Questions Further Mechanics B AS (25 questions)

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OCR MEI Further Mechanics B AS 2019 June Q1
1 A small object of mass 5 kg is attached to one end of each of two identical parallel light elastic strings. The upper ends of both strings are attached to a horizontal ceiling.
The object hangs in equilibrium at R , with the extension of each string being 0.1 m , as shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-2_620_394_580_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the stiffness of each string. One of the strings is now removed and the object initially falls downwards. The object does not return to R at any point in the subsequent motion.
  2. Suggest a reason why the object does not return to \(R\).
OCR MEI Further Mechanics B AS 2019 June Q2
2 A particle P of mass \(m\) travels in a straight line on a smooth horizontal surface.
At time \(t , \mathrm { P }\) is a distance \(x\) from a fixed point O and is moving with speed \(v\) away from O . A horizontal force of magnitude \(3 m t\) acts on P , in a direction away from O .
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = 3 t\).
  2. Verify that the general solution of this differential equation is \(x = \frac { 1 } { 2 } t ^ { 3 } + A t + k\), where \(A\) and \(k\) are constants.
  3. Given that \(x = 6\) and \(v = 12\) when \(t = 1\), find the values of \(A\) and \(k\).
OCR MEI Further Mechanics B AS 2019 June Q3
3 A particle Q of mass \(m\) moves in a horizontal plane under the action of a single force \(\mathbf { F }\). At time \(t , \mathrm { Q }\) has velocity \(\binom { 2 } { 3 t - 2 }\).
  1. Find an expression for \(\mathbf { F }\) in terms of \(m\). At time \(t\), the displacement of Q is given by \(\mathbf { r } = \binom { x } { y }\). When \(t = 1 , \mathrm { Q }\) is at the point with position vector \(\binom { 4 } { - 4 }\).
  2. Find the equation of the path of Q , giving your answer in the form \(y = a x ^ { 2 } + b x + c\), where \(a\), \(b\) and \(c\) are constants to be determined.
  3. What can you deduce about the path of Q from the value of the constant \(c\) you found in part (b)?
OCR MEI Further Mechanics B AS 2019 June Q4
4 Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg , collide with smooth contact while moving on a smooth horizontal surface.
Immediately before the collision, A is moving with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) with the line of centres, where \(\sin \alpha = 0.6\), and B is moving with speed \(0.3 \mathrm {~ms} ^ { - 1 }\) at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B , perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-3_725_1131_1361_242} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Find the speeds of A and B immediately after the collision.
  2. Explain why there could be a second collision between A and B if B rebounds from the wall with sufficient speed.
  3. Find the range of values of the coefficient of restitution between B and the wall for which there will be a second collision between A and B .
  4. How does your answer to part (b) change if the contact between B and the wall is not smooth?
OCR MEI Further Mechanics B AS 2019 June Q5
5 Fig. 5 shows the curve with equation \(y = - x ^ { 2 } + 4 x + 2\).
The curve intersects the \(x\)-axis at P and Q . The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 4\) is occupied by a uniform lamina L . The horizontal base of L is OA , where A is the point \(( 4,0 )\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-4_533_930_466_242} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
    1. Explain why the centre of mass of L lies on the line \(x = 2\).
    2. In this question you must show detailed reasoning. Find the \(y\)-coordinate of the centre of mass of \(L\).
  2. L is freely suspended from A . Find the angle AO makes with the vertical. The region bounded by the curve and the \(x\)-axis is now occupied by a uniform lamina M . The horizontal base of M is PQ.
  3. Explain how the position of the centre of mass of M differs from the position of the centre of mass of \(L\).
OCR MEI Further Mechanics B AS 2019 June Q6
6 A smooth solid hemisphere of radius \(a\) is fixed with its plane face in contact with a horizontal surface.
The highest point on the hemisphere is H , and the centre of its base is O . A particle of mass \(m\) is held at a point S on the surface of the hemisphere such that angle HOS is \(30 ^ { \circ }\), as shown in Fig. 6. The particle is projected from S with speed \(0.8 \sqrt { a g }\) along the surface of the hemisphere towards H . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4acb019b-e630-4766-9d7f-39bc0e174ba1-5_358_1056_497_244} \captionsetup{labelformat=empty} \caption{Fig. 6}
\end{figure}
  1. Show that the particle passes through H without leaving the surface of the hemisphere. After passing through H , the particle passes through a point Q on the surface of the hemisphere, where angle \(\mathrm { HOQ } = \theta ^ { \circ }\).
  2. State, in terms of \(g\) and \(\theta\), the tangential component of the acceleration of the particle when it is at Q . The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L .
  3. Find the value of \(\cos \theta\) correct to 3 significant figures.
  4. Show that \(\mathrm { OL } = k a\), where \(k\) is to be found correct to 3 significant figures.
OCR MEI Further Mechanics B AS 2022 June Q1
1 A small smooth ring of mass 0.5 kg is travelling round a smooth circular wire, with centre O and radius 0.8 m . The circle of wire is in a horizontal plane. The speed of the ring, \(v \mathrm {~ms} ^ { - 1 }\), at time \(t \mathrm {~s}\) after passing through a point A on the wire is given by \(\mathrm { v } = 0.2 \mathrm { t } ^ { 2 } + 0.4 \mathrm { t } + 0.1\).
  1. Find the angular speed of the ring 5 seconds after it passes through A .
  2. Find the distance the ring travels along the wire in the first second after passing through A . At time \(T\) s after the ring passes through A the magnitude of the force exerted on the ring by the wire is 6.4 N . You may assume that any forces acting on the ring other than the force exerted on the ring by the wire and gravity can be ignored.
    1. Determine the value of \(T\).
    2. Hence find the tangential acceleration of the ring at this time.
OCR MEI Further Mechanics B AS 2022 June Q2
2 A light elastic string has natural length \(a\) and modulus of elasticity \(k m g\), where \(k > 2\). One end of the string is attached to a fixed point O . A particle P of mass \(m\) is attached to the other end of the string. P is held at rest a distance \(\frac { 3 } { 2 } a\) vertically below O . At time \(t\) after P is released, its vertical distance below O is \(y\).
  1. Show that, while the string is in tension, the equation of motion of P is given by the differential equation \(\frac { d ^ { 2 } y } { d t ^ { 2 } } = ( k + 1 ) g - \frac { k g } { a } y\). A student transforms the differential equation in part (a) into the standard SHM equation \(\frac { d ^ { 2 } x } { d t ^ { 2 } } = - \omega ^ { 2 } x\).
  2. - Find an expression for \(x\) in terms of \(y , k\) and \(a\).
    • Find an expression for \(\omega\) in terms of \(k , a\) and \(g\).
OCR MEI Further Mechanics B AS 2022 June Q3
3 Fig. 3.1 shows the curve with equation \(y = x ^ { 2 } + 3\). The region \(R\), shown shaded, is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\). A uniform solid of revolution S is formed by rotating the region R through \(2 \pi\) about the \(x\)-axis. The volume of \(S\) is \(\frac { 202 } { 5 } \pi\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{feb9a438-26b0-41d3-b044-6acd6efccde0-3_392_547_511_246} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure} \section*{(a) In this question you must show detailed reasoning.} Show that the \(x\)-coordinate of the centre of mass of S is \(\frac { 395 } { 303 }\). S is fixed to a cylinder of base radius 3 units and height 2 units to form the uniform solid D . The smaller circular face of S is joined to the top circular face of the cylinder, as shown in Fig. 3.2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{feb9a438-26b0-41d3-b044-6acd6efccde0-3_394_556_1491_244} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure} (b) Find the distance of the centre of mass of D from its smaller circular face. D is placed with its smaller circular face in contact with a rough plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is given that D does not slip.
(c) Determine whether D topples.
OCR MEI Further Mechanics B AS 2022 June Q4
4 A plane is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. A particle is projected from a point A on the plane with speed \(V \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(\phi ^ { \circ }\) with a line of greatest slope of the plane. The particle lands at a point B on the plane, as shown in the diagram, and the time of flight is \(T\) seconds.
\includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-4_332_872_461_246}
  1. By considering the motion of the particle perpendicular to the plane, show that \(\mathrm { T } = \frac { 2 \mathrm {~V} \sin \phi } { \mathrm {~g} \cos \theta }\). Consider the case when \(\theta = 30 , \phi = 25\) and \(V = 20\).
    1. Calculate the distance AB .
    2. State, with reasons but without any detailed calculations, what effect each of the following actions would have on the distance AB .
      • Increasing \(V\) while leaving \(\theta\) and \(\phi\) unchanged.
  3. Increasing \(\phi\) while leaving \(\theta\) and \(V\) unchanged.
OCR MEI Further Mechanics B AS 2022 June Q5
5 Two small uniform discs, A of mass \(2 m \mathrm {~kg}\) and B of mass \(3 m \mathrm {~kg}\), slide on a smooth horizontal surface and collide obliquely with smooth contact. Immediately before the collision, A is moving towards B along the line of centres with speed \(2 \mathrm {~ms} ^ { - 1 }\) and B is moving towards A with speed \(\sqrt { 3 } \mathrm {~ms} ^ { - 1 }\) in a direction making an angle of \(30 ^ { \circ }\) with the line of centres, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-5_366_976_539_244}
  1. Explain how you know that the motion of A will be along the line of centres after the collision.
  2. - Determine the maximum possible speed of A after the collision.
    • Find the value of the coefficient of restitution in this case.
    • - Determine the minimum possible speed of B after the collision.
    • Find the value of the coefficient of restitution in this case.
    When the speed of B after the collision is a minimum, the loss of kinetic energy in the collision is 1.4625 J .
  3. Determine the value of \(m\).
OCR MEI Further Mechanics B AS 2022 June Q6
6 Two identical light elastic strings, each of length \(l\) and modulus of elasticity \(\lambda m g\) are attached to a particle \(P\) of mass \(m\). The other end of the first string is attached to a fixed point A , and the other end of the second string is attached to a fixed point B . The points A and B are such that A is above and to the right of B and both strings are taut. The string attached to A makes an angle of \(30 ^ { \circ }\) with the vertical, and the string attached to B makes an angle of \(\theta ^ { \circ }\) with the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{feb9a438-26b0-41d3-b044-6acd6efccde0-6_546_533_699_242} The system is in equilibrium in a vertical plane. The extension of the string attached to A is 0.9 l and the extension of the string attached to B is \(0.5 l\).
  1. Explain how you know that APB is not a straight line.
  2. Show that the elastic potential energy stored in string AP is \(k m g l\), where the value of \(k\) is to be determined correct to \(\mathbf { 3 }\) significant figures.
OCR MEI Further Mechanics B AS 2021 November Q1
1 The end O of a light elastic string OA is attached to a fixed point.
Fiona attaches a mass of 1 kg to the string at A . The system hangs vertically in equilibrium and the length of the stretched string is 70 cm . Fiona removes the 1 kg mass and attaches a mass of 2 kg to the string at A . The system hangs vertically in equilibrium and the length of the stretched string is now 80 cm . Fiona then removes the 2 kg mass and attaches a mass of 5 kg to the string at A . The system hangs vertically in equilibrium.
  1. Use the information given in the question to determine expected values for
    • the length of the stretched string when the 5 kg mass is attached,
    • the elastic potential energy stored in the string in this case.
    Fiona discovers that, when the mass of 5 kg is attached to the string at A , the length of the stretched string is greater than the expected length.
  2. Suggest a reason why this has happened.
OCR MEI Further Mechanics B AS 2021 November Q2
3 marks
2 A particle, Q , moves so that its velocity, \(\mathbf { v }\), at time \(t\) is given by \(\mathbf { v } = ( 6 t - 6 ) \mathbf { i } + \left( 3 - 2 t + t ^ { 2 } \right) \mathbf { j } + 4 \mathbf { k }\), where \(0 \leqslant t \leqslant 6\).
  1. Explain how you know that Q is never stationary. When Q is at a point A the direction of the acceleration of Q is parallel to the \(\mathbf { i }\) direction. When Q is at a point B the direction of the acceleration of Q makes an angle of \(45 ^ { \circ }\) with the \(\mathbf { i }\) direction.
  2. Determine the straight-line distance AB .
OCR MEI Further Mechanics B AS 2021 November Q4
4 Two small smooth discs, A of mass 0.5 kg and B of mass 0.4 kg , collide while sliding on a smooth horizontal plane. Immediately before the collision A and B are moving towards each other, A with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Before the collision the direction of motion of A makes an angle \(\alpha\) with the line of centres, where \(\tan \alpha = 0.75\), and the direction of motion of B makes an angle of \(60 ^ { \circ }\) with the line of centres, as shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-4_506_938_687_244} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} After the collision, one of the discs moves in a direction perpendicular to the line of centres, and the other disc moves in a direction making an angle \(\beta\) with the line of centres.
  1. Explain why the disc which moves perpendicular to the line of centres must be A .
  2. Determine the value of \(\beta\).
  3. Determine the kinetic energy lost in the collision.
  4. Determine the value of the coefficient of restitution between A and B .
OCR MEI Further Mechanics B AS 2021 November Q5
5 On a fairground ride, the centre of a horizontal circular frame is attached to the top of a vertical pole, OP . When the frame and pole rotate, OP remains vertical and the frame remains horizontal. Chairs of mass 10 kg are attached to the frame by means of chains of length 2.5 m . The chains are modelled as being both light and inextensible. A side view of the situation when the ride is stationary is shown in Fig. 5. A chain fixed to point A on the circular frame supports a chair. The distance OA is 2 m . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-5_839_1074_641_240} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} A child of mass 40 kg sits in a chair and, after a short time, the ride is rotating at a steady angular speed of \(\omega\) radians per second, with the chain inclined at an angle of \(50 ^ { \circ }\) to the downward vertical. The motion of the child and chair is in a horizontal circle.
  1. Draw a sketch showing the forces acting on the chair when the ride is moving at this angular speed.
  2. - Determine the tension in the chain.
    • Determine the value of \(\omega\).
    On another occasion, a man of mass 90 kg sits in the chair; after a short time, the ride is rotating in a horizontal circle at a steady speed of \(\omega\) radians per second, with the chain inclined at the same angle of \(50 ^ { \circ }\) to the downward vertical.
  3. Without any detailed calculations, explain how your answers to part (b) for the child would compare with those for the man.
  4. Explain why the chain is modelled as light.
  5. State two other modelling assumptions that were used in answering part (b).
OCR MEI Further Mechanics B AS 2021 November Q6
6 A section of a golf practice ground is inclined at \(15 ^ { \circ }\) to the horizontal. A golfer is hitting a ball up and down a line of greatest slope of this section of the practice ground. The golfer hits the ball up the slope, so that the ball initially makes an angle of \(30 ^ { \circ }\) with the slope. The ball first bounces on the slope 50 m from its point of projection.
  1. Determine the initial speed of the ball. The golfer now hits the ball down the slope. The ball initially moves with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the ball initially travels at an angle \(\theta\) above the horizontal, as shown in Fig. 6. The ball first bounces at a point a distance \(L\) m down the slope. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-6_545_791_794_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
    \end{figure}
  2. Show that \(\mathrm { L } = \frac { 800 } { \mathrm {~g} } \left( \frac { \sin \theta \cos \theta } { \cos 15 ^ { \circ } } + \frac { \sin 15 ^ { \circ } \cos ^ { 2 } \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)\). You are given that \(\frac { \mathrm { dL } } { \mathrm { d } \theta } = \frac { 800 } { \mathrm {~g} } \left( \frac { \cos 2 \theta } { \cos 15 ^ { \circ } } - \frac { \sin 15 ^ { \circ } \sin 2 \theta } { \cos ^ { 2 } 15 ^ { \circ } } \right)\).
  3. Determine the value of \(\theta\) for which \(\frac { \mathrm { d } L } { \mathrm {~d} \theta } = 0\).
  4. Hence determine the maximum distance the golfer can hit the ball down the slope.
OCR MEI Further Mechanics B AS Specimen Q1
1 A particle, P , has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds given by \(\mathbf { v } = \left( \begin{array} { c } 6 \left( t ^ { 2 } - 3 t + 2 \right)
2 ( 1 - t )
3 \left( t ^ { 2 } - 1 \right) \end{array} \right)\), where \(0 \leq t \leq 3\).
  1. Show that there is just one time at which P is instantaneously at rest and state this value of \(t\). P has a mass of 5 kg and is acted on by a single force \(\mathbf { F }\) N.
  2. Find \(\mathbf { F }\) when \(t = 2\).
  3. Find an expression for the position, \(\mathbf { r } \mathrm { m }\), of P at time \(t \mathrm {~s}\), given that \(\mathbf { r } = \left( \begin{array} { c } - 5
    2
    6 \end{array} \right)\) when \(t = 0\).
OCR MEI Further Mechanics B AS Specimen Q2
2 A smooth wire is bent to form a circle of radius 2.5 m ; the circle is in a horizontal plane. A small ring of mass 0.2 kg is travelling round the wire.
  1. At one instant the ring is travelling at an angular speed of 120 revolutions per minute.
    (A) Calculate the angular speed in radians per second.
    (B) Calculate the component towards the centre of the circle of the force exerted on the ring by the wire.
  2. Why must the contact between the wire and the ring be smooth if your answer to part (i) ( \(B\) ) is also the total horizontal component of the force exerted on the ring by the wire?
OCR MEI Further Mechanics B AS Specimen Q3
3 A young woman wishes to make a bungee jump. One end of an elastic rope is attached to her safety harness. The other end is attached to the bridge from which she will jump. She calculates that the stretched length of the rope at the bottom of her motion should be 20 m , she knows that her weight is 576 N and the stiffness of the elastic rope is \(90 \mathrm { Nm } ^ { - 1 }\). She has to calculate the unstretched length of rope required to perform the jump safely. She models the situation by assuming the following.
  • The rope is of negligible mass.
  • Air resistance may be neglected.
  • She is a particle.
  • She moves vertically downwards from rest.
  • Her starting point is level with the fixed end of the rope.
  • The length she calculates for the rope does not include any extra for attaching the ends.
    1. (A) Show that the greatest extension of the rope, \(X\), satisfies the equation \(X ^ { 2 } = 256\).
      (B) Hence determine the natural length of rope she needs.
    2. To remain safe she wishes to be sure that, if air resistance is taken into account, the stretched length of the rope of natural length determined in part (i) will not be more than 20 m . Advise her on this point.
OCR MEI Further Mechanics B AS Specimen Q4
4 Two uniform circular discs with the same radius, A of mass 1 kg and B of mass 5.25 kg , slide on a smooth horizontal surface and collide obliquely with smooth contact. Fig. 4 gives information about the velocities of the discs just before and just after the collision.
  • The line XY passes through the centres of the discs at the moment of collision
  • The components parallel and perpendicular to XY of the velocities of A are shown
  • Before the collision, B is at rest and after it is moving at \(2 \mathrm {~ms} ^ { - 1 }\) in the direction XY
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-4_582_1716_721_155} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The coefficient of restitution between the two discs is \(\frac { 2 } { 3 }\).
  1. Find the values of \(U\) and \(u\).
  2. What information in the question tells you that \(v = V\) ? The speed of disc A before the collision is \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the speed of disc A after the collision. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_398_396_397_475} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_399_332_399_945} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-5_305_326_493_1354} \captionsetup{labelformat=empty} \caption{Fig. 5.3}
    \end{figure} Fig. 5.1 shows a vertical light elastic spring. It is fixed to a horizontal table at one end. Fig 5.2 shows the spring with a particle of mass \(m \mathrm {~kg}\) attached to it at the other end. The system is in equilibrium when the spring is compressed by a distance \(h \mathrm {~m}\).
OCR MEI Further Mechanics B AS Specimen Q7
7 A plane is inclined at \(30 ^ { \circ }\) above the horizontal. A particle is projected up the plane from a point C on the plane with a velocity of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(40 ^ { \circ }\) above a line of greatest slope of the plane. The particle hits the plane at D. See Fig. 7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-7_305_766_484_589} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Using the standard model for projectile motion, show that the time of flight, \(T\), is given by $$T = \frac { 28 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
  2. Calculate the distance CD. \section*{END OF QUESTION PAPER} OCR is committed to seeking permission to reproduce all third-party content that it uses in the assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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OCR MEI Further Mechanics B AS Specimen Q5
  1. Find an expression for the stiffness of the spring, \(k \mathrm { Nm } ^ { - 1 }\), in terms of \(m , h\) and \(g\). The particle is pushed down a further distance from the equilibrium position and released from rest. At time \(t\) seconds, the displacement of the particle from the equilibrium position of the system is \(y \mathrm {~m}\) in the downward direction, as shown in Fig. 5.3. You are given that \(| y | \leq h\).
  2. Show that the motion of the particle is modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { g y } { h } = 0\).
  3. Find an expression for the period of the motion of the particle.
  4. Would the model for the motion of the particle be valid for large values of \(m\) ? Justify your answer.
OCR MEI Further Mechanics B AS 2021 November Q3
3 marks
3 In this question you must show detailed reasoning. [In this question you may use the formula: Volume of cone \(= \frac { 1 } { 3 } \times\) base area × height.]
The region between the line \(\mathrm { y } = - 3 \mathrm { x } + 3 \mathrm { a }\), where \(a > 0\), the \(x\)-axis and the \(y\)-axis is rotated about the \(y\)-axis to form a uniform right circular cone C with base radius \(a\).
  1. Show that the centre of mass of C is \(\frac { 3 } { 4 } a\) from its base. The cone C is fixed on top of a uniform cube, B , of edge length \(2 a\), so that there is no overlap. Fig. 3.1 shows a side view of C and B fixed together; Fig. 3.2 shows a view of C and B from above. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_570_323_785_246} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_309_319_982_753} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} The centre of mass of the combined shape lies on the boundary of C and B .
    The density of \(B\) is not equal to the density of \(C\).
  2. Determine the exact value of \(\frac { \text { density of } \mathrm { C } } { \text { density of } \mathrm { B } }\).
    [0pt] [3]
OCR MEI Further Mechanics B AS Specimen Q6
6 In this question you must show detailed reasoning. As shown in Fig. 6.1, the region R is bounded by the lines \(x = 1 , x = 2 , y = 0\) and the curve \(y = 2 x ^ { 2 }\) for \(1 \leq x \leq 2\). A uniform solid of revolution, S , is formed when R is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_725_449_539_751} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
\end{figure}
  1. Show that the volume of S is \(\frac { 124 \pi } { 5 }\).
  2. Show that the distance of the centre of mass of S from the centre of its smaller circular plane surface is \(\frac { 43 } { 62 }\). Fig. 6.2 shows S placed so that its smaller circular plane surface is in contact with a slope inclined at \(\alpha ^ { \circ }\) to the horizontal. S does not slip but is on the point of tipping. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_458_565_2014_694} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
    \end{figure}
  3. Find the value of \(\alpha\), giving your answer in degrees correct to 3 significant figures.