Questions — OCR MEI (4455 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR MEI Further Mechanics Major 2024 June Q8
10 marks Standard +0.8
A particle P of mass \(3m\) kg is attached to one end of a light elastic string of modulus of elasticity \(4mg\) N and natural length 0.4 m. The other end of the string is attached to a fixed point O. The particle P rests in equilibrium at a point A with the string vertical.
  1. Find the distance OA. [2]
At time \(t = 0\) seconds, P is given a speed of \(2.5 \text{ m s}^{-1}\) vertically downwards from A.
  1. Show that P initially performs simple harmonic motion with amplitude \(a\) m, where \(a\) is to be determined correct to 3 significant figures. [5]
  2. Determine the smallest distance between P and O in the subsequent motion. [3]
OCR MEI Further Mechanics Major 2024 June Q9
11 marks Standard +0.8
A particle P of mass 5 kg is released from rest at a point O and falls vertically. A resistance of magnitude \(0.05v^2\) N acts vertically upwards on P, where \(v \text{ m s}^{-1}\) is the velocity of P when it has fallen a distance \(x\) m.
  1. Show that \(\left(\frac{100v}{980-v^2}\right)\frac{dv}{dx} = 1\). [2]
  2. Verify that \(v^2 = 980(1-e^{-0.02x})\). [4]
  3. Determine the work done against the resistance while P is falling from O to the point where P's acceleration is \(8.36 \text{ m s}^{-2}\). [5]
OCR MEI Further Mechanics Major 2024 June Q10
10 marks Challenging +1.2
A particle P of mass 2 kg is projected vertically upwards from horizontal ground with an initial speed of \(14 \text{ m s}^{-1}\). At the same instant a particle Q of mass 8 kg is released from rest 5 m vertically above P. During the subsequent motion P and Q collide. The coefficient of restitution between P and Q is \(\frac{11}{14}\). Determine the time between this collision and P subsequently hitting the ground. [10]
OCR MEI Further Mechanics Major 2024 June Q11
16 marks Challenging +1.2
A particle P of mass 1 kg is fixed to one end of a light inextensible string of length 0.5 m. The other end of the string is attached to a fixed point O, which is 1.75 m above a horizontal plane. P is held with the string horizontal and taut. P is then projected vertically downwards with a speed of \(3.2 \text{ m s}^{-1}\).
  1. Find the tangential acceleration of P when OP makes an angle of \(20°\) with the horizontal. [2]
The string breaks when the tension in it is 32 N. At this point the angle between OP and the horizontal is \(\theta\).
  1. Show that \(\theta = 23.1°\), correct to 1 decimal place. [5]
Particle P subsequently hits the plane at a point A.
  1. Determine the speed of P when it arrives at A. [4]
  2. Show that A is almost vertically below O. [5]
OCR MEI Further Mechanics Major 2024 June Q12
15 marks Challenging +1.2
Two small uniform discs A and B, of equal radius, have masses 3 kg and 5 kg respectively. The discs are sliding on a smooth horizontal surface and collide obliquely. The contact between the discs is smooth and A is stationary after the collision. Immediately before the collision B is moving with speed \(2 \text{ m s}^{-1}\) in a direction making an angle of \(60°\) with the line of centres, XY (see diagram below). \includegraphics{figure_12}
  1. Explain how you can tell that A must have been moving along XY before the collision. [1]
The coefficient of restitution between A and B is 0.8.
  1. • Determine the speed of A immediately before the collision. • Determine the speed of B immediately after the collision. [7]
  2. Determine the angle turned through by the direction of B in the collision. [3]
Disc B subsequently collides with a smooth wall, which is parallel to XY. The kinetic energy of B after the collision with the wall is 95% of the kinetic energy of B before the collision with the wall.
  1. Determine the coefficient of restitution between B and the wall. [4]
OCR MEI Further Mechanics Major 2024 June Q13
10 marks Challenging +1.8
\includegraphics{figure_13} A conical shell, of semi-vertical angle \(\alpha\), is fixed with its axis vertical and its vertex V upwards. A light inextensible string passes through a small smooth hole at V and a particle P of mass 4 kg hangs in equilibrium at one end of the string. The other end of the string is attached to a particle Q of mass 25 kg which moves in a horizontal circle at constant angular speed \(2.8 \text{ rad s}^{-1}\) on the smooth outer surface of the shell at a vertical depth \(h\) m below V (see diagram).
  1. Show that \(k_1 h \sin^2 \alpha + k_2 \cos^2 \alpha = k_3 \cos \alpha\), where \(k_1\), \(k_2\) and \(k_3\) are integers to be determined. [7]
  2. Determine the greatest value of \(h\) for which Q remains in contact with the shell. [3]
OCR MEI Further Mechanics Major 2020 November Q1
5 marks Standard +0.3
A particle P of mass \(0.5\) kg is attached to a fixed point O by a light elastic string of natural length \(3\) m and modulus of elasticity \(75\) N. P is released from rest at O and is allowed to fall freely. Determine the length of the string when P is at its lowest point in the subsequent motion. [5]
OCR MEI Further Mechanics Major 2020 November Q2
5 marks Standard +0.3
A student conducts an experiment by first stretching a length of wire and fixing its ends. The student then plucks the wire causing it to vibrate. The frequency of these vibrations, \(f\), is modelled by the formula $$f = kC^\alpha l^\beta \sigma^\gamma,$$ where \(C\) is the tension in the wire, \(l\) is the length of the stretched wire, \(\sigma\) is the mass per unit length of the stretched wire and \(k\) is a dimensionless constant. Use dimensional analysis to find \(\alpha\), \(\beta\) and \(\gamma\). [5]
OCR MEI Further Mechanics Major 2020 November Q3
7 marks Standard +0.3
The vertices of a triangular lamina, which is in the \(x\)–\(y\) plane, are at the origin O and the points A\((2, 3)\) and B\((-2, 1)\). Forces \(2\mathbf{i} + \mathbf{j}\) and \(-3\mathbf{i} + 2\mathbf{j}\) are applied to the lamina at A and B, respectively, and a force \(\mathbf{F}\), whose line of action is in the \(x\)–\(y\) plane, is applied at O. The three forces form a couple.
  1. Determine the magnitude and the direction of \(\mathbf{F}\). [4]
  2. Determine the magnitude and direction of the additional couple that must be applied to the lamina in order to keep it in equilibrium. [3]
OCR MEI Further Mechanics Major 2020 November Q4
10 marks Moderate -0.3
A particle P moves so that its position vector \(\mathbf{r}\) at time \(t\) is given by $$\mathbf{r} = (5 + 20t)\mathbf{i} + (95 + 10t - 5t^2)\mathbf{j}.$$
  1. Determine the initial velocity of P. [3] At time \(t = T\), P is moving in a direction perpendicular to its initial direction of motion.
  2. Determine the value of \(T\). [3]
  3. Determine the distance of P from its initial position at time \(T\). [4]
OCR MEI Further Mechanics Major 2020 November Q5
8 marks Standard +0.3
A car of mass \(900\) kg moves along a straight level road. The power developed by the car is constant and equal to \(60\) kW. The resistance to the motion of the car is constant and equal to \(1500\) N. At time \(t\) seconds the velocity of the car is denoted by \(v\) m s\(^{-1}\). Initially the car is at rest.
  1. Show that \(\frac{3v\,dv}{5\,dt} = 40 - v\). [3]
  2. Verify that \(t = 24\ln\left(\frac{40}{40-v}\right) - \frac{3}{5}v\). [5]
OCR MEI Further Mechanics Major 2020 November Q6
10 marks Challenging +1.8
A small ball of mass \(m\) kg is held at a height of \(78.4\) m above horizontal ground. The ball is released from rest, falls vertically and rebounds from the ground. The coefficient of restitution between the ball and ground is \(e\). The ball continues to bounce until it comes to rest after \(6\) seconds.
  1. Determine the value of \(e\). [8]
  2. Given that the magnitude of the impulse that the ground exerts on the ball at the first bounce is \(23.52\) Ns, determine the value of \(m\). [2]
OCR MEI Further Mechanics Major 2020 November Q7
13 marks Challenging +1.2
\includegraphics{figure_7} A particle P of mass \(m\) is attached to one end of a light elastic string of natural length \(6a\) and modulus of elasticity \(3mg\). The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of \(30°\) to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A, as shown in Fig. 7. P is now pulled a further distance \(2a\) down the line of greatest slope through A and released from rest. At time \(t\) later, the displacement of P from A is \(x\), where the positive direction of \(x\) is down the plane.
  1. Show that, until the string slackens, \(x\) satisfies the differential equation $$\frac{d^2x}{dt^2} + \frac{gx}{2a} = 0.$$ [6]
  2. Determine, in terms of \(a\) and \(g\), the time at which the string slackens. [5]
  3. Find, in terms of \(a\) and \(g\), the speed of P when the string slackens. [2]
OCR MEI Further Mechanics Major 2020 November Q8
13 marks Standard +0.8
[In this question, you may use the fact that the volume of a right circular cone of base radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\).]
  1. By using integration, show that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac{3}{4}h\) from the vertex. [5]
\includegraphics{figure_8} Fig. 8 shows the side view of a toy formed by joining a uniform solid circular cylinder of radius \(r\) and height \(2r\) to a uniform solid right circular cone, made of the same material as the cylinder, of radius \(r\) and height \(r\). The toy is placed on a horizontal floor with the curved surface of the cone in contact with the floor.
  1. Determine whether the toy will topple. [7]
  2. Explain why it is not necessary to know whether the floor is rough or smooth in answering part (b). [1]
OCR MEI Further Mechanics Major 2020 November Q9
10 marks Challenging +1.2
\includegraphics{figure_9} Fig. 9 shows a uniform rod AB of length \(2a\) and weight \(8W\) which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight \(W\) is threaded on the bar and is connected to the rod at B by a light inextensible string of length \(2a\). The system is in equilibrium with the rod inclined at an angle \(\theta\) to the horizontal.
  1. Determine, in terms of \(W\) and \(\theta\), the tension in the string. [4] It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is \(2.4a\).
  2. Determine the coefficient of friction between the bar and the ring. [6]
OCR MEI Further Mechanics Major 2020 November Q10
14 marks Challenging +1.8
\includegraphics{figure_10} Fig. 10 shows a small bead P of mass \(m\) which is threaded on a smooth thin wire. The wire is in the form of a circle of radius \(a\) and centre O. The wire is fixed in a vertical plane. The bead is initially at the lowest point A of the wire and is projected along the wire with a velocity which is just sufficient to carry it to the highest point on the wire. The angle between OP and the downward vertical is denoted by \(\theta\).
  1. Determine the value of \(\theta\) when the magnitude of the reaction of the wire on the bead is \(\frac{7}{5}mg\). [7]
  2. Show that the angular velocity of P when OP makes an angle \(\theta\) with the downward vertical is given by \(k\sqrt{\frac{g}{a}\cos\left(\frac{\theta}{2}\right)}\), stating the value of the constant \(k\). [4]
  3. Hence determine, in terms of \(g\) and \(a\), the angular acceleration of P when \(\theta\) takes the value found in part (a). [3]
OCR MEI Further Mechanics Major 2020 November Q11
13 marks Challenging +1.2
Two uniform small smooth spheres A and B have equal radii and equal masses. The spheres are on a smooth horizontal surface. Sphere A is moving at an acute angle \(\alpha\) to the line of centres, when it collides with B, which is stationary. After the impact A is moving at an acute angle \(\beta\) to the line of centres. The coefficient of restitution between A and B is \(\frac{1}{3}\).
  1. Show that \(\tan\beta = 3\tan\alpha\). [5]
  2. Explain why the assumption that the contact between the spheres is smooth is needed in answering part (a). [1] It is given that A is deflected through an angle \(\gamma\).
  3. Determine, in terms of \(\alpha\), an expression for \(\tan\gamma\). [2]
  4. Determine the maximum value of \(\gamma\). You do not need to justify that this value is a maximum. [5]
OCR MEI Further Mechanics Major 2020 November Q12
12 marks Challenging +1.2
\includegraphics{figure_12} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P, of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.
  1. Show that the normal contact force between P and the bowl is of magnitude \(mg + 2mr\omega^2\cos^2\alpha\). [9]
  2. Deduce that \(g < r\omega^2(k_1 + k_2\cos^2\alpha)\), stating the value of the constants \(k_1\) and \(k_2\). [3]
OCR MEI Further Mechanics Major Specimen Q1
4 marks Moderate -0.3
A particle P has position vector \(\mathbf{r}\) m at time \(t\) s given by \(\mathbf{r} = (t^3 - 3t^2)\mathbf{i} - (4t^2 + 1)\mathbf{j}\) for \(t \geq 0\). Find the magnitude of the acceleration of P when \(t = 2\). [4]
OCR MEI Further Mechanics Major Specimen Q2
3 marks Moderate -0.8
A particle of mass 5 kg is moving with velocity \(2\mathbf{i} + 5\mathbf{j}\) m s\(^{-1}\). It receives an impulse of magnitude 15 N s in the direction \(\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\). Find the velocity of the particle immediately afterwards. [3]
OCR MEI Further Mechanics Major Specimen Q3
5 marks Standard +0.3
The fixed points E and F are on the same horizontal level with EF = 1.6 m. A light string has natural length 0.7 m and modulus of elasticity 29.4 N. One end of the string is attached to E and the other end is attached to a particle of mass \(M\) kg. A second string, identical to the first, has one end attached to F and the other end attached to the particle. The system is in equilibrium in a vertical plane with each string stretched to a length of 1 m, as shown in Fig. 3. \includegraphics{figure_3}
  1. Find the tension in each string. [2]
  2. Find \(M\). [3]
OCR MEI Further Mechanics Major Specimen Q4
6 marks Standard +0.3
A fixed smooth sphere has centre O and radius \(a\). A particle P of mass \(m\) is placed at the highest point of the sphere and given an initial horizontal speed \(u\). For the first part of its motion, P remains in contact with the sphere and has speed \(v\) when OP makes an angle \(\theta\) with the upward vertical. This is shown in Fig. 4. \includegraphics{figure_4}
  1. By considering the energy of P, show that \(v^2 = u^2 + 2ga(1 - \cos\theta)\). [2]
  2. Show that the magnitude of the normal contact force between the sphere and particle P is $$mg(3\cos\theta - 2) - \frac{mv^2}{a}.$$ [2]
The particle loses contact with the sphere when \(\cos\theta = \frac{3}{4}\).
  1. Find an expression for \(u\) in terms of \(a\) and \(g\). [2]
OCR MEI Further Mechanics Major Specimen Q5
8 marks Standard +0.8
Fig. 5 shows a light inextensible string of length 3.3 m passing through a small smooth ring R. The ends of the string are attached to fixed points A and B, where A is vertically above B. The ring R has mass 0.27 kg and is moving with constant speed in a horizontal circle of radius 1.2 m. The distances AR and BR are 2 m and 1.3 m respectively. \includegraphics{figure_5}
  1. Show that the tension in the string is 6.37 N. [4]
  2. Find the speed of R. [4]
OCR MEI Further Mechanics Major Specimen Q6
10 marks Standard +0.8
Fig. 6 shows a pendulum which consists of a rod AB freely hinged at the end A with a weight at the end B. The pendulum is oscillating in a vertical plane. The total energy, \(E\), of the pendulum is given by $$E = \frac{1}{2}I\omega^2 - mgh\cos\theta,$$ where
  • \(\omega\) is its angular speed
  • \(m\) is its mass
  • \(h\) is the distance of its centre of mass from A
  • \(\theta\) is the angle the rod makes with the downward vertical
  • \(g\) is the acceleration due to gravity
  • \(I\) is a quantity known as the moment of inertia of the pendulum.
\includegraphics{figure_6}
  1. Use the expression for \(E\) to deduce the dimensions of \(I\). [4]
It is suggested that the period of oscillation, \(T\), of the pendulum is given by \(T = kI^\alpha(mg)^\beta h^\gamma\), where \(k\) is a dimensionless constant.
  1. Use dimensional analysis to find the values of \(\alpha\), \(\beta\) and \(\gamma\). [5]
A class experiment finds that, when all other quantities are fixed, \(T\) is proportional to \(\frac{1}{\sqrt{m}}\).
  1. Determine whether this result is consistent with your answer to part (ii). [1]
OCR MEI Further Mechanics Major Specimen Q7
9 marks Standard +0.3
A uniform ladder of length 8 m and weight 180 N stands on a rough horizontal surface and rests against a smooth vertical wall. The ladder makes an angle of 20° with the wall. A woman of weight 720 N stands on the ladder. Fig. 7 shows this situation modelled with the woman's weight acting at a distance \(x\) m from the lower end of the ladder. The system is in equilibrium. \includegraphics{figure_7}
  1. Show that the frictional force between the ladder and the horizontal surface is \(F\) N, where \(F = 90(1 + x)\tan 20°\). [4]
    1. State with a reason whether \(F\) increases, stays constant or decreases as \(x\) increases. [1]
    2. Hence determine the set of values of the coefficient of friction between the ladder and the surface for which the woman can stand anywhere on the ladder without it slipping. [4]