Questions — OCR MEI Further Mechanics Major (88 questions)

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OCR MEI Further Mechanics Major 2021 November Q1
3 marks Moderate -0.3
1 A small ball of mass 0.25 kg is held above a horizontal floor. The ball is released from rest and hits the floor with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It rebounds from the floor with a speed of \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The situation is modelled by assuming that the ball is in contact with the floor for 0.02 s and during this time the normal contact force the floor exerts on the ball is constant. Determine the magnitude of the normal contact force that the floor exerts on the ball.
OCR MEI Further Mechanics Major 2021 November Q2
4 marks Moderate -0.8
2 The diagram shows a system of three particles of masses \(3 m , 5 m\) and \(2 m\) situated in the \(x - y\) plane at the points \(\mathrm { A } ( 1,2 ) , \mathrm { B } ( 2 , - 2 )\) and \(\mathrm { C } ( 5,3 )\) respectively.
[diagram]
Determine the coordinates of the centre of mass of the system of particles.
OCR MEI Further Mechanics Major 2021 November Q3
4 marks Moderate -0.8
3 One end of a light elastic spring of natural length 0.3 m is attached to a fixed point. A mass of 4 kg is attached to the other end of the spring. When the spring hangs vertically in equilibrium the extension of the spring is 0.02 m .
  1. Determine the modulus of elasticity of the spring. A student calculates that if the mass of 4 kg is removed and replaced with a mass of 20 kg the extension of the spring will be 0.1 m .
  2. Suggest a reason why this extension may not be 0.1 m .
OCR MEI Further Mechanics Major 2021 November Q4
6 marks Challenging +1.2
4 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_646_812_312_242} The diagram shows parts of the curves \(y = 3 \sqrt { x }\) and \(y = 4 - x ^ { 2 }\), which intersect at the point ( 1,3 ). The shaded region, bounded by the two curves and the \(y\)-axis, is occupied by a uniform lamina. Determine the exact \(x\)-coordinate of the centre of mass of the lamina.
OCR MEI Further Mechanics Major 2021 November Q5
6 marks Standard +0.8
5 Two small uniform smooth spheres A and B , of equal radius, have masses 2 kg and 4 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, A has speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and B has speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along a line which is perpendicular to the line of centres (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-03_389_764_1592_244} The direction of motion of B after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Determine the coefficient of restitution between A and B .
OCR MEI Further Mechanics Major 2021 November Q6
11 marks Moderate -0.8
6
  1. Write down the dimensions of force. The force \(F\) of gravitational attraction between two objects with masses \(m _ { 1 }\) and \(m _ { 2 }\), at a distance \(d\) apart, is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is the universal gravitational constant.
    In SI units the value of \(G\) is \(6.67 \times 10 ^ { - 11 } \mathrm {~kg} ^ { - 1 } \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 2 }\).
  2. Write down the dimensions of \(G\).
  3. Determine the value of \(G\) in imperial units based on pounds, feet, and seconds. Use the facts that 1 pound \(= 0.454 \mathrm {~kg}\) and 1 foot \(= 0.305 \mathrm {~m}\). For a planet of mass \(M\) and radius \(r\), it is suggested that the velocity \(v\) needed for an object to escape the gravitational pull of the planet, the 'escape velocity', is given by the following formula. \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\),
    where \(k\) is a dimensionless constant.
  4. Show that this formula is dimensionally consistent. Information regarding the planets Earth and Mars can be found in the table below.
    EarthMars
    Radius (m)63710003389500
    Mass (kg)\(5.97 \times 10 ^ { 24 }\)\(6.39 \times 10 ^ { 23 }\)
    Escape velocity ( \(\mathrm { m } \mathrm { s } ^ { - 1 }\) )11186
  5. Using the formula \(\mathrm { v } = \sqrt { \frac { \mathrm { kGM } } { \mathrm { r } } }\), determine the escape velocity for planet Mars.
OCR MEI Further Mechanics Major 2021 November Q7
12 marks Challenging +1.2
7 A box B of mass \(m \mathrm {~kg}\) is raised vertically by an engine working at a constant rate of \(k m g \mathrm {~W}\). Initially B is at rest. The speed of B when it has been raised a distance \(x \mathrm {~m}\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } \frac { d v } { d x } = ( k - v ) g\).
  2. Verify that \(\mathrm { gx } = \mathrm { k } ^ { 2 } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { v } } \right) - \mathrm { kv } - \frac { 1 } { 2 } \mathrm { v } ^ { 2 }\).
  3. By using the work-energy principle, show that the time taken for B to reach a speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest is given by \(\frac { \mathrm { k } } { \mathrm { g } } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { V } } \right) - \frac { \mathrm { V } } { \mathrm { g } }\).
OCR MEI Further Mechanics Major 2021 November Q8
12 marks Challenging +1.8
8 A capsule consists of a uniform hollow right circular cylinder of radius \(r\) and length \(2 h\) attached to two uniform hollow hemispheres of radius \(r\).
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17e92314-d7df-49b8-a441-8d18c91dbbb0-06_702_684_445_244} \captionsetup{labelformat=empty} \caption{Fig. 8}
\end{figure} Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall. The total weight of the capsule is \(W\) and acts at a point midway between A and B . The plane containing \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D is vertical, with AB making an acute angle \(\theta\) with the downward vertical.
  1. Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule. The coefficient of friction at each point of contact is \(\frac { 1 } { 3 }\).
  2. By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of \(W\).
  3. By determining an expression for \(r\) in terms of \(h\) and \(\theta\), show that \(\tan \theta > \frac { 3 } { 4 }\).
OCR MEI Further Mechanics Major 2021 November Q9
15 marks Challenging +1.2
9 A small ball P is projected with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(( \alpha + \theta )\) from a point O at the bottom of a plane inclined at \(\alpha\) to the horizontal. P subsequently hits the plane at a point R , where OR is a line of greatest slope, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-07_456_862_406_242}
  1. By deriving an expression, in terms of \(\theta\), \(\alpha\) and \(g\), for the time of flight of P , show that the distance OR, in metres, is $$\frac { 50 \sin \theta \cos ( \theta + \alpha ) } { g \cos ^ { 2 } \alpha }$$
  2. By using the identity \(2 \sin \mathrm {~A} \cos \mathrm {~B} \equiv \sin ( \mathrm {~A} + \mathrm { B } ) - \sin ( \mathrm { B } - \mathrm { A } )\), determine, in terms of \(g\) and \(\sin \alpha\), an expression for the maximum range of P up the plane, as \(\theta\) varies.
  3. Given that OR is the maximum range of P up the plane and is equal to 1.8 m , determine the value of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{17e92314-d7df-49b8-a441-8d18c91dbbb0-08_625_1180_255_239} A rigid wire ABC is fixed in a vertical plane. The section AB of the wire, of length \(b\), is straight and horizontal. The section BC of the wire is smooth and in the form of a circular arc of radius \(a\) and length \(\frac { 1 } { 2 } a \pi\). The centre of the arc is O , which is vertically above B . A bead P of mass \(m\) is threaded on the wire and projected from B with speed \(u\) towards C . The angle BOP when P is between B and C is denoted by \(\theta\), as shown in the diagram.
OCR MEI Further Mechanics Major 2021 November Q11
16 marks Challenging +1.2
11 Two small uniform smooth spheres A and B , of equal radius, have masses 4 kg and 3 kg respectively. The spheres are placed in a smooth horizontal circular groove. The coefficient of restitution between the spheres is \(e\), where \(e > \frac { 2 } { 5 }\). At a given instant B is at rest and A is set moving along the groove with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It may be assumed that in the subsequent motion the two spheres do not leave the groove.
  1. Determine, in terms of \(e\) and \(V\), the speeds of A and B immediately after the first collision.
  2. Show that the arc through which A moves between the first and second collisions subtends an angle at the centre of the circular groove of $$\frac { 2 \pi ( 4 - 3 e ) } { 7 e } \text { radians. }$$
    1. Determine, in terms of \(e\) and \(V\), the speed of B immediately after the second collision.
    2. What can be said about the motion of A and B if the collisions between A and B are perfectly elastic?
OCR MEI Further Mechanics Major 2021 November Q12
18 marks Challenging +1.2
12 A particle P of mass \(m\) is fixed to one end of a light elastic string of natural length \(l\) and modulus of elasticity 12 mg . The other end of the string is attached to a fixed point O . Particle P is held next to O and then released from rest.
  1. Show that P next comes instantaneously to rest when the length of the string is \(\frac { 3 } { 2 } l\). The string first becomes taut at time \(t = 0\). At time \(t \geqslant 0\), the length of the string is \(l + x\), where \(x\) is the extension in the string.
  2. Show that when the string is taut, \(x\) satisfies the differential equation $$\ddot { \mathrm { x } } + \omega ^ { 2 } \mathrm { x } = \mathrm { g } \text {, where } \omega ^ { 2 } = \frac { 12 \mathrm {~g} } { \mathrm { I } } \text {. }$$
  3. By using the substitution \(x = y + \frac { g } { \omega ^ { 2 } }\), solve the differential equation to show that the time when the string first becomes slack satisfies the equation $$\cos \omega \mathrm { t } - \sqrt { \mathrm { k } } \sin \omega \mathrm { t } = 1$$ where \(k\) is an integer to be determined.
OCR MEI Further Mechanics Major 2021 November Q10
13 marks Challenging +1.2
  1. Determine the magnitude of the normal reaction of the wire on P in terms of \(m , g , a , u\) and \(\theta\), when P is between B and C . P collides with a fixed barrier at C . The coefficient of restitution between P and the fixed barrier is \(e\). After this collision P moves back towards B . On the straight portion BA , the motion of P is resisted by a constant horizontal force \(F\).
  2. Show that P will reach A if $$F b \leqslant \frac { 1 } { 2 } m \left[ e ^ { 2 } u ^ { 2 } + k \left( 1 - e ^ { 2 } \right) g a \right] ,$$ where \(k\) is an integer to be determined.
OCR MEI Further Mechanics Major 2019 June Q1
5 marks Standard +0.3
Three forces represented by the vectors \(-4\mathbf{i} + \mathbf{j} + 2\mathbf{j}\) and \(k\mathbf{i} - 2\mathbf{j}\) act at the points with coordinates \((0, 0)\), \((3, 0)\) and \((0, 4)\) respectively.
  1. Given that the three forces form a couple, find the value of \(k\). [2]
  2. Find the magnitude and direction of the couple. [3]
OCR MEI Further Mechanics Major 2019 June Q2
4 marks Moderate -0.5
The Reynolds number, \(R\), is an important dimensionless quantity in fluid dynamics; it can be used to predict flow patterns when a fluid is in motion relative to a surface. The Reynolds number is defined as $$R = \frac{\rho ul}{\mu},$$ where \(\rho\) is the density of the fluid, \(u\) is the velocity of the fluid relative to the surface, \(l\) is the distance travelled by the fluid and \(\mu\) is the viscosity of the fluid. Find the dimensions of \(\mu\). [4]
OCR MEI Further Mechanics Major 2019 June Q3
5 marks Moderate -0.3
A ball of mass \(2\)kg is moving with velocity \((3\mathbf{i} - 2\mathbf{j})\)ms\(^{-1}\) when it is struck by a bat. The impulse on the ball is \((-8\mathbf{i} + 10\mathbf{j})\)Ns.
  1. Find the speed of the ball immediately after the impact. [4]
  2. State one modelling assumption you have used in answering part (a). [1]
OCR MEI Further Mechanics Major 2019 June Q4
6 marks Standard +0.8
\includegraphics{figure_4} Fig. 4 shows a uniform lamina ABCDE such that ABDE is a rectangle and BCD is an isosceles triangle. AB = 5a, AE = 4a and BC = CD. The point F is the midpoint of BD and FC = a.
  1. Find, in terms of \(a\), the exact distance of the centre of mass of the lamina from AE. [4]
The lamina is freely suspended from B and hangs in equilibrium.
  1. Find the angle between AB and the downward vertical. [2]
OCR MEI Further Mechanics Major 2019 June Q5
7 marks Standard +0.3
A particle P of mass 4 kilograms moves in such a way that its position vector at time \(t\) seconds is \(\mathbf{r}\) metres, where $$\mathbf{r} = 3t\mathbf{i} + 2e^{-3t}\mathbf{j}.$$
  1. Find the initial kinetic energy of P. [4]
  2. Find the time when the acceleration of P is 2 metres per second squared. [3]
OCR MEI Further Mechanics Major 2019 June Q6
7 marks Challenging +1.2
\includegraphics{figure_6} The rim of a smooth hemispherical bowl is a circle of centre O and radius \(a\). The bowl is fixed with its rim horizontal and uppermost. A particle P of mass \(m\) is released from rest at a point A on the rim as shown in Fig. 6. When P reaches the lowest point of the bowl it collides directly with a stationary particle Q of mass \(\frac{1}{2}m\). After the collision Q just reaches the rim of the bowl. Find the coefficient of restitution between P and Q. [7]
OCR MEI Further Mechanics Major 2019 June Q7
8 marks Challenging +1.8
In this question you must show detailed reasoning. \includegraphics{figure_7} Fig. 7 shows the curve with equation \(y = \frac{2}{3}\ln x\). The region R, shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis. Find the exact \(x\)-coordinate of the centre of mass of the solid. [8]
OCR MEI Further Mechanics Major 2019 June Q8
11 marks Standard +0.3
A car of mass 800kg travels up a line of greatest slope of a straight road inclined at \(5°\) to the horizontal. The power developed by the car is constant and equal to 25kW. The resistance to the motion of the car is constant and equal to 750N. The car passes through a point A on the road with speed \(7\)ms\(^{-1}\).
  1. Find
    [5]
The car later passes through a point B on the road where AB = 131m. The time taken to travel from A to B is 10.4s.
  1. Calculate the speed of the car at B. [6]
OCR MEI Further Mechanics Major 2019 June Q9
12 marks Challenging +1.2
\includegraphics{figure_9} A particle P of mass \(m\) is joined to a fixed point O by a light inextensible string of length \(l\). P is released from rest with the string taut and making an acute angle \(\alpha\) with the downward vertical, as shown in Fig. 9. At a time \(t\) after P is released the string makes an angle \(\theta\) with the downward vertical and the tension in the string is \(T\). Angles \(\alpha\) and \(\theta\) are measured in radians.
  1. Show that $$\left(\frac{\mathrm{d}\theta}{\mathrm{d}t}\right)^2 = \frac{2g}{l}\cos\theta + k_1,$$ where \(k_1\) is a constant to be determined in terms of \(g\), \(l\) and \(\alpha\). [4]
  2. Show that $$T = 3mg\cos\theta + k_2,$$ where \(k_2\) is a constant to be determined in terms of \(m\), \(g\) and \(\alpha\). [3]
It is given that \(\alpha\) is small enough for \(\alpha^2\) to be negligible.
  1. Find, in terms of \(m\) and \(g\), the approximate tension in the string. [2]
  2. Show that the motion of P is approximately simple harmonic. [3]
OCR MEI Further Mechanics Major 2019 June Q10
8 marks Challenging +1.2
A particle P, of mass \(m\), moves on a rough horizontal table. P is attracted towards a fixed point O on the table by a force of magnitude \(\frac{kmg}{x^2}\), where \(x\) is the distance OP. The coefficient of friction between P and the table is \(\mu\). P is initially projected in a direction directly away from O. The velocity of P is first zero at a point A which is a distance \(a\) from O.
  1. Show that the velocity \(v\) of P, when P is moving away from O, satisfies the differential equation $$\frac{\mathrm{d}}{\mathrm{d}x}(v^2) + \frac{2kg}{x^2} + 2\mu g = 0.$$ [3]
  2. Verify that $$v^2 = 2gk\left(\frac{1}{x} - \frac{1}{a}\right) + 2\mu g(a-x).$$ [3]
  3. Find, in terms of \(k\) and \(a\), the range of values of \(\mu\) for which P remains at A. [2]
OCR MEI Further Mechanics Major 2019 June Q11
14 marks Standard +0.8
Two uniform smooth spheres A and B have equal radii and are moving on a smooth horizontal surface. The mass of A is 0.2kg and the mass of B is 0.6kg. The spheres collide obliquely. When the spheres collide the line joining their centres is parallel to \(\mathbf{i}\). Immediately before the collision the velocity of A is \(\mathbf{u}_A\)ms\(^{-1}\) and the velocity of B is \(\mathbf{u}_B\)ms\(^{-1}\). The coefficient of restitution between A and B is 0.5. Immediately after the collision the velocity of A is \((-4\mathbf{i} + 2\mathbf{j})\)ms\(^{-1}\) and the velocity of B is \((2\mathbf{i} + 3\mathbf{j})\)ms\(^{-1}\).
  1. Find \(\mathbf{u}_A\) and \(\mathbf{u}_B\). [7]
After the collision B collides with a smooth vertical wall which is parallel to \(\mathbf{j}\). The loss in kinetic energy of B caused by the collision with the wall is 1.152J.
  1. Find the coefficient of restitution between B and the wall. [3]
  2. Find the angle through which the direction of motion of B is deflected as a result of the collision with the wall. [4]
OCR MEI Further Mechanics Major 2019 June Q12
16 marks Challenging +1.2
\includegraphics{figure_12} The ends of a light inextensible string are fixed to two points A and B in the same vertical line, with A above B. The string passes through a small smooth ring of mass \(m\). The ring is fastened to the string at a point P. When the string is taut the angle APB is a right angle, the angle BAP is \(\theta\) and the perpendicular distance of P from AB is \(r\). The ring moves in a horizontal circle with constant angular velocity \(\omega\) and the string taut as shown in Fig. 12.
  1. By resolving horizontally and vertically, show that the tension in the part of the string BP is \(m(r\omega^2\cos\theta - g\sin\theta)\). [6]
  2. Find a similar expression, in terms of \(r\), \(\omega\), \(m\), \(g\) and \(\theta\), for the tension in the part of the string AP. [2]
It is given that AB = 5a and AP = 4a.
  1. Show that \(16a\omega^2 > 5g\). [3]
The ring is now free to move on the string but remains in the same position on the string as before. The string remains taut and the ring continues to move in a horizontal circle.
  1. Find the period of the motion of the ring, giving your answer in terms of \(a\), \(g\) and \(\pi\). [5]
OCR MEI Further Mechanics Major 2019 June Q13
17 marks Challenging +1.3
\includegraphics{figure_13} A step-ladder has two sides AB and AC, each of length \(4a\). Side AB has weight \(W\) and its centre of mass is at the half-way point; side AC is light. The step-ladder is smoothly hinged at A and the two parts of the step-ladder, AB and AC, are connected by a light taut rope DE, where D is on AB, E is on AC and AD = AE = \(a\). A man of weight \(4W\) stands at a point F on AB, where BF = \(x\). The system is in equilibrium with B and C on a smooth horizontal floor and the sides AB and AC are each at an angle \(\theta\) to the vertical, as shown in Fig. 13.
  1. By taking moments about A for side AB of the step-ladder and then for side AC of the step-ladder show that the tension in the rope is $$W\left(1 + \frac{2x}{a}\right)\tan\theta.$$ [7]
The rope is elastic with natural length \(\frac{1}{2}a\) and modulus of elasticity \(W\).
  1. Show that the condition for equilibrium is that $$x = \frac{1}{2}a(8\cos\theta - \cot\theta - 1).$$ [5]
In this question you must show detailed reasoning.
  1. Hence determine, in terms of \(a\), the maximum value of \(x\) for which equilibrium is possible. [5]
END OF QUESTION PAPER