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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]
OCR MEI Further Mechanics Major Specimen Q8
16 marks Standard +0.3
A tractor has a mass of 6000 kg. When developing a power of 5 kW, the tractor is travelling at a steady speed of 2.5 m s\(^{-1}\) across a horizontal field.
  1. Calculate the magnitude of the resistance to the motion of the tractor. [2]
The tractor comes to horizontal ground where the resistance to motion is different. The power developed by the tractor during the next 10 s has an average value of 8 kW. During this time, the tractor accelerates uniformly from 2.5 m s\(^{-1}\) to 3 m s\(^{-1}\).
    1. Show that the work done against the resistance to motion during the 10 s is 71 750 J. [4]
    2. Assuming that the resistance to motion is constant, calculate its value. [3]
The tractor can usually travel up a straight track inclined at an angle \(\alpha\) to the horizontal, where \(\sin\alpha = \frac{1}{20}\), while accelerating uniformly from 3 m s\(^{-1}\) to 3.25 m s\(^{-1}\) over a distance of 100 m against a resistance to motion of constant magnitude of 2000 N. The tractor develops a fault which limits its maximum power to 16kW.
  1. Determine whether the tractor could now perform the same motion up the track. [You should assume that the mass of the tractor and the resistance to motion remain the same.] [7]
OCR MEI Further Mechanics Major Specimen Q9
14 marks Challenging +1.2
\includegraphics{figure_9} Fig. 9 shows the instant of impact of two identical uniform smooth spheres, A and B, each with mass \(m\). Immediately before they collide, the spheres are sliding towards each other on a smooth horizontal table in the directions shown in the diagram, each with speed \(v\). The coefficient of restitution between the spheres is \(\frac{1}{2}\).
  1. Show that, immediately after the collision, the speed of A is \(\frac{1}{8}v\). Find its direction of motion. [6]
  2. Find the percentage of the original kinetic energy that is lost in the collision. [7]
  3. State where in your answer to part (i) you have used the assumption that the contact between the spheres is smooth. [1]
OCR MEI Further Mechanics Major Specimen Q10
14 marks Standard +0.3
In this question take \(g = 10\). A smooth ball of mass 0.1 kg is projected from a point on smooth horizontal ground with speed 65 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal, where \(\tan\alpha = \frac{3}{4}\). While it is in the air the ball is modelled as a particle moving freely under gravity. The ball bounces on the ground repeatedly. The coefficient of restitution for the first bounce is 0.4.
  1. Show that the ball leaves the ground after the first bounce with a horizontal speed of 52 m s\(^{-1}\) and a vertical speed of 15.6 m s\(^{-1}\). Explain your reasoning carefully. [4]
  2. Calculate the magnitude of the impulse exerted on the ball by the ground at the first bounce. [2]
Each subsequent bounce is modelled by assuming that the coefficient of restitution is 0.4 and that the bounce takes no time. The ball is in the air for \(T_1\) seconds between projection and bouncing the first time, \(T_2\) seconds between the first and second bounces, and \(T_n\) seconds between the \((n-1)\)th and \(n\)th bounces.
    1. Show that \(T_1 = \frac{39}{5}\). [2]
    2. Find an expression for \(T_n\) in terms of \(n\). [2]
  2. According to the model, how far does the ball travel horizontally while it is still bouncing? [3]
  3. According to the model, what is the motion of the ball after it has stopped bouncing? [1]
OCR MEI Further Mechanics Major Specimen Q11
16 marks Challenging +1.2
The region bounded by the \(x\)-axis and the curve \(y = \frac{1}{2}k(1-x^2)\) for \(-1 \leq x \leq 1\) is occupied by a uniform lamina, as shown in Fig. 11.1. \includegraphics{figure_11_1}
  1. In this question you must show detailed reasoning. Show that the centre of mass of the lamina is at \(\left(0, \frac{1}{5}k\right)\). [7]
A shop sign is modelled as a uniform lamina in the form of the lamina in part (i) attached to a rectangle ABCD, where AB = 2 and BC = 1. The sign is suspended by two vertical wires attached at A and D, as shown in Fig. 11.2. \includegraphics{figure_11_2}
  1. Show that the centre of mass of the sign is at a distance $$\frac{2k^2 + 10k + 15}{10k + 30}$$ from the midpoint of CD. [4]
The tension in the wire at A is twice the tension in the wire at D.
  1. Find the value of \(k\). [5]
OCR MEI Further Mechanics Major Specimen Q12
15 marks Challenging +1.2
Fig. 12 shows \(x\)- and \(y\)- coordinate axes with origin O and the trajectory of a particle projected from O with speed 28 m s\(^{-1}\) at an angle \(\alpha\) to the horizontal. After \(t\) seconds, the particle has horizontal and vertical displacements \(x\) m and \(y\) m. Air resistance should be neglected. \includegraphics{figure_12}
  1. Show that the equation of the trajectory is given by $$\tan^2\alpha - \frac{160}{x}\tan\alpha + \frac{160y}{x^2} + 1 = 0.$$ (*) [5]
    1. Show that if (*) is treated as an equation with \(\tan\alpha\) as a variable and with \(x\) and \(y\) as constants, then (*) has two distinct real roots for \(\tan\alpha\) when \(y < 40 - \frac{x^2}{160}\). [3]
    2. Show the inequality in part (ii)(A) as a locus on the graph of \(y = 40 - \frac{x^2}{160}\) in the Printed Answer Booklet and label it R. [1]
S is the locus of points \((x, y)\) where (*) has one real root for \(\tan\alpha\). T is the locus of points \((x, y)\) where (*) has no real roots for \(\tan\alpha\).
  1. Indicate S and T on the graph in the Printed Answer Booklet. [2]
  2. State the significance of R, S and T for the possible trajectories of the particle. [3]
A machine can fire a tennis ball from ground level with a maximum speed of 28 m s\(^{-1}\).
  1. State, with a reason, whether a tennis ball fired from the machine can achieve a range of 80 m. [1]
OCR MEI Further Statistics Minor Specimen Q1
4 marks Moderate -0.8
A darts player is trying to hit the bullseye on a dart board. On each throw the probability that she hits it is \(0.05\), independently of any other throw.
  1. Find the probability that she hits the bullseye for the first time on her \(10\)th throw. [2]
  2. Find the probability that she does not hit the bullseye in her first \(10\) throws. [1]
  3. Write down the expected number of throws which it takes her to hit the bullseye for the first time. [1]
OCR MEI Further Statistics Minor Specimen Q2
8 marks Moderate -0.8
The number of televisions of a particular model sold per week at a retail store can be modelled by a random variable \(X\) with the probability function shown in the table.
\(x\)\(0\)\(1\)\(2\)\(3\)\(4\)
\(P(X = x)\)\(0.05\)\(0.2\)\(0.5\)\(0.2\)\(0.05\)
    1. Explain why \(\text{E}(X) = 2\). [1]
    2. Find \(\text{Var}(X)\). [3]
  2. The profit, measured in pounds made in a week, on the sales of this model of television is given by \(Y\), where \(Y = 250X - 80\). Find
The remote controls for the televisions are quality tested by the manufacturer to see how long they last before they fail.
  1. Explain why it would be inappropriate to test all the remote controls in this way. [1]
  2. State an advantage of using random sampling in this context. [1]