OCR M2 (Mechanics 2) 2010 June

1 A particle is projected horizontally with a speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 10 m above horizontal ground. The particle moves freely under gravity. Calculate the speed and direction of motion of the particle at the instant it hits the ground.

2

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4acdc4d7-01f0-496f-9390-de5f136bc5f9-2_333_551_488_836} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A uniform piece of wire, \(A B C\), forms a semicircular arc of radius \(6 \mathrm {~cm} . O\) is the mid-point of \(A C\) (see Fig. 1). Show that the distance from \(O\) to the centre of mass of the wire is 3.82 cm , correct to 3 significant figures.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4acdc4d7-01f0-496f-9390-de5f136bc5f9-2_579_721_1178_753} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Two semicircular pieces of wire, \(A B C\) and \(A D C\), are joined together at their ends to form a circular hoop of radius 6 cm . The mass of \(A B C\) is 3 grams and the mass of \(A D C\) is 5 grams. The hoop is freely suspended from \(A\) (see Fig. 2). Calculate the angle which the diameter \(A C\) makes with the vertical, giving your answer correct to the nearest degree.

3 The maximum power produced by the engine of a small aeroplane of mass 2 tonnes is 128 kW . Air resistance opposes the motion directly and the lift force is perpendicular to the direction of motion. The magnitude of the air resistance is proportional to the square of the speed and the maximum steady speed in level flight is \(80 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the magnitude of the air resistance when the speed is \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The aeroplane is climbing at a constant angle of \(2 ^ { \circ }\) to the horizontal.
2. Find the maximum acceleration at an instant when the speed of the aeroplane is \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4
\includegraphics[max width=\textwidth, alt={}, center]{4acdc4d7-01f0-496f-9390-de5f136bc5f9-3_360_1006_260_571} A non-uniform beam \(A B\) of length 4 m and mass 5 kg has its centre of mass at the point \(G\) of the beam where \(A G = 2.5 \mathrm {~m}\). The beam is freely suspended from its end \(A\) and is held in a horizontal position by means of a wire attached to the end \(B\). The wire makes an angle of \(20 ^ { \circ }\) with the vertical and the tension is \(T \mathrm {~N}\) (see diagram).

1. Calculate \(T\).
2. Calculate the magnitude and the direction of the force acting on the beam at \(A\).
   \includegraphics[max width=\textwidth, alt={}, center]{4acdc4d7-01f0-496f-9390-de5f136bc5f9-3_488_1095_1169_525} One end of a light inextensible string of length \(l\) is attached to the vertex of a smooth cone of semivertical angle \(45 ^ { \circ }\). The cone is fixed to the ground with its axis vertical. The other end of the string is attached to a particle of mass \(m\) which rotates in a horizontal circle in contact with the outer surface of the cone. The angular speed of the particle is \(\omega\) (see diagram). The tension in the string is \(T\) and the contact force between the cone and the particle is \(R\).
3. By resolving horizontally and vertically, find two equations involving \(T\) and \(R\) and hence show that \(T = \frac { 1 } { 2 } m \left( \sqrt { 2 } g + l \omega ^ { 2 } \right)\).
4. When the string has length 0.8 m , calculate the greatest value of \(\omega\) for which the particle remains in contact with the cone.

6 A particle \(A\) of mass \(2 m\) is moving with speed \(u\) on a smooth horizontal surface when it collides with a stationary particle \(B\) of mass \(m\). After the collision the speed of \(A\) is \(v\), the speed of \(B\) is \(3 v\) and the particles move in the same direction.

1. Find \(v\) in terms of \(u\).
2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac { 4 } { 5 }\).
   \(B\) subsequently hits a vertical wall which is perpendicular to the direction of motion. As a result of the impact, \(B\) loses \(\frac { 3 } { 4 }\) of its kinetic energy.
3. Show that the speed of \(B\) after hitting the wall is \(\frac { 3 } { 5 } u\).
4. \(B\) then hits \(A\). Calculate the speeds of \(A\) and \(B\), in terms of \(u\), after this collision and state their directions of motion.

7
\includegraphics[max width=\textwidth, alt={}, center]{4acdc4d7-01f0-496f-9390-de5f136bc5f9-4_559_1057_1025_543} A small ball of mass 0.2 kg is projected with speed \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a roof from a point \(A\) at the bottom of the roof. The ball remains in contact with the roof and moves up the line of greatest slope to the top of the roof at \(B\). The roof is rough and the coefficient of friction is \(\frac { 1 } { 2 }\). The distance \(A B\) is 5 m and \(A B\) is inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).

1. Show that the speed of the ball when it reaches \(B\) is \(5.44 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 2 decimal places. The ball leaves the roof at \(B\) and moves freely under gravity. The point \(C\) is at the lower edge of the roof. The distance \(B C\) is 5 m and \(B C\) is inclined at \(30 ^ { \circ }\) to the horizontal.
2. Determine whether or not the ball hits the roof between \(B\) and \(C\). {www.ocr.org.uk}) after the live examination series.
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