Questions M2 (1537 questions)

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OCR M2 2005 June Q5
10 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_319_650_1219_749} A uniform \(\operatorname { rod } A B\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).
  1. Show that the tension in the string is 4.5 N .
  2. Find the magnitude and direction of the force acting on the rod at \(A\).
OCR M2 2005 June Q6
10 marks Standard +0.3
6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5 ^ { \circ }\) to the horizontal. At a certain point \(P\) on the hill the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is \(15 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
  2. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW , correct to 3 significant figures.
  3. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\).
OCR M2 2005 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_76_243_269_365} \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_332_1427_322_360} A barrier is modelled as a uniform rectangular plank of wood, \(A B C D\), rigidly joined to a uniform square metal plate, \(D E F G\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that \(C D E\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(C H\) is 0.25 m .
  1. Calculate the contact force at \(H\) when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above \(D\).
  2. Calculate the angle between \(A B\) and the horizontal when the barrier is in the open position.
OCR M2 2005 June Q8
13 marks Standard +0.3
8 A particle is projected with speed \(49 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\) from a point \(O\) on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at time \(t\) seconds after projection are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac { x ^ { 2 } \left( 1 + \tan ^ { 2 } \theta \right) } { 490 } .$$
    \includegraphics[max width=\textwidth, alt={}]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_627_1249_1699_447}
    The particle passes through the point where \(x = 70\) and \(y = 30\). The two possible values of \(\theta\) are \(\theta _ { 1 }\) and \(\theta _ { 2 }\), and the corresponding points where the particle returns to the plane are \(A _ { 1 }\) and \(A _ { 2 }\) respectively (see diagram).
  2. Find \(\theta _ { 1 }\) and \(\theta _ { 2 }\).
  3. Calculate the distance between \(A _ { 1 }\) and \(A _ { 2 }\).
OCR M2 2006 June Q1
4 marks Easy -1.8
1 A child of mass 35 kg runs up a flight of stairs in 10 seconds. The vertical distance climbed is 4 m . Assuming that the child's speed is constant, calculate the power output.
OCR M2 2006 June Q2
5 marks Moderate -0.8
2 A small sphere of mass 0.3 kg is dropped from rest at a height of 2 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.4 m above the ground. Ignoring air resistance, calculate the magnitude of the impulse which the ground exerts on the sphere when it rebounds.
OCR M2 2006 June Q3
7 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-2_710_572_721_788} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point \(O\), the centre of the plane face, and the other string is attached to the point \(A\) on the rim of the plane face. The hemisphere hangs in equilibrium and \(O A\) makes an angle of \(60 ^ { \circ }\) with the vertical (see diagram).
  1. Find the horizontal distance from the centre of mass of the hemisphere to the vertical through \(O\).
  2. Calculate the tensions in the strings.
OCR M2 2006 June Q4
9 marks Moderate -0.3
4 A car of mass 900 kg is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a level road. The total resistance to motion is 450 N .
  1. Calculate the power output of the car's engine. A roof box of mass 50 kg is mounted on the roof of the car. The total resistance to motion of the vehicle increases to 500 N .
  2. The car's engine continues to work at the same rate. Calculate the maximum speed of the car on the level road. The power output of the car's engine increases to 15000 W . The resistance to motion of the car, with roof box, remains 500 N .
  3. Calculate the instantaneous acceleration of the car on the level road when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. The car climbs a hill which is at an angle of \(5 ^ { \circ }\) to the horizontal. Calculate the instantaneous retardation of the car when its speed is \(26 \mathrm {~ms} ^ { - 1 }\).
OCR M2 2006 June Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-3_657_549_1219_799} A uniform lamina \(A B C D E\) consists of a square and an isosceles triangle. The square has sides of 18 cm and \(B C = C D = 15 \mathrm {~cm}\) (see diagram).
  1. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
  2. The lamina is freely suspended from \(B\). Calculate the angle that \(B D\) makes with the vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_441_1355_265_394} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A light inextensible string of length 1 m passes through a small smooth hole \(A\) in a fixed smooth horizontal plane. One end of the string is attached to a particle \(P\), of mass 0.5 kg , which hangs in equilibrium below the plane. The other end of the string is attached to a particle \(Q\), of mass 0.3 kg , which rotates with constant angular speed in a circle of radius 0.2 m on the surface of the plane (see Fig. 1).
OCR M2 2006 June Q7
13 marks Standard +0.3
7 A small ball is projected at an angle of \(50 ^ { \circ }\) above the horizontal, from a point \(A\), which is 2 m above ground level. The highest point of the path of the ball is 15 m above the ground, which is horizontal. Air resistance may be ignored.
  1. Find the speed with which the ball is projected from \(A\). The ball hits a net at a point \(B\) when it has travelled a horizontal distance of 45 m .
  2. Find the height of \(B\) above the ground.
  3. Find the speed of the ball immediately before it hits the net.
OCR M2 2006 June Q8
14 marks Standard +0.3
8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 2 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a smooth horizontal surface, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), 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 \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the greatest possible value of \(m\). It is given that \(m = 1\).
  2. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
  3. Find the kinetic energy lost due to the collision.
AQA M2 2006 January Q1
8 marks Moderate -0.8
1 A stone, of mass 0.4 kg , is thrown vertically upwards with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 6 metres above ground level.
  1. Calculate the initial kinetic energy of the stone.
    1. Show that the kinetic energy of the stone when it hits the ground is 36.3 J , correct to three significant figures.
    2. Hence find the speed at which the stone hits the ground.
    3. State one assumption that you have made.
AQA M2 2006 January Q2
7 marks Moderate -0.8
2 A particle, of mass 2 kg , is attached to one end of a light inextensible string. The other end is fixed to the point \(O\). The particle is set into motion, so that it describes a horizontal circle of radius 0.6 metres, with the string at an angle of \(30 ^ { \circ }\) to the vertical. The centre of the circle is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-2_344_340_1418_842}
  1. Show that the tension in the string is 22.6 N , correct to three significant figures.
  2. Find the speed of the particle.
AQA M2 2006 January Q3
9 marks Moderate -0.3
3 A particle moves in a straight line and at time \(t\) has velocity \(v\), where $$v = 2 t - 12 \mathrm { e } ^ { - t } , \quad t \geqslant 0$$
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. State the range of values of the acceleration of the particle.
  2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).
AQA M2 2006 January Q4
10 marks Standard +0.3
4 The diagram shows a uniform lamina \(A B C D E F G H\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-3_346_933_1123_577}
  1. Explain why the centre of mass is 25 cm from \(A H\).
  2. Show that the centre of mass is 4.375 cm from \(H G\).
  3. The lamina is freely suspended from \(A\). Find the angle between \(A B\) and the vertical when the lamina is in equilibrium.
  4. Explain, briefly, how you have used the fact that the lamina is uniform.
AQA M2 2006 January Q5
8 marks Moderate -0.3
5 A particle moves such that at time \(t\) seconds its acceleration is given by $$( 2 \cos t \mathbf { i } - 5 \sin t \mathbf { j } ) \mathrm { m } \mathrm {~s} ^ { - 2 }$$
  1. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 0\).
  2. When \(t = 0\), the velocity of the particle is \(( 2 \mathbf { i } + 10 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find an expression for the velocity of the particle at time \(t\).
AQA M2 2006 January Q6
10 marks Standard +0.3
6 A student is modelling the motion of a small boat as it moves on a lake. When the speed of the boat is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the engine is switched off. At time \(t\) seconds later, it has a velocity of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and experiences a resistance force of magnitude \(20 v\) newtons. The mass of the boat is 80 kg . To set up a simple model for the motion of the boat, the student assumes that the water in the lake is still and that the boat travels in a straight line.
  1. Explain how these two assumptions allow the student to create a simple model.
  2. State one other assumption that the student should make.
    1. Express \(\frac { \mathrm { d } v } { \mathrm {~d} t }\) in terms of \(v\).
    2. Find an expression for \(v\) in terms of \(t\).
AQA M2 2006 January Q7
9 marks Standard +0.3
7 A particle \(P\), of mass \(m \mathrm {~kg}\), is placed at the point \(Q\) on the top of a smooth upturned hemisphere of radius 3 metres and centre \(O\). The plane face of the hemisphere is fixed to a horizontal table. The particle is set into motion with an initial horizontal velocity of \(2 \mathrm {~ms} ^ { - 1 }\). When the particle is on the surface of the hemisphere, the angle between \(O P\) and \(O Q\) is \(\theta\) and the particle has speed \(v \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-4_415_1007_1573_513}
  1. Show that \(v ^ { 2 } = 4 + 6 g ( 1 - \cos \theta )\).
  2. Find the value of \(\theta\) when the particle leaves the hemisphere.
AQA M2 2006 January Q8
14 marks Standard +0.3
8 A particle, of mass 10 kg , is attached to one end of a light elastic string of natural length 0.4 metres and modulus of elasticity 100 N . The other end of the string is fixed to the point \(O\).
  1. Find the length of the elastic string when the particle hangs in equilibrium directly below \(O\).
  2. The particle is pulled down and held at a point \(P\), which is 1 metre vertically below \(O\). Show that the elastic potential energy of the string when the particle is in this position is 45 J .
  3. The particle is released from rest at the point \(P\). In the subsequent motion, the particle has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is \(x\) metres below \(\boldsymbol { O }\).
    1. Show that, while the string is taut, $$v ^ { 2 } = 39.6 x - 25 x ^ { 2 } - 14.6$$
    2. Find the value of \(x\) when the particle comes to rest for the first time after being released, given that the string is still taut.
AQA M2 2008 January Q1
10 marks Moderate -0.8
1 A ball is thrown vertically upwards from ground level with an initial speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has a mass of 0.6 kg . Assume that the only force acting on the ball after it is thrown is its weight.
  1. Calculate the initial kinetic energy of the ball.
  2. By using conservation of energy, find the maximum height above ground level reached by the ball.
  3. By using conservation of energy, find the kinetic energy and the speed of the ball when it is at a height of 3 m above ground level.
  4. State one modelling assumption which has been made.
AQA M2 2008 January Q2
8 marks Moderate -0.8
2 A particle moves in a straight line and at time \(t\) it has velocity \(v\), where $$v = 3 t ^ { 2 } - 2 \sin 3 t + 6$$
    1. Find an expression for the acceleration of the particle at time \(t\).
    2. When \(t = \frac { \pi } { 3 }\), show that the acceleration of the particle is \(2 \pi + 6\).
  2. When \(t = 0\), the particle is at the origin. Find an expression for the displacement of the particle from the origin at time \(t\).
AQA M2 2008 January Q3
11 marks Standard +0.3
3 A uniform ladder of length 4 metres and mass 20 kg rests in equilibrium with its foot, \(A\), on a rough horizontal floor and its top leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the angle between the ladder and the floor is \(60 ^ { \circ }\). A man of mass 80 kg is standing at point \(C\) on the ladder. With the man in this position, the ladder is on the point of slipping. The coefficient of friction between the ladder and the floor is 0.4 . The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-3_567_448_708_788}
  1. Draw a diagram to show the forces acting on the ladder.
  2. Show that the magnitude of the frictional force between the ladder and the ground is 392 N .
  3. Find the distance \(A C\).
AQA M2 2008 January Q4
9 marks Standard +0.3
4 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, the position vector, \(\mathbf { r }\) metres, of the particle is given by $$\mathbf { r } = \left( t ^ { 3 } - 3 t ^ { 2 } + 4 \right) \mathbf { i } + \left( 4 t + t ^ { 2 } \right) \mathbf { j }$$
  1. Find an expression for the velocity of the particle at time \(t\).
  2. The mass of the particle is 3 kg .
    1. Find an expression for \(\mathbf { F }\) at time \(t\).
    2. Find the magnitude of \(\mathbf { F }\) when \(t = 3\).
  3. Find the value of \(t\) when \(\mathbf { F }\) acts due north.
AQA M2 2008 January Q5
9 marks Standard +0.3
5 Two light inextensible strings, of lengths 0.4 m and 0.2 m , each have one end attached to a particle, \(P\), of mass 4 kg . The other ends of the strings are attached to the points \(A\) and \(B\) respectively. The point \(A\) is vertically above the point \(B\). The particle moves in a horizontal circle, centre \(B\) and radius 0.2 m , at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle and strings are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_396_558_587_735} $$\text { ← } 0.2 \mathrm {~m} \longrightarrow$$
  1. Calculate the magnitude of the acceleration of the particle.
  2. Show that the tension in string \(P A\) is 45.3 N , correct to three significant figures.
  3. Find the tension in string \(P B\).
AQA M2 2008 January Q6
10 marks Standard +0.3
6 A light elastic string has one end attached to a point \(A\) fixed on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle of mass 6 kg . The elastic string has natural length 4 metres and modulus of elasticity 300 newtons. The particle is pulled down the plane in the direction of the line of greatest slope through \(A\). The particle is released from rest when it is 5.5 metres from \(A\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_314_713_1900_660}
  1. Calculate the elastic potential energy of the string when the particle is 5.5 metres from the point \(A\).
  2. Show that the speed of the particle when the string becomes slack is \(3.66 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  3. Show that the particle will not reach point \(A\) in the subsequent motion.