Questions — OCR M2 (155 questions)

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OCR M2 Q1
5 marks Standard +0.3
A uniform solid cone has vertical height 20 cm and base radius \(r\) cm. It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of inclination is \(24°\) (see diagram). \includegraphics{figure_1}
  1. Find \(r\), correct to 1 decimal place. [4]
A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is inclined at an angle of \(24°\).
  1. State, with justification, whether this cone will topple. [1]
OCR M2 Q2
6 marks Moderate -0.5
A particle is projected horizontally with a speed of 6 m 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. [6]
OCR M2 Q3
8 marks Standard +0.3
\includegraphics{figure_3} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(PQ = 0.8\) m. A small smooth bead \(B\), of mass 0.01 kg, is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius 0.6 m. \(QB\) rotates with constant angular speed \(\omega\) rad s\(^{-1}\) (see diagram).
  1. Show that the tension in the string is 0.1225 N. [3]
  2. Find \(\omega\). [3]
  3. Calculate the kinetic energy of the bead. [2]
OCR M2 Q4
9 marks Standard +0.3
\includegraphics{figure_4} Three smooth spheres \(A\), \(B\) and \(C\), of equal radius and of masses \(m\) kg, \(2m\) kg and \(3m\) kg respectively, lie in a straight line and are free to move on a smooth horizontal table. Sphere \(A\) is moving with speed 5 m s\(^{-1}\) when it collides directly with sphere \(B\) which is stationary. As a result of the collision \(B\) starts to move with speed 2 m s\(^{-1}\).
  1. Find the coefficient of restitution between \(A\) and \(B\). [4]
  2. Find, in terms of \(m\), the magnitude of the impulse that \(A\) exerts on \(B\), and state the direction of this impulse. [2]
Sphere \(B\) subsequently collides with sphere \(C\) which is stationary. As a result of this impact \(B\) and \(C\) coalesce.
  1. Show that there will be another collision. [3]
OCR M2 Q5
10 marks Standard +0.3
\includegraphics{figure_5} A uniform rod \(AB\) 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. [4]
  2. Find the magnitude and direction of the force acting on the rod at \(A\). [6]
OCR M2 Q6
10 marks Standard +0.3
A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5°\) to the horizontal. At a certain point \(P\) on the hill the car's speed is 20 m s\(^{-1}\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is 15 m s\(^{-1}\).
  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. [4]
Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
  1. 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]
  2. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\). [3]
OCR M2 Q7
11 marks Standard +0.8
\includegraphics{figure_7} A barrier is modelled as a uniform rectangular plank of wood, \(ABCD\), rigidly joined to a uniform square metal plate, \(DEFG\). 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 \(CDE\) 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 \(CH\) is 0.25 m.
  1. Calculate the contact force at \(H\) when the barrier is in the closed position. [3]
In the open position, the centre of mass of the barrier is vertically above \(D\).
  1. Calculate the angle between \(AB\) and the horizontal when the barrier is in the open position. [8]
OCR M2 Q8
13 marks Standard +0.3
A particle is projected with speed 49 m 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 \(x\) m and \(y\) m respectively.
  1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac{x^2(1 + \tan^2 \theta)}{490}.$$ [4]
\includegraphics{figure_8} 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).
  1. Find \(\theta_1\) and \(\theta_2\). [4]
  2. Calculate the distance between \(A_1\) and \(A_2\). [5]
OCR M2 2013 January Q1
5 marks Easy -1.2
A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at 15° below the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find
  1. the work done by the force, [3]
  2. the power with which the force is working. [2]
OCR M2 2013 January Q2
7 marks Standard +0.3
A car of mass 1500 kg travels along a straight horizontal road. The resistance to the motion of the car is \(kv^{\frac{3}{2}}\) N, where \(v\) ms\(^{-1}\) is the speed of the car and \(k\) is a constant. At the instant when the engine produces a power of 15000 W, the car has speed 15 ms\(^{-1}\) and is accelerating at 0.4 ms\(^{-2}\).
  1. Find the value of \(k\). [4]
It is given that the greatest steady speed of the car on this road is 30 ms\(^{-1}\).
  1. Find the greatest power that the engine can produce. [3]
OCR M2 2013 January Q3
9 marks Standard +0.3
A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of 30° with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is 2 ms\(^{-1}\). \(B\) has velocity 1 ms\(^{-1}\) down the plane immediately after the collision. Find
  1. the speed of \(A\) immediately after the collision, [4]
  2. the distance \(A\) moves up the plane after the collision. [2]
The masses of \(A\) and \(B\) are 0.5 kg and \(m\) kg, respectively.
  1. Find the value of \(m\). [3]
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]