Questions Further Mechanics Major (88 questions)

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OCR MEI Further Mechanics Major 2022 June Q1
5 marks Moderate -0.8
\includegraphics{figure_1} Three forces of magnitudes 4 N, 7 N and \(P\) N act at a point in the directions shown in the diagram. The forces are in equilibrium.
  1. Draw a closed figure to represent the three forces. [1]
  2. Hence, or otherwise, find the following.
    1. The value of \(\theta\). [2]
    2. The value of \(P\). [2]
OCR MEI Further Mechanics Major 2022 June Q2
4 marks Standard +0.3
\includegraphics{figure_2} A particle is projected with speed \(v\) from a point O on horizontal ground. The angle of projection is \(\theta\) above the horizontal. The particle passes, in succession, through two points A and B, each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram. You are given that \(d^2 = \frac{v^\alpha \sin^2 2\theta}{g^\beta} - \frac{8h^2 \cos^2 \theta}{g}\). Use dimensional analysis to find \(\alpha\) and \(\beta\). [4]
OCR MEI Further Mechanics Major 2022 June Q3
6 marks Standard +0.3
A particle, of mass 2 kg, is placed at a point A on a rough horizontal surface. There is a straight vertical wall on the surface and the point on the wall nearest to A is B. The distance AB is 5 m. The particle is projected with speed 4.2 m s\(^{-1}\) along the surface from A towards B. The particle hits the wall directly and rebounds. The coefficient of friction between the particle and the surface is 0.1.
  1. Determine the speed of the particle immediately before impact with the wall. [4]
The magnitude of the impulse that the wall exerts on the particle is 9.8 N s.
  1. Find the speed of the particle immediately after impact with the wall. [2]
OCR MEI Further Mechanics Major 2022 June Q4
7 marks Standard +0.3
\includegraphics{figure_4} The diagram shows a particle P, of mass 0.1 kg, which is attached by a light inextensible string of length 0.5 m to a fixed point O. P moves with constant angular speed 5 rad s\(^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical.
  1. Determine the tension in the string. [3]
  2. Find the value of \(\theta\). [2]
  3. Find the kinetic energy of P. [2]
OCR MEI Further Mechanics Major 2022 June Q5
7 marks Standard +0.3
At time \(t\) seconds, where \(t \geq 0\), a particle P of mass 2 kg is moving on a smooth horizontal surface. The particle moves under the action of a constant horizontal force of \((-2\mathbf{i} + 6\mathbf{j})\) N and a variable horizontal force of \((2\cos 2t \mathbf{i} + 4\sin t \mathbf{j})\) N. The acceleration of P at time \(t\) seconds is \(\mathbf{a}\) m s\(^{-2}\).
  1. Find \(\mathbf{a}\) in terms of \(t\). [2]
The particle P is at rest when \(t = 0\).
  1. Determine the speed of P at the instant when \(t = 2\). [5]
OCR MEI Further Mechanics Major 2022 June Q6
7 marks Standard +0.3
In this question the box should be modelled as a particle. A box of mass \(m\) kg is placed on a rough slope which makes an angle of \(\alpha\) with the horizontal.
  1. Show that the box is on the point of slipping if \(\mu = \tan \alpha\), where \(\mu\) is the coefficient of friction between the box and the slope. [2]
A box of mass 5 kg is pulled up a rough slope which makes an angle of 15° with the horizontal. The box is subject to a constant frictional force of magnitude 3 N. The speed of the box increases from 2 m s\(^{-1}\) at a point A on the slope to 5 m s\(^{-1}\) at a point B on the slope with B higher up the slope than A. The distance AB is 10 m. \includegraphics{figure_6} The pulling force has constant magnitude \(P\) N and acts at a constant angle of 25° above the slope, as shown in the diagram.
  1. Use the work–energy principle to determine the value of \(P\). [5]
OCR MEI Further Mechanics Major 2022 June Q7
12 marks Standard +0.3
Two small uniform smooth spheres A and B, of masses 2 kg and 3 kg respectively, are moving in opposite directions along the same straight line towards each other on a smooth horizontal surface. Sphere A has speed 2 m s\(^{-1}\) and B has speed 1 m s\(^{-1}\) before they collide. The coefficient of restitution between A and B is \(e\).
  1. Show that the velocity of B after the collision, in the original direction of motion of A, is \(\frac{1}{5}(1 + 6e)\) m s\(^{-1}\) and find a similar expression for the velocity of A after the collision. [5]
  2. The following three parts are independent of each other, and each considers a different scenario regarding the collision between A and B.
    1. In the collision between A and B the spheres coalesce to form a combined body C. State the speed of C after the collision. [1]
    2. In the collision between A and B the direction of motion of A is reversed. Find the range of possible values of \(e\). [2]
    3. The total loss in kinetic energy due to the collision is 3 J. Determine the value of \(e\). [4]
OCR MEI Further Mechanics Major 2022 June Q8
13 marks Standard +0.8
A particle P is projected from a fixed point O with initial velocity \(u\mathbf{i} + ku\mathbf{j}\), where \(k\) is a positive constant. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the horizontal and vertically upward directions respectively. P moves with constant gravitational acceleration of magnitude \(g\). At time \(t \geq 0\), particle P has position vector \(\mathbf{r}\) relative to O.
  1. Starting from an expression for \(\ddot{\mathbf{r}}\), use integration to derive the formula $$\mathbf{r} = ut\mathbf{i} + \left(kut - \frac{1}{2}gt^2\right)\mathbf{j}.$$ [4]
The position vector \(\mathbf{r}\) of P at time \(t \geq 0\) can be expressed as \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\), where the axes Ox and Oy are horizontally and vertically upwards through O respectively. The axis Ox lies on horizontal ground.
  1. Show that the path of P has cartesian equation $$gy^2 - 2ku^2x + 2u^2y = 0.$$ [3]
  2. Hence find, in terms of \(g\), \(k\) and \(u\), the maximum height of P above the ground during its motion. [3]
The maximum height P reaches above the ground is equal to the distance OA, where A is the point where P first hits the ground.
  1. Determine the value of \(k\). [3]
OCR MEI Further Mechanics Major 2022 June Q9
11 marks Challenging +1.8
[In this question you may use the facts that for a uniform solid right circular cone of height \(h\) and base radius \(r\) the volume is \(\frac{1}{3}\pi r^2 h\) and the centre of mass is \(\frac{1}{4}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_9} The diagram shows the shaded region S bounded by the curve \(y = e^{\frac{1}{4}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis, and the line \(y = \frac{1}{4}e(6-x)\). The line \(y = \frac{1}{4}e(6-x)\) meets the curve \(y = e^{\frac{1}{4}x}\) at the point A with coordinates \((2, e)\). The region S is rotated through \(2\pi\) radians about the \(x\)-axis to form a uniform solid of revolution T.
  1. Show that the \(x\)-coordinate of the centre of mass of T is \(\frac{3(5e^2 + 1)}{7e^2 - 3}\). [8]
Solid T is freely suspended from A and hangs in equilibrium.
  1. Determine the angle between AO, where O is the origin, and the vertical. [3]
OCR MEI Further Mechanics Major 2022 June Q10
10 marks Standard +0.8
\includegraphics{figure_10} A small toy car runs along a track in a vertical plane. The track consists of three sections: a curved section AB, a horizontal section BC which rests on the floor, and a circular section that starts at C with centre O and radius \(r\) m. The section BC is tangential to the curved section at B and tangential to the circular section at C, as shown in the diagram. The car, of mass \(m\) kg, is placed on the track at A, at a height \(h\) m above the floor, and released from rest. The car runs along the track from A to C and enters the circular section at C. It can be assumed that the track does not obstruct the car moving on to the circular section at C. The track is modelled as being smooth, and the car is modelled as a particle P.
  1. Show that, while P remains in contact with the circular section of the track, the magnitude of the normal contact force between P and the circular section is $$mg\left(3\cos\theta - 2 + \frac{2h}{r}\right)\text{N},$$ where \(\theta\) is the angle between OC and OP. [7]
  2. Hence determine, in terms of \(r\), the least possible value of \(h\) so that P can complete a vertical circle. [2]
  3. Apart from not modelling the car as a particle, state one refinement that would make the model more realistic. [1]
OCR MEI Further Mechanics Major 2022 June Q11
8 marks Challenging +1.2
\includegraphics{figure_11} The diagram shows two small identical uniform smooth spheres, A and B, just before A collides with B. Sphere B is at rest on a horizontal table with its centre vertically above the edge of the table. Sphere A is projected vertically upwards so that, just before it collides with B, the speed of A is \(U\) m s\(^{-1}\) and it is in contact with the vertical side of the table. The point of contact of A with the vertical side of the table and the centres of the spheres are in the same vertical plane.
  1. Show that on impact the line of centres makes an angle of 30° with the vertical. [1]
The coefficient of restitution between A and B is \(\frac{1}{2}\). After the impact B moves freely under gravity.
  1. Determine, in terms of \(U\) and \(g\), the time taken for B to first return to the table. [7]
OCR MEI Further Mechanics Major 2022 June Q12
13 marks Challenging +1.8
\includegraphics{figure_12} The diagram shows a uniform square lamina ABCD, of weight \(W\) and side-length \(a\). The lamina is in equilibrium in a vertical plane that also contains the point O. The vertex A rests on a smooth plane inclined at an angle of 30° to the horizontal. The vertex B rests on a smooth plane inclined at an angle of 60° to the horizontal. OA is a line of greatest slope of the plane inclined at 30° to the horizontal and OB is a line of greatest slope of the plane inclined at 60° to the horizontal. The side AB is inclined at an angle \(\theta\) to the horizontal and the lamina is kept in equilibrium in this position by a clockwise couple of magnitude \(\frac{1}{8}aW\).
  1. By resolving horizontally and vertically, determine, in terms of \(W\), the magnitude of the normal contact force between the plane and the lamina at B. [6]
  2. By taking moments about A, show that \(\theta\) satisfies the equation $$2(\sqrt{3} + 2)\sin\theta - 2\cos\theta = 1.$$ [5]
  3. Verify that \(\theta = 22.4°\), correct to 1 decimal place. [2]
OCR MEI Further Mechanics Major 2022 June Q13
17 marks Challenging +1.3
In this question take \(g = 10\). A particle P of mass 0.15 kg is attached to one end of a light elastic string of modulus of elasticity 13.5 N and natural length 0.45 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. Show that the distance OA is 0.5 m. [3]
At time \(t = 0\), P is projected vertically downwards from A with a speed of 1.25 m s\(^{-1}\). Throughout the subsequent motion, P experiences a variable resistance \(R\) newtons which is of magnitude 0.6 times its speed (in m s\(^{-1}\)).
  1. Given that the downward displacement of P from A at time \(t\) seconds is \(x\) metres, show that, while the string remains taut, \(x\) satisfies the differential equation $$\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 200x = 0.$$ [3]
  2. Verify that \(x = \frac{5}{56}e^{-2t}\sin(14t)\). [6]
  3. Determine whether the string becomes slack during the motion. [5]
OCR MEI Further Mechanics Major 2023 June Q1
4 marks Moderate -0.8
A car of mass 800 kg moves in a straight line along a horizontal road. There is a constant resistance to the motion of the car of magnitude 600 N. When the car is travelling at a speed of \(15 \text{ m s}^{-1}\) the power developed by the car is 27 kW. Determine the acceleration of the car when it is travelling at \(15 \text{ m s}^{-1}\). [4]
OCR MEI Further Mechanics Major 2023 June Q2
4 marks Standard +0.3
\includegraphics{figure_2} Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere A is moving towards B with speed \(6 \text{ m s}^{-1}\) and B is moving away from A with speed \(2 \text{ m s}^{-1}\) (see diagram). Spheres A and B collide. After this collision A moves with speed \(0.2 \text{ m s}^{-1}\). Determine the possible speeds with which B moves after the collision. [4]
OCR MEI Further Mechanics Major 2023 June Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows a particle P, of mass 0.2 kg, which is attached by a light inextensible string of length 0.75 m to a fixed point O. Particle P moves with constant angular speed \(\omega \text{ rad s}^{-1}\) in a horizontal circle with centre vertically below O. The string is inclined at an angle \(\theta\) to the vertical. The greatest tension that the string can withstand without breaking is 15 N.
  1. Find the greatest possible value of \(\theta\), giving your answer to the nearest degree. [2]
  2. Determine the greatest possible value of \(\omega\). [3]
OCR MEI Further Mechanics Major 2023 June Q4
6 marks Standard +0.8
\includegraphics{figure_4} A rigid lamina of negligible mass is in the form of a rhombus ABCD, where AC = 6 m and BD = 8 m. Forces of magnitude 2 N, 4 N, 3 N and 5 N act along its sides AB, BC, CD and DA, respectively, as shown in the diagram. A further force F N, acting at A, and a couple of magnitude G N m are also applied to the lamina so that it is in equilibrium.
  1. Determine the magnitude and direction of F. [4]
  2. Determine the value of G. [2]
OCR MEI Further Mechanics Major 2023 June Q5
7 marks Standard +0.3
A particle P of mass \(m\) kg is projected with speed \(u \text{ m s}^{-1}\) along a rough horizontal surface. During the motion of P, a constant frictional force of magnitude \(F\) N acts on P. When the velocity of P is \(v \text{ m s}^{-1}\), it experiences a force of magnitude \(kv\) N due to air resistance, where \(k\) is a constant.
  1. Determine the dimensions of \(k\). [3]
At time \(T\) s after projection P comes to rest. A formula approximating the value of \(T\) is $$T = \frac{mu}{F} - \frac{kmu^2}{2F^2} + \frac{1}{3}k^2m^{\alpha}u^{\beta}F^{\gamma}.$$
  1. Use dimensional analysis to find \(\alpha\), \(\beta\) and \(\gamma\). [4]
OCR MEI Further Mechanics Major 2023 June Q6
9 marks Standard +0.3
At time \(t\) seconds, where \(t \geq 0\), a particle P has position vector \(\mathbf{r}\) metres, where $$\mathbf{r} = (2t^2 - 12t + 6)\mathbf{i} + (t^3 + 3t^2 - 8t)\mathbf{j}.$$ The velocity of P at time \(t\) seconds is \(v \text{ m s}^{-1}\).
  1. Find \(v\) in terms of \(t\). [1]
  2. Determine the speed of P at the instant when it is moving parallel to the vector \(\mathbf{i} - 4\mathbf{j}\). [5]
  3. Determine the value of \(t\) when the magnitude of the acceleration of P is \(20.2 \text{ m s}^{-2}\). [3]
OCR MEI Further Mechanics Major 2023 June Q7
9 marks Standard +0.3
One end of a rope is attached to a block A of mass 2 kg. The other end of the rope is attached to a second block B of mass 4 kg. Block A is held at rest on a fixed rough ramp inclined at \(30°\) to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P, at a distance \(d\) m above the ground, as shown in the diagram. \includegraphics{figure_7} Block A is more than \(d\) m from P. The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is \(\frac{1}{2\sqrt{3}}\). The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.
  1. Determine, in terms of \(g\) and \(d\), the work done against friction as A moves \(d\) m up the ramp. [3]
  2. Given that the speed of B immediately before it hits the ground is \(1.75 \text{ m s}^{-1}\), use the work–energy principle to determine the value of \(d\). [5]
  3. Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic. [1]
OCR MEI Further Mechanics Major 2023 June Q8
8 marks Challenging +1.8
\includegraphics{figure_8} The diagram shows the shaded region R bounded by the curve \(y = \sqrt{3x + 4}\), the \(x\)-axis, the \(y\)-axis, and the straight line that passes through the points \((k, 0)\) and \((4, 4)\), where \(0 < k < 4\). Region R is occupied by a uniform lamina.
  1. Determine, in terms of \(k\), an expression for the \(y\)-coordinate of the centre of mass of the lamina. Give your answer in the form \(\frac{a + bk}{c + dk}\), where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]
  2. Show that the \(y\)-coordinate of the centre of mass of the lamina cannot be \(\frac{3}{2}\). [2]
OCR MEI Further Mechanics Major 2023 June Q9
12 marks Challenging +1.3
In this question take \(g = 10\). A small ball P is projected with speed \(20 \text{ m s}^{-1}\) at an angle of elevation of \((\alpha + \theta)\) from a point O at the bottom of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\) and \(\tan \theta = \frac{3}{4}\). The ball subsequently hits the plane at a point A, where OA is a line of greatest slope of the plane, as shown in the diagram. \includegraphics{figure_9}
  1. Determine the following, in either order.
    [9]
After P hits the plane at A it continues to move away from O. Immediately after hitting the plane at A the direction of motion of P makes an angle \(\beta\) with the horizontal.
  1. Determine the maximum possible value of \(\beta\), giving your answer to the nearest degree. [3]
OCR MEI Further Mechanics Major 2023 June Q10
16 marks Challenging +1.8
\includegraphics{figure_10} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O. The point A lies on the rim of the bowl such that AO makes an angle of \(30°\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A, with speed \(u\), where \(u > \sqrt{\frac{gr}{2}}\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A. The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.
  1. Determine, in terms of \(m\), \(g\), \(u\), \(r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B. [7]
The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4mg\).
  1. Determine, in terms of \(r\), the vertical distance of B above the floor. [4]
It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.
  1. Show that \(u^2 > 2gr\). [5]
OCR MEI Further Mechanics Major 2023 June Q11
12 marks Challenging +1.8
\includegraphics{figure_11} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC, with AB perpendicular to AC, which lies in a vertical plane. The length of AB is 3 cm, and the length of AC is 12 cm. The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle \(\theta\) with the horizontal. A horizontal force of magnitude \(P\) N is now applied to the prism at B. This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each \(\frac{1}{2}\).
  1. Show that the least value of \(P\) needed to move the prism is given by $$P = \frac{40 \cos \theta + 95 \sin \theta}{16 \sin \theta - 13 \cos \theta}.$$ [8]
  2. Determine the range in which the value of \(\theta\) must lie. [4]
OCR MEI Further Mechanics Major 2023 June Q12
13 marks Challenging +1.2
Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u_1 \mathbf{i} + u_2 \mathbf{j}\) towards B, which is at rest. The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf{i}\). The coefficient of restitution between A and B is \(e\).
    1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf{i}\). [1]
    2. Determine this impulse in terms of \(\lambda\), \(m\), \(e\) and \(u_1\). [6]
The loss in kinetic energy due to the collision between A and B is \(\frac{1}{8}mu_1^2\).
  1. Determine the range of possible values of \(\lambda\). [6]