Questions M2 (1537 questions)

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OCR M2 2013 January Q4
8 marks Standard +0.3
\includegraphics{figure_4} A uniform square lamina \(ABCD\) of side 6 cm has a semicircular piece, with \(AB\) as diameter, removed (see diagram).
  1. Find the distance of the centre of mass of the remaining shape from \(CD\). [6]
The remaining shape is suspended from a fixed point by a string attached at \(C\) and hangs in equilibrium.
  1. Find the angle between \(CD\) and the vertical. [2]
OCR M2 2013 January Q5
8 marks Standard +0.3
\includegraphics{figure_5} A uniform rod \(AB\), of mass 3 kg and length 4 m, is in limiting equilibrium with \(A\) on rough horizontal ground. The rod is at an angle of 60° to the horizontal and is supported by a small smooth peg \(P\), such that the distance \(AP\) is 2.5 m (see diagram). Find
  1. the force acting on the rod at \(P\), [3]
  2. the coefficient of friction between the ground and the rod. [5]
OCR M2 2013 January Q6
10 marks Moderate -0.3
A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 N s directed up a line of greatest slope of the plane.
  1. Find the speed at which the particle starts to move. [2]
The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).
  1. Given that the plane is smooth, find the speed of the particle at \(Q\). [4]
It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed 3 ms\(^{-1}\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).
  1. Given that the plane is inclined at 30° to the horizontal, find the magnitude of the frictional force on the particle. [4]
OCR M2 2013 January Q7
11 marks Standard +0.3
A particle is projected with speed \(u\) ms\(^{-1}\) at an angle of \(\theta\) above the horizontal from a point \(O\). At time \(t\) s after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and \(\theta\) and hence obtain the equation of trajectory $$y = x \tan \theta - \frac{gx^2 \sec^2 \theta}{2u^2}.$$ [4]
In a shot put competition, a shot is thrown from a height of 2.1 m above horizontal ground. It has initial velocity of 14 ms\(^{-1}\) at an angle of \(\theta\) above the horizontal. The shot travels a horizontal distance of 22 m before hitting the ground.
  1. Show that \(12.1 \tan^2 \theta - 22 \tan \theta + 10 = 0\), and find the value of \(\theta\). [5]
  2. Find the time of flight of the shot. [2]
OCR M2 2013 January Q8
14 marks Challenging +1.2
\includegraphics{figure_8} A conical shell has radius 6 m and height 8 m. The shell, with its vertex \(V\) downwards, is rotating about its vertical axis. A particle, of mass 0.4 kg, is in contact with the rough inner surface of the shell. The particle is 4 m above the level of \(V\) (see diagram). The particle and shell rotate with the same constant angular speed. The coefficient of friction between the particle and the shell is \(\mu\).
  1. The frictional force on the particle is \(F\) N, and the normal force of the shell on the particle is \(R\) N. It is given that the speed of the particle is 4.5 ms\(^{-1}\), which is the smallest possible speed for the particle not to slip.
    1. By resolving vertically, show that \(4F + 3R = 19.6\). [2]
    2. By finding another equation connecting \(F\) and \(R\), find the values of \(F\) and \(R\) and show that \(\mu = 0.336\), correct to 3 significant figures. [6]
  2. Find the largest possible angular speed of the shell for which the particle does not slip. [6]
OCR M2 2010 June Q1
6 marks Moderate -0.8
A particle is projected horizontally with a speed of \(7 \text{ ms}^{-1}\) 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. [6]
OCR M2 2010 June Q2
7 marks Standard +0.3
  1. A uniform piece of wire, \(ABC\), forms a semicircular arc of radius 6 cm. \(O\) is the mid-point of \(AC\) (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]
  2. Two semicircular pieces of wire, \(ABC\) and \(ADC\), are joined together at their ends to form a circular hoop of radius 6 cm. The mass of \(ABC\) is 3 grams and the mass of \(ADC\) is 5 grams. The hoop is freely suspended from \(A\) (see Fig. 2). Calculate the angle which the diameter \(AC\) makes with the vertical, giving your answer correct to the nearest degree. [5]
OCR M2 2010 June Q3
9 marks Standard +0.8
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 \(80 \text{ ms}^{-1}\).
  1. Calculate the magnitude of the air resistance when the speed is \(60 \text{ ms}^{-1}\). [5]
The aeroplane is climbing at a constant angle of \(2°\) to the horizontal.
  1. Find the maximum acceleration at an instant when the speed of the aeroplane is \(60 \text{ ms}^{-1}\). [4]
OCR M2 2010 June Q4
10 marks Moderate -0.3
A non-uniform beam \(AB\) of length 4 m and mass 5 kg has its centre of mass at the point \(G\) of the beam where \(AG = 2.5\) 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 \(20°\) with the vertical and the tension is \(T\) N (see diagram).
  1. Calculate \(T\). [3]
  2. Calculate the magnitude and the direction of the force acting on the beam at \(A\). [7]
OCR M2 2010 June Q5
10 marks Standard +0.3
One end of a light inextensible string of length \(l\) is attached to the vertex of a smooth cone of semi-vertical angle \(45°\). 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 \(R\).
  1. By resolving horizontally and vertically, find two equations involving \(T\) and \(R\) and hence show that \(T = \frac{1}{2}ml(\sqrt{2}g + l\omega^2)\). [6]
  2. When the string has length 0.8 m, calculate the greatest value of \(\omega\) for which the particle remains in contact with the cone. [4]
OCR M2 2010 June Q6
17 marks Standard +0.3
A particle \(A\) of mass \(2m\) 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 \(3v\) and the particles move in the same direction.
  1. Find \(v\) in terms of \(u\). [3]
  2. Show that the coefficient of restitution between \(A\) and \(B\) is \(\frac{1}{3}\). [2]
\(B\) 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.
  1. Show that the speed of \(B\) after hitting the wall is \(\frac{3}{4}u\). [4]
  2. \(B\) then hits \(A\). Calculate the speeds of \(A\) and \(B\), in terms of \(u\), after this collision and state their directions of motion. [8]
OCR M2 2010 June Q7
13 marks Standard +0.8
A small ball of mass 0.2 kg is projected with speed \(11 \text{ ms}^{-1}\) 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 \(\frac{1}{4}\). The distance \(AB\) is 5 m and \(AB\) is inclined at \(30°\) to the horizontal (see diagram).
  1. Show that the speed of the ball when it reaches \(B\) is \(5.44 \text{ ms}^{-1}\), correct to 2 decimal places. [6]
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 \(BC\) is 5 m and \(BC\) is inclined at \(30°\) to the horizontal.
  1. Determine whether or not the ball hits the roof between \(B\) and \(C\). [7]
OCR M2 2016 June Q1
6 marks Moderate -0.3
A car of mass 1400 kg is travelling on a straight horizontal road against a constant resistance to motion of 600 N. At a certain instant the car is accelerating at \(0.3 \text{ m s}^{-2}\) and the engine of the car is working at a rate of 23 kW.
  1. Find the speed of the car at this instant. [3]
Subsequently the car moves up a hill inclined at \(10°\) to the horizontal at a steady speed of \(12 \text{ m s}^{-1}\). The resistance to motion is still a constant 600 N.
  1. Calculate the power of the car's engine as it moves up the hill. [3]
OCR M2 2016 June Q2
7 marks Standard +0.3
\(A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(55°\) to the horizontal. \(A\) is below the level of \(B\) and \(AB = 4\) m. A particle \(P\) of mass 2.5 kg is projected up the plane from \(A\) towards \(B\) and the speed of \(P\) at \(B\) is \(6.7 \text{ m s}^{-1}\). The coefficient of friction between the plane and \(P\) is 0.15. Find
  1. the work done against the frictional force as \(P\) moves from \(A\) to \(B\), [3]
  2. the initial speed of \(P\) at \(A\). [4]
OCR M2 2016 June Q3
12 marks Standard +0.3
\includegraphics{figure_1} A uniform lamina \(ABDC\) is bounded by two semicircular arcs \(AB\) and \(CD\), each with centre \(O\) and of radii \(3a\) and \(a\) respectively, and two straight edges, \(AC\) and \(DB\), which lie on the line \(AOB\) (see Fig. 1).
  1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac{13a}{3\pi}\). [5]
\includegraphics{figure_2} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(AB\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40°\) below the horizontal (see Fig. 2).
  1. Calculate the tension in the string. [3]
  2. Find the direction of the force acting on the lamina at \(A\). [4]
OCR M2 2016 June Q4
9 marks Standard +0.8
A smooth solid cone of semi-vertical angle \(60°\) is fixed to the ground with its axis vertical. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. \(P\) rotates in a horizontal circle on the surface of the cone with constant angular velocity \(\omega\). The string is inclined to the downward vertical at an angle of \(30°\) (see diagram).
  1. Show that the magnitude of the contact force between the cone and the particle is \(\frac{1}{4}m(2\sqrt{3}g - 3a\omega^2)\). [6]
  2. Given that \(a = 0.5\) m and \(m = 3.5\) kg, find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string. [3]
OCR M2 2016 June Q5
11 marks Standard +0.3
A uniform ladder \(AB\), of weight \(W\) and length \(2a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{12}{13}\). A man of weight \(6W\) is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.
  1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{5W}{24}\left(1 + \frac{6x}{a}\right)\). [5]
The coefficient of friction between the ladder and the ground is \(\frac{1}{3}\).
  1. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. [3]
The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight \(6W\) can now stand at the top of the ladder \(B\) without the ladder slipping.
  1. Find the least possible value of \(\tan \alpha\). [3]
OCR M2 2016 June Q6
10 marks Standard +0.8
The masses of two particles \(A\) and \(B\) are 4 kg and 3 kg respectively. The particles are moving towards each other along a straight line on a smooth horizontal surface. \(A\) has speed \(8 \text{ m s}^{-1}\) and \(B\) has speed \(10 \text{ m s}^{-1}\) before they collide. The kinetic energy lost due to the collision is 121.5 J.
  1. Find the speed and direction of motion of each particle after the collision. [8]
  2. Find the coefficient of restitution between \(A\) and \(B\). [2]
OCR M2 2016 June Q7
17 marks Challenging +1.8
A particle \(P\) is projected with speed \(32 \text{ m s}^{-1}\) at an angle of elevation \(\alpha\), where \(\sin \alpha = \frac{3}{4}\), from a point \(A\) on horizontal ground. At the same instant a particle \(Q\) is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation \(\beta\), where \(\sin \beta = \frac{24}{25}\), from a point \(B\) on the same horizontal ground. The particles move freely under gravity in the same vertical plane and collide with each other at the point \(C\) at the instant when they are travelling horizontally (see diagram).
  1. Calculate the height of \(C\) above the ground and the distance \(AB\). [4]
Immediately after the collision \(P\) falls vertically. \(P\) hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height 5 m above the ground.
  1. Given that the mass of \(P\) is 3 kg, find the magnitude and direction of the impulse exerted on \(P\) by the ground. [4]
The coefficient of restitution between the two particles is \(\frac{1}{2}\).
  1. Find the distance of \(Q\) from \(C\) at the instant when \(Q\) is travelling in a direction of \(25°\) below the horizontal. [9]
OCR MEI M2 2007 January Q1
17 marks Moderate -0.3
A sledge and a child sitting on it have a combined mass of 29.5 kg. The sledge slides on horizontal ice with negligible resistance to its movement.
  1. While at rest, the sledge is hit directly from behind by a ball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The coefficient of restitution in the collision is 0.8. After the impact the speeds of the sledge and the ball are \(V_1 \text{ m s}^{-1}\) and \(V_2 \text{ m s}^{-1}\) respectively. Calculate \(V_1\) and \(V_2\) and state the direction in which the ball is travelling after the impact. [7]
  2. While at rest, the sledge is hit directly from behind by a snowball of mass 0.5 kg travelling horizontally at \(10 \text{ m s}^{-1}\). The snowball sticks to the sledge.
    1. Calculate the velocity with which the combined sledge and snowball start to move. [3]
    2. The child scoops up the 0.5 kg of snow and drops it over the back of the sledge. What happens to the velocity of the sledge? Give a reason for your answer. [2]
  3. In another situation, the sledge is travelling over the ice at \(2 \text{ m s}^{-1}\) with 10.5 kg of snow on it (giving a total mass of 40 kg). The child throws a snowball of mass 0.5 kg from the sledge, parallel to the ground and in the positive direction of the motion of the sledge. Immediately after the snowball is thrown, the sledge has a speed of \(V \text{ m s}^{-1}\) and the snowball and sledge are separating at a speed of \(10 \text{ m s}^{-1}\). Draw a diagram showing the velocities of the sledge and snowball before and after the snowball is thrown. Calculate \(V\). [5]
OCR MEI M2 2007 January Q2
20 marks Standard +0.8
\includegraphics{figure_2} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods AB, BC, AD, BD, BE, CE and DE. [The triangles ABD, BDE and BCE are all equilateral.] The rods AB, BC and DE are horizontal. The rods are freely pin-jointed to each other at A, B, C, D and E. The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD. The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(LN\) at C; the normal reaction force \(RN\) of the plane on the framework at D; the horizontal and vertical forces \(XN\) and \(YN\), respectively, acting at A.
  1. Write down equations for the horizontal and vertical equilibrium of the framework. [3]
  2. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt{3}L\) and \(Y = 0\). [4]
  3. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods. [2]
  4. Show that the internal force in the rod AD is zero. [2]
  5. Find the forces internal to AB, CE and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.] [7]
  6. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust. [2]
OCR MEI M2 2007 January Q3
18 marks Standard +0.8
A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \includegraphics{figure_3.1}
  1. The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. [1]
The base OABC is added to the vertical faces.
  1. Write down the \(x\)- and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875. [5]
The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.
  1. Show that the coordinates of the centre of mass of the box in this situation are \((10, 2.4, 2.1)\). [6]
[This question is continued on the facing page.] The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30°\) to the horizontal, as shown in Fig. 3.2. \includegraphics{figure_3.2} The weight of the box is 40 N. A force \(P\) N acts parallel to the plane and is applied to the mid-point of FG at \(90°\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO.
  1. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm. [4]
  2. Calculate the value of \(P\). [2]
OCR MEI M2 2007 January Q4
17 marks Standard +0.3
Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}
  1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
  2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
    1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
    2. Calculate the work done against friction per tile. [2]
    3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
  3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]
OCR MEI M2 2008 January Q1
19 marks Moderate -0.3
  1. A battering-ram consists of a wooden beam fixed to a trolley. The battering-ram runs along horizontal ground and collides directly with a vertical wall, as shown in Fig. 1.1. The battering-ram has a mass of 4000 kg. \includegraphics{figure_1} Initially the battering-ram is at rest. Some men push it for 8 seconds and let go just as it is about to hit the wall. While the battering-ram is being pushed, the constant overall force on it in the direction of its motion is 1500 N.
    1. At what speed does the battering-ram hit the wall? [3]
    The battering-ram hits a loose stone block of mass 500 kg in the wall. Linear momentum is conserved and the coefficient of restitution in the impact is 0.2.
    1. Calculate the speeds of the stone block and of the battering-ram immediately after the impact. [6]
    2. Calculate the energy lost in the impact. [3]
  2. Small objects A and B are sliding on smooth, horizontal ice. Object A has mass 4 kg and speed 18 m s\(^{-1}\) in the \(\mathbf{i}\) direction. B has mass 8 kg and speed 9 m s\(^{-1}\) in the direction shown in Fig. 1.2, where \(\mathbf{i}\) and \(\mathbf{j}\) are the standard unit vectors. \includegraphics{figure_2}
    1. Write down the linear momentum of A and show that the linear momentum of B is \((36\mathbf{i} + 36\sqrt{3}\mathbf{j})\) N s. [2]
    After the objects meet they stick together (coalesce) and move with a common velocity of \((u\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\).
    1. Calculate \(u\) and \(v\). [3]
    2. Find the angle between the direction of motion of the combined object and the \(\mathbf{i}\) direction. Make your method clear. [2]
OCR MEI M2 2008 January Q2
17 marks Moderate -0.3
A cyclist and her bicycle have a combined mass of 80 kg.
  1. Initially, the cyclist accelerates from rest to 3 m s\(^{-1}\) against negligible resistances along a horizontal road.
    1. How much energy is gained by the cyclist and bicycle? [2]
    2. The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle? [2]
  2. While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from 4 m s\(^{-1}\) to 10 m s\(^{-1}\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h\) m. Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\). [5]
  3. The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N.
    1. When the power of the driving force on the bicycle is a constant 200 W, what constant speed can the cyclist maintain? [3]
    2. Find the power of the driving force on the bicycle when travelling at a speed of 0.5 m s\(^{-1}\) with an acceleration of 2 m s\(^{-2}\). [5]