OCR MEI Further Mechanics B AS (Further Mechanics B AS) Specimen

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 \(t = 0\).

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?

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.

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] \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 \(h \mathrm {~m}\).

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.

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] \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.

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] \end{figure}

1. Using the standard model for projectile motion, show that the time of flight, \(T\), is given by $$T = \frac { 28 \sin 40 ^ { \circ } } { g \cos 30 ^ { \circ } }$$
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