Questions — OCR M2 (149 questions)

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OCR M2 2008 June Q4
4 A golfer hits a ball from a point \(O\) on horizontal ground with a velocity of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal. The horizontal range of the ball is \(R\) metres and the time of flight is \(t\) seconds.
  1. Express \(t\) in terms of \(\theta\), and hence show that \(R = 125 \sin 2 \theta\). The golfer hits the ball so that it lands 110 m from \(O\).
  2. Calculate the two possible values of \(t\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_672_403_267_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A toy is constructed by attaching a small ball of mass 0.01 kg to one end of a uniform rod of length 10 cm whose other end is attached to the centre of the plane face of a uniform solid hemisphere with radius 3 cm . The rod has mass 0.02 kg , the hemisphere has mass 0.5 kg and the rod is perpendicular to the plane face of the hemisphere (see Fig. 1).
OCR M2 2008 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_794_735_264_705} A particle \(P\) of mass 0.5 kg is attached to points \(A\) and \(B\) on a fixed vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical (see diagram). The particle moves with constant speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.4 m .
  1. Calculate the tensions in the two strings. The particle now moves with constant angular speed \(\omega\) rad s \(^ { - 1 }\) and the string \(B P\) is on the point of becoming slack.
  2. Calculate \(\omega\).
OCR M2 2008 June Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-4_305_1301_1708_424} Two small spheres \(A\) and \(B\) of masses 2 kg and 3 kg respectively lie at rest on a smooth horizontal platform which is fixed at a height of 4 m above horizontal ground (see diagram). Sphere \(A\) is given an impulse of 6 N s towards \(B\), and \(A\) then strikes \(B\) directly. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\).
  1. Show that the speed of \(B\) after it has been hit by \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\). Sphere \(B\) leaves the platform and follows the path of a projectile.
  2. Calculate the speed and direction of motion of \(B\) at the instant when it hits the ground.
OCR M2 2008 June Q8
8
  1. Fig. 1 A uniform lamina \(A B C D\) is in the form of a right-angled trapezium. \(A B = 6 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(A D = 17 \mathrm {~cm}\) (see Fig. 1). Taking \(x\) - and \(y\)-axes along \(A D\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-5_481_1079_991_575} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The lamina is smoothly pivoted at \(A\) and it rests in a vertical plane in equilibrium against a fixed smooth block of height 7 cm . The mass of the lamina is 3 kg . \(A D\) makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2). Calculate the magnitude of the force which the block exerts on the lamina.
OCR M2 2009 June Q1
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 \(R\).
OCR M2 2009 June Q2
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.
OCR M2 2009 June Q3
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 \(A\).
OCR M2 2009 June Q4
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.
OCR M2 2009 June Q6
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 \(B\).
OCR M2 2009 June Q7
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.
OCR M2 2010 June Q1
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.
OCR M2 2010 June Q2
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.
OCR M2 2010 June Q3
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 \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR M2 2010 June Q4
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.
OCR M2 2010 June Q6
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.
OCR M2 2010 June Q7
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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OCR M2 2011 June Q1
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.
OCR M2 2011 June Q2
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.
OCR M2 2011 June Q3
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.
OCR M2 2011 June Q4
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.
OCR M2 2011 June Q5
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.
OCR M2 2011 June Q6
6 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-3_538_478_758_836} \captionsetup{labelformat=empty} \caption{Fig. 1}
\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.]}
OCR M2 2011 June Q7
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-4_474_912_260_493} \captionsetup{labelformat=empty} \caption{Fig. 1}
\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 \(\mu\).
OCR M2 2012 June Q1
1 A particle, of mass 0.8 kg , moves along a smooth horizontal surface. It hits a vertical wall, which is at right angles to the direction of motion of the particle, and rebounds. The speed of the particle as it hits the wall is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of restitution between the particle and the wall is 0.3 . Find
  1. the impulse that the wall exerts on the particle,
  2. the kinetic energy lost in the impact.
OCR M2 2012 June Q2
2 A car of mass 1600 kg moves along a straight horizontal road. The resistance to the motion of the car has constant magnitude 800 N and the car's engine is working at a constant rate of 20 kW .
  1. Find the acceleration of the car at an instant when the car's speed is \(20 \mathrm {~ms} ^ { - 1 }\). The car now moves up a hill inclined at \(4 ^ { \circ }\) to the horizontal. The car's engine continues to work at 20 kW and the magnitude of the resistance to motion remains at 800 N .
  2. Find the greatest steady speed at which the car can move up the hill.