^ { - 1 }\) and the final speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude, \(R \mathrm {~N}\), of the resistance to motion is constant. By considering the change in energy, calculate \(R\).

2 A car of mass 1100 kg has maximum power of 44000 W . The resistive forces have constant magnitude 1400 N .

1. Calculate the maximum steady speed of the car on the level. The car is moving on a hill of constant inclination \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\).
2. Calculate the maximum steady speed of the car when ascending the hill.
3. Calculate the acceleration of the car when it is descending the hill at a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) working at half the maximum power.

3
\includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-2_497_951_1123_598} A uniform beam \(A B\) has weight 70 N and length 2.8 m . The beam is freely hinged to a wall at \(A\) and is supported in a horizontal position by a strut \(C D\) of length 1.3 m . One end of the strut is attached to the beam at \(C , 0.5 \mathrm {~m}\) from \(A\), and the other end is attached to the wall at \(D\), vertically below \(A\). The strut exerts a force on the beam in the direction \(D C\). The beam carries a load of weight 50 N at its end \(B\) (see diagram).

1. Calculate the magnitude of the force exerted by the strut on the beam.
2. Calculate the magnitude of the force acting on the beam at \(A\).

4 A light inextensible string of length 0.6 m has one end fixed to a point \(A\) on a smooth horizontal plane. The other end of the string is attached to a particle \(B\), of mass 0.4 kg , which rotates about \(A\) with constant angular speed \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) on the surface of the plane.

1. Calculate the tension in the string. A particle \(P\) of mass 0.1 kg is attached to the mid-point of the string. The line \(A P B\) is straight and rotation continues at \(2 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Calculate the tension in the section of the string \(A P\).
3. Calculate the total kinetic energy of the system.

1. Fig. 1 Fig. 1 shows a uniform lamina \(B C D\) in the shape of a quarter circle of radius 6 cm . Show that the distance of the centre of mass of the lamina from \(B\) is 3.60 cm , correct to 3 significant figures. A uniform rectangular lamina \(A B D E\) has dimensions \(A B = 12 \mathrm {~cm}\) and \(A E = 6 \mathrm {~cm}\). A single plane object is formed by attaching the rectangular lamina to the lamina \(B C D\) along \(B D\) (see Fig. 2). The mass of \(A B D E\) is 3 kg and the mass of \(B C D\) is 2 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-3_959_447_1123_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the object. The object is freely suspended at \(C\) and rests in equilibrium.
3. Calculate the angle that \(A C\) makes with the vertical.

6 Two uniform spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 0.4 kg and the mass of \(B\) is 0.2 kg . The spheres \(A\) and \(B\) are travelling in the same direction in a straight line on a smooth horizontal surface, \(A\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and \(B\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v < 5\). A collides directly with \(B\) and the impulse between them has magnitude 0.9 Ns . Immediately after the collision, the speed of \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate \(v\).
   \(B\) subsequently collides directly with a stationary sphere \(C\) of mass 0.1 kg and the same radius as \(A\) and \(B\). The coefficient of restitution between \(B\) and \(C\) is 0.6 .
2. Determine whether there will be a further collision between \(A\) and \(B\).

7
\includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-4_440_657_906_744} A ball is projected with an initial speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(25 ^ { \circ }\) below the horizontal from a point on the top of a vertical wall. The point of projection is 8 m above horizontal ground. The ball hits a vertical fence which is at a horizontal distance of 9 m from the wall (see diagram).

1. Calculate the height above the ground of the point where the ball hits the fence.
2. Calculate the direction of motion of the ball immediately before it hits the fence.
3. It is given that \(30 \%\) of the kinetic energy of the ball is lost when it hits the fence. Calculate the speed of the ball immediately after it hits the fence.