Questions — OCR M3 (138 questions)

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OCR M3 2010 January Q6
13 marks Challenging +1.2
  1. By considering the total energy of the system, obtain an expression for \(v ^ { 2 }\) in terms of \(\theta\).
  2. Show that the magnitude of the force exerted on \(P\) by the cylinder is \(( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }\).
  3. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(1.53 < \alpha < 1.54\).
OCR M3 2007 June Q6
13 marks Challenging +1.8
  1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
  2. Verify that \(P\) and \(Q\) together just reach the safety net.
  3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
  4. State two additional modelling assumptions made when answering this question.
OCR M3 2006 January Q10
Standard +0.8
10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes
  • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  • Answer all the questions.
  • Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
  • The acceleration due to gravity is denoted by \(\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).
  • You are permitted to use a graphical calculator in this paper.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total number of marks for this paper is 72.
  • Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
  • You are reminded of the need for clear presentation in your answers.
OCR M3 2009 June Q1
6 marks Moderate -0.3
A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \text{ m s}^{-1}\) and \(6 \text{ m s}^{-1}\) respectively. The speed of the sphere immediately after it bounces is \(5 \text{ m s}^{-1}\).
  1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \text{ m s}^{-1}\), and hence find the coefficient of restitution between the surface and the sphere. [3]
  2. State the direction of the impulse on the sphere and find its magnitude. [3]
OCR M3 2009 June Q2
8 marks Standard +0.3
\includegraphics{figure_2} Two uniform rods, \(AB\) and \(BC\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is \(1.5\) m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is \(3\) m (see diagram). The weight of \(AB\) is \(80\) N and the frictional force acting on \(AB\) at \(A\) is \(14\) N.
  1. Write down the horizontal component of the force acting on \(AB\) at \(B\) and show that the vertical component of this force is \(33\) N upwards. [4]
  2. Given that the force acting on \(BC\) at \(C\) has magnitude \(50\) N, find the weight of \(BC\). [4]
OCR M3 2009 June Q3
10 marks Standard +0.8
\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(4\) kg and \(2\) kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \text{ m s}^{-1}\). The spheres are moving in opposite directions, each at \(60°\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.
  1. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres. [8]
  2. Find the coefficient of restitution between the spheres. [2]
OCR M3 2009 June Q4
11 marks Challenging +1.2
A motor-cycle, whose mass including the rider is \(120\) kg, is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \text{ m s}^{-1}\) and when it has travelled a distance of \(x\) m beyond \(A\) its speed is \(v \text{ m s}^{-1}\). The engine develops a constant power of \(8\) kW and resistances are modelled by a force of \(0.25v^2\) N opposing the motion.
  1. Show that \(\frac{480v^2}{v^3 - 32000} \frac{dv}{dx} = -1\). [5]
  2. Find the speed of the motor-cycle when it has travelled \(500\) m beyond \(A\). [6]
OCR M3 2009 June Q5
11 marks Challenging +1.2
\includegraphics{figure_5} Each of two identical strings has natural length \(1.5\) m and modulus of elasticity \(18\) N. One end of one of the strings is attached to \(A\) and one end of the other string is attached to \(B\), where \(A\) and \(B\) are fixed points which are \(3\) m apart and at the same horizontal level. \(M\) is the mid-point of \(AB\). A particle \(P\) of mass \(m\) kg is attached to the other end of each of the strings. \(P\) is held at rest at the point \(0.8\) m vertically above \(M\), and then released. The lowest point reached by \(P\) in the subsequent motion is \(2\) m below \(M\) (see diagram).
  1. Find the maximum tension in each of the strings during \(P\)'s motion. [3]
  2. By considering energy,
    1. show that the value of \(m\) is \(0.42\), correct to 2 significant figures, [5]
    2. find the speed of \(P\) at \(M\). [3]
OCR M3 2009 June Q6
13 marks Standard +0.3
\includegraphics{figure_6} A particle \(P\) of mass \(m\) kg is attached to one end of a light inextensible string of length \(L\) m. The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(OP\), at time \(t\) s, is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is \(0.05\).
  1. Show that \(\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin \theta\). [2]
  2. Hence show that the motion of \(P\) is approximately simple harmonic. [2]
  3. Given that the period of the approximate simple harmonic motion is \(\frac{4}{3}\pi\) s, find the value of \(L\). [2]
  4. Find the value of \(\theta\) when \(t = 0.7\) s, and the value of \(t\) when \(\theta\) next takes this value. [4]
  5. Find the speed of \(P\) when \(t = 0.7\) s. [3]
OCR M3 2009 June Q7
13 marks Standard +0.3
\includegraphics{figure_7} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(OP\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).
  1. Find \(v^2\) in terms of \(u\), \(a\), \(g\) and \(\theta\) and show that \(R = \frac{mu^2}{a} + mg(3\cos\theta - 2)\). [7]
  2. Given that \(P\) just reaches the highest point of the circle, find \(u^2\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v^2\) is \(ag\). [4]
  3. Given instead that \(P\) oscillates between \(\theta = \pm\frac{1}{5}\pi\) radians, find \(u^2\) in terms of \(a\) and \(g\). [2]
OCR M3 2010 June Q1
6 marks Moderate -0.3
A small ball of mass \(0.8\) kg is moving with speed \(10.5\) m s\(^{-1}\) when it receives an impulse of magnitude \(4\) N s. The speed of the ball immediately afterwards is \(8.5\) m s\(^{-1}\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\). [6]
OCR M3 2010 June Q2
7 marks Standard +0.3
\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\) are of equal length and each has weight \(100\) N. The rods are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(AB\) horizontal and \(C\) resting on a rough horizontal surface. \(C\) is vertically below the mid-point of \(AB\) (see diagram).
  1. By taking moments about \(A\) for \(AB\), find the vertical component of the force on \(AB\) at \(B\). Hence find the vertical component of the contact force on \(BC\) at \(C\). [3]
  2. Calculate the magnitude of the frictional force on \(BC\) at \(C\) and state its direction. [4]
OCR M3 2010 June Q3
8 marks Standard +0.3
A uniform smooth sphere \(A\) moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude \(4\) m s\(^{-1}\). Immediately after hitting the wall the components have magnitudes \(u\) m s\(^{-1}\) and \(v\) m s\(^{-1}\), respectively (see Fig. 1). \includegraphics{figure_1}
  1. Given that the coefficient of restitution between the sphere and the wall is \(\frac{1}{4}\), state the values of \(u\) and \(v\). [2]
Shortly after hitting the wall the sphere \(A\) comes into contact with another uniform smooth sphere \(B\), which has the same mass and radius as \(A\). The sphere \(B\) is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \includegraphics{figure_2}
  1. Find, for each sphere, its speed and its direction of motion immediately after the contact. [6]
OCR M3 2010 June Q4
11 marks Standard +0.3
\(O\) is a fixed point on a horizontal plane. A particle \(P\) of mass \(0.25\) kg is released from rest at \(O\) and moves in a straight line on the plane. At time \(t\) s after release the only horizontal force acting on \(P\) has magnitude $$\frac{1}{2400}(144 - t^2) \text{ N} \quad \text{for } 0 \leqslant t \leqslant 12$$ and $$\frac{1}{2400}(t^2 - 144) \text{ N} \quad \text{for } t \geqslant 12.$$ The force acts in the direction of \(P\)'s motion. \(P\)'s velocity at time \(t\) s is \(v\) m s\(^{-1}\).
  1. Find an expression for \(v\) in terms of \(t\), valid for \(t \geqslant 12\), and hence show that \(v\) is three times greater when \(t = 24\) than it is when \(t = 12\). [8]
  2. Sketch the \((t, v)\) graph for \(0 \leqslant t \leqslant 24\). [3]
OCR M3 2010 June Q5
11 marks Standard +0.8
\includegraphics{figure_5} Particles \(P_1\) and \(P_2\) are each moving with simple harmonic motion along the same straight line. \(P_1\)'s motion has centre \(C_1\), period \(2\pi\) s and amplitude \(3\) m; \(P_2\)'s motion has centre \(C_2\), period \(\frac{4}{3}\pi\) s and amplitude \(4\) m. The points \(C_1\) and \(C_2\) are \(6.5\) m apart. The displacements of \(P_1\) and \(P_2\) from their centres of oscillation at time \(t\) s are denoted by \(x_1\) m and \(x_2\) m respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x_1 = 3\) and \(x_2 = 4\).
  1. State expressions for \(x_1\) and \(x_2\) in terms of \(t\), which are valid until the particles collide. [3]
The particles collide when \(t = 5.99\), correct to \(3\) significant figures.
  1. Find the distance travelled by \(P_2\) before the collision takes place. [4]
  2. Find the velocities of \(P_1\) and \(P_2\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions. [4]
OCR M3 2010 June Q6
12 marks Standard +0.8
A bungee jumper of weight \(W\) N is joined to a fixed point \(O\) by a light elastic rope of natural length \(20\) m and modulus of elasticity \(32\,000\) N. The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.
  1. Given that the jumper just reaches a point \(25\) m below \(O\), find the value of \(W\). [5]
  2. Find the maximum speed reached by the jumper. [4]
  3. Find the maximum value of the deceleration of the jumper during the downward motion. [3]
OCR M3 2010 June Q7
17 marks Challenging +1.2
\includegraphics{figure_7} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length \(0.7\) m. A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(OP\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6\) m s\(^{-1}\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9\) m s\(^{-1}\).
  1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\). [3]
  2. Given that the speed of \(Q\) is \(v\) m s\(^{-1}\) when \(OQ\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v^2\) in terms of \(\theta\), and show that the tension in the string \(OQ\) is \(14.7m(1 + 2\cos\theta)\) N, where \(m\) kg is the mass of \(Q\). [6]
  3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(OQ\) becomes slack. [4]
  4. Show that \(V^2 = 0.8575\), where \(V\) m s\(^{-1}\) is the speed of \(Q\) when it reaches its greatest height (after the string \(OQ\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position. [4]
OCR M3 2011 June Q1
4 marks Moderate -0.3
\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving in a straight line with speed \(4\) m s\(^{-1}\) when it is deflected through an angle \(\theta\) by an impulse of magnitude \(I\) N s. The impulse acts at right angles to the initial direction of motion of \(P\) (see diagram). The speed of \(P\) immediately after the impulse acts is \(5\) m s\(^{-1}\). Show that \(\cos \theta = 0.8\) and find the value of \(I\). [4]
OCR M3 2011 June Q2
10 marks Standard +0.8
\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), of lengths \(3\) m and \(4\) m respectively, have weights \(300\) N and \(400\) N respectively. The rods are freely jointed at \(A\). The mid-points of the rods are joined by a light inextensible string. The rods are in equilibrium in a vertical plane with the string taut and \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is \(2.4\) m above the surface (see diagram).
  1. Show that the force exerted by the surface on \(AB\) is \(374\) N and find the force exerted by the surface on \(AC\). [4]
  2. Find the tension in the string. [3]
  3. Find the horizontal and vertical components of the force exerted on \(AB\) at \(A\) and state their directions. [3]
OCR M3 2011 June Q3
10 marks Standard +0.8
A particle \(P\) of mass \(0.25\) kg is projected horizontally with speed \(5\) m s\(^{-1}\) from a fixed point \(O\) on a smooth horizontal surface and moves in a straight line on the surface. The only horizontal force acting on \(P\) has magnitude \(0.2v^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s after it is projected from \(O\). This force is directed towards \(O\).
  1. Find an expression for \(v\) in terms of \(t\). [5]
The particle \(P\) passes through a point \(X\) with speed \(0.2\) m s\(^{-1}\).
  1. Find the average speed of \(P\) for its motion between \(O\) and \(X\). [5]
OCR M3 2011 June Q4
11 marks Standard +0.3
One end of a light inextensible string of length \(2\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.2\) kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(OP\) makes an angle of \(0.15\) radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(OP\) makes an angle of \(\theta\) radians with the downward vertical.
  1. Show that \(\frac{d^2\theta}{dt^2} = -4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. [3]
Using the simple harmonic approximation,
  1. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = -0.1\), [5]
  2. find the angular speed of \(OP\) and the linear speed of \(P\) when \(t = 0.5\). [3]
OCR M3 2011 June Q5
12 marks Standard +0.8
\includegraphics{figure_5} Two uniform smooth identical spheres \(A\) and \(B\) are moving towards each other on a horizontal surface when they collide. Immediately before the collision \(A\) and \(B\) are moving with speeds \(u_A\) m s\(^{-1}\) and \(u_B\) m s\(^{-1}\) respectively, at acute angles \(\alpha\) and \(\beta\), respectively, to the line of centres. Immediately after the collision \(A\) and \(B\) are moving with speeds \(v_A\) m s\(^{-1}\) and \(v_B\) m s\(^{-1}\) respectively, at right angles and at acute angle \(\gamma\), respectively, to the line of centres (see diagram).
  1. Given that \(\sin \beta = 0.96\) and \(\frac{v_B}{u_B} = 1.2\), find the value of \(\sin \gamma\). [2]
  2. Given also that, before the collision, the component of \(A\)'s velocity parallel to the line of centres is \(2\) m s\(^{-1}\), find the values of \(u_B\) and \(v_B\). [5]
  3. Find the coefficient of restitution between the spheres. [3]
  4. Given that the kinetic energy of \(A\) immediately before the collision is \(6.5m\) J, where \(m\) kg is the mass of \(A\), find the value of \(v_A\). [2]
OCR M3 2011 June Q6
11 marks Challenging +1.8
\includegraphics{figure_6} A particle \(P\) of weight \(6\) N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius \(0.8\) m. The string has natural length \(\frac{1}{10}\pi\) m and modulus of elasticity \(9\) N. \(P\) is released from rest at a point \(X\) on the sphere where \(OX\) makes an angle of \(\frac{1}{3}\pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(OP\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac{1}{4}\pi\), \(P\) passes through the point \(Y\).
  1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2.4(\sqrt{3} - \sqrt{2})\) J and the elastic potential energy in the string decreases by \(0.4\pi\) J. [5]
  2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac{1}{4}\pi\), and hence find the maximum speed of \(P\). [6]
OCR M3 2011 June Q7
14 marks Standard +0.3
One end of a light inextensible string of length \(0.8\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.5\) kg is attached to the other end of the string. \(P\) is projected horizontally from the point \(0.8\) m vertically below \(O\) with speed \(5.6\) m s\(^{-1}\). \(P\) starts to move in a vertical circle with centre \(O\). The speed of \(P\) is \(v\) m s\(^{-1}\) when the string makes an angle \(\theta\) with the downward vertical.
  1. While the string remains taut, show that \(v^2 = 15.68(1 + \cos \theta)\), and find the tension in the string in terms of \(\theta\). [7]
  2. For the instant when the string becomes slack, find the value of \(\theta\) and the value of \(v\). [3]
  3. Find, in either order, the speed of \(P\) when it is at its greatest height after the string becomes slack, and the greatest height reached by \(P\) above its point of projection. [4]
OCR M3 2015 June Q1
6 marks Moderate -0.3
A particle \(P\) of mass \(0.2\) kg is moving on a smooth horizontal surface with speed \(3\text{ ms}^{-1}\), when it is struck by an impulse of magnitude \(I\) Ns. The impulse acts horizontally in a direction perpendicular to the original direction of motion of \(P\), and causes the direction of motion of \(P\) to change by an angle \(\alpha\), where \(\tan \alpha = \frac{5}{12}\).
  1. Show that \(I = 0.25\). [4]
  2. Find the speed of \(P\) after the impulse acts. [2]