Questions Further Mechanics Major (88 questions)

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OCR MEI Further Mechanics Major 2023 June Q13
15 marks Challenging +1.2
A particle P of mass \(m\) is fixed to one end of a light spring of natural length \(a\) and modulus of elasticity \(man^2\), where \(n > 0\). The other end of the spring is attached to the ceiling of a lift. The lift is at rest and P is hanging vertically in equilibrium.
  1. Find, in terms of \(g\) and \(n\), the extension in the spring. [3]
At time \(t = 0\) the lift begins to accelerate upwards from rest. At time \(t\), the upward displacement of the lift from its initial position is \(y\) and the extension of the spring is \(x\).
  1. Express, in terms of \(g\), \(n\), \(x\) and \(y\), the upward displacement of P from its initial position at time \(t\). [2]
  2. Given that \(\ddot{y} = kt\), where \(k\) is a positive constant, express the upward acceleration of P in terms of \(\ddot{x}\), \(k\) and \(t\). [1]
  3. Show that \(x\) satisfies the differential equation $$\ddot{x} + n^2 x = kt + g.$$ [3]
  4. Verify that \(x = \frac{1}{n^2}(knt + gn - k \sin(nt))\). [4]
  5. By considering \(\ddot{x}\) comment on the motion of P relative to the ceiling of the lift for all times after the lift begins to move. [2]
OCR MEI Further Mechanics Major 2024 June Q1
4 marks Moderate -0.3
A car A of mass 1200 kg is about to tow another car B of mass 800 kg in a straight line along a horizontal road by means of a tow-rope attached between A and B. The tow-rope is modelled as being light and inextensible. Just before the tow-rope tightens, A is travelling at a speed of \(1.5 \text{ m s}^{-1}\) and B is at rest. Just after the tow-rope tightens, both cars have a speed of \(v \text{ m s}^{-1}\).
  1. Find the value of \(v\). [2]
  2. Calculate the magnitude of the impulse on A when the tow-rope tightens. [2]
OCR MEI Further Mechanics Major 2024 June Q2
9 marks Moderate -0.3
One end of a light spring is attached to a fixed point. A mass of 2 kg is attached to the other end of the spring. The spring hangs vertically in equilibrium. The extension of the spring is 0.05 m.
  1. Find the stiffness of the spring. [2]
  2. Find the energy stored in the spring. [2]
  3. Find the dimensions of stiffness of a spring. [1]
A particle P of mass \(m\) is performing complete oscillations with amplitude \(a\) on the end of a light spring with stiffness \(k\). The spring hangs vertically and the maximum speed \(v\) of P is given by the formula $$v = Cm^{\alpha}a^{\beta}k^{\gamma},$$ where C is a dimensionless constant.
  1. Use dimensional analysis to determine \(\alpha\), \(\beta\), and \(\gamma\). [4]
OCR MEI Further Mechanics Major 2024 June Q3
5 marks Challenging +1.2
\includegraphics{figure_3} A circular hole with centre C and radius \(r\) m, where \(r < 0.5\), is cut in a uniform circular disc with centre O and radius 0.5 m. The hole touches the rim of the disc at A (see diagram). The centre of mass, G, of the remainder of the disc is on the rim of the hole. Determine the value of \(r\). [5]
OCR MEI Further Mechanics Major 2024 June Q4
8 marks Standard +0.8
\includegraphics{figure_4} A uniform rod AB has mass 3 kg and length 4 m. The end A of the rod is in contact with rough horizontal ground. The rod rests in equilibrium on a smooth horizontal peg 1.5 m above the ground, such that the rod is inclined at an angle of \(25°\) to the ground (see diagram). The rod is in a vertical plane perpendicular to the peg.
  1. Determine the magnitude of the normal contact force between the peg and the rod. [3]
  2. Determine the range of possible values of the coefficient of friction between the rod and the ground. [5]
OCR MEI Further Mechanics Major 2024 June Q5
7 marks Standard +0.3
A car of mass 850 kg is travelling along a straight horizontal road. The power developed by the car is constant and is equal to 18 kW. There is a constant resistance to motion of magnitude 600 N.
  1. Find the greatest steady speed at which the car can travel. [2]
Later in the journey, while travelling at a speed of \(15 \text{ m s}^{-1}\), the car comes to the bottom of a straight hill which is inclined at an angle of \(\sin^{-1}\left(\frac{1}{40}\right)\) to the horizontal. The power developed by the car remains constant at 18 kW. The magnitude of the resistance force is no longer constant but changes such that the total work done against the resistance force in ascending the hill is 103 000 J. The car takes 10 seconds to ascend the hill and at the top of the hill the car is travelling at \(18 \text{ m s}^{-1}\).
  1. Determine the distance the car travels from the bottom to the top of the hill. [5]
OCR MEI Further Mechanics Major 2024 June Q6
6 marks Challenging +1.2
In this question you must show detailed reasoning. In this question, positions are given relative to a fixed origin, O. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the \(x\)- and \(y\)-directions respectively in a horizontal plane. Distances are measured in centimetres and the time, \(t\), is measured in seconds, where \(0 \leq t \leq 5\). A small radio-controlled toy car C moves on a horizontal surface which contains O. The acceleration of C is given by \(2\mathbf{i} + t\mathbf{j} \text{ cm s}^{-2}\). When \(t = 4\), the displacement of C from O is \(16\mathbf{i} + \frac{32}{3}\mathbf{j}\) cm, and the velocity of C is \(8\mathbf{i} \text{ cm s}^{-1}\). Determine a cartesian equation for the path of C for \(0 < t < 5\). You are not required to simplify your answer. [6]
OCR MEI Further Mechanics Major 2024 June Q7
9 marks Standard +0.8
The region bounded by the curve \(y = x^3 - 3x^2 + 4\), the positive \(x\)-axis and the positive \(y\)-axis is occupied by a uniform lamina L. The vertices of L are O, A and B, where O is the origin, A is a point on the positive \(x\)-axis and B is a point on the positive \(y\)-axis (see diagram). \includegraphics{figure_7}
  1. Determine the coordinates of the centre of mass of L. [5]
The lamina L is the cross-section through the centre of mass of a uniform solid prism M. The prism M is placed on an inclined plane, which makes an angle of \(30°\) with the horizontal, so that OA lies along a line of greatest slope of the plane with O lower down the plane than A. It is given that M does not slip on the plane.
  1. Determine whether M will topple in this case. Give a reason to support your answer. [2]
The prism M is now placed on the same inclined plane so that OB lies along a line of greatest slope of the plane with O lower down the plane than B. It is given that M still does not slip on the plane.
  1. Determine whether M will topple in this case. Give a reason to support your answer. [2]
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]