Questions — OCR MEI (4301 questions)

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OCR MEI M3 2012 June Q4
18 marks Challenging +1.2
4
  1. A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = 3 - \sqrt { x }\) for \(0 \leqslant x \leqslant 9\). Find the coordinates of the centre of mass of this lamina.
  2. Fig. 4.1 shows the region bounded by the line \(x = 2\) and the part of the circle \(y ^ { 2 } = 25 - x ^ { 2 }\) for which \(2 \leqslant x \leqslant 5\). This region is rotated about the \(x\)-axis to form a uniform solid of revolution \(S\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_675_659_479_705} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure}
    1. Find the \(x\)-coordinate of the centre of mass of \(S\). The solid \(S\) rests in equilibrium with its curved surface in contact with a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. Fig. 4.2 shows a vertical section containing AB , which is a diameter and also a line of greatest slope of the flat surface of \(S\). This section also contains XY, which is a line of greatest slope of the plane. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_494_560_1615_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
    2. Find the angle \(\theta\) that AB makes with the horizontal.
OCR MEI M3 2013 June Q1
18 marks Standard +0.3
1
  1. A particle P of mass 1.5 kg is connected to a fixed point by a light inextensible string of length 3.2 m . The particle P is moving as a conical pendulum in a horizontal circle at a constant angular speed of \(2.5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
    1. Find the tension in the string.
    2. Find the angle that the string makes with the vertical.
  2. A particle Q of mass \(m\) moves on a smooth horizontal surface, and is connected to a fixed point on the surface by a light elastic string of natural length \(d\) and stiffness \(k\). With the string at its natural length, Q is set in motion with initial speed \(u\) perpendicular to the string. In the subsequent motion, the maximum length of the string is \(2 d\), and the string first returns to its natural length after time \(t\). You are given that \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) and \(t = A k ^ { \alpha } d ^ { \beta } m ^ { \gamma }\), where \(A\) is a dimensionless constant.
    1. Show that the dimensions of \(k\) are \(\mathrm { MT } ^ { - 2 }\).
    2. Show that the equation \(u = \sqrt { \frac { 4 k d ^ { 2 } } { 3 m } }\) is dimensionally consistent.
    3. Find \(\alpha , \beta\) and \(\gamma\). You are now given that Q has mass 5 kg , and the string has natural length 0.7 m and stiffness \(60 \mathrm { Nm } ^ { - 1 }\).
    4. Find the initial speed \(u\), and use conservation of energy to find the speed of Q at the instant when the length of the string is double its natural length.
OCR MEI M3 2013 June Q2
18 marks Standard +0.3
2 A particle P of mass 0.25 kg is connected to a fixed point O by a light inextensible string of length \(a\) metres, and is moving in a vertical circle with centre O and radius \(a\) metres. When P is vertically below O , its speed is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When OP makes an angle \(\theta\) with the downward vertical, and the string is still taut, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the tension in the string is \(T \mathrm {~N}\), as shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-3_483_551_447_749} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Find an expression for \(v ^ { 2 }\) in terms of \(a\) and \(\theta\), and show that $$T = \frac { 17.64 } { a } + 7.35 \cos \theta - 4.9 .$$
  2. Given that \(a = 0.9\), show that P moves in a complete circle. Find the maximum and minimum magnitudes of the tension in the string.
  3. Find the largest value of \(a\) for which P moves in a complete circle.
  4. Given that \(a = 1.6\), find the speed of P at the instant when the string first becomes slack.
OCR MEI M3 2013 June Q3
18 marks Standard +0.8
3 A light spring, with modulus of elasticity 686 N , has one end attached to a fixed point A . The other end is attached to a particle P of mass 18 kg which hangs in equilibrium when it is 2.2 m vertically below A .
  1. Find the natural length of the spring AP . Another light spring has natural length 2.5 m and modulus of elasticity 145 N . One end of this spring is now attached to the particle P , and the other end is attached to a fixed point B which is 2.5 m vertically below P (so leaving the equilibrium position of P unchanged). While in its equilibrium position, P is set in motion with initial velocity \(3.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards, as shown in Fig. 3. It now executes simple harmonic motion along part of the vertical line AB . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-4_721_383_726_831} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} At time \(t\) seconds after it is set in motion, P is \(x\) metres below its equilibrium position.
  2. Show that the tension in the spring AP is \(( 176.4 + 392 x ) \mathrm { N }\), and write down an expression for the thrust in the spring BP.
  3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 25 x\).
  4. Find the period and the amplitude of the motion.
  5. Find the magnitude and direction of the velocity of P when \(t = 2.4\).
  6. Find the total distance travelled by P during the first 2.4 seconds of its motion.
OCR MEI M3 2013 June Q4
18 marks Challenging +1.2
4
  1. A uniform solid of revolution \(S\) is formed by rotating the region enclosed between the \(x\)-axis and the curve \(y = x \sqrt { 4 - x }\) for \(0 \leqslant x \leqslant 4\) through \(2 \pi\) radians about the \(x\)-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point \(( 4,0 )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure}
    1. Find the \(x\)-coordinate of the centre of mass of the solid \(S\). The solid \(S\) has weight \(W\), and it is freely hinged to a fixed point at O . A horizontal force, of magnitude \(W\) acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. \(S\) is in equilibrium. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
    2. Find the angle that OA makes with the vertical.
      [0pt] [Question 4(b) is printed overleaf]
  2. Fig. 4.3 shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\) and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leqslant y \leqslant 8\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
    \end{figure} Find the coordinates of the centre of mass of a uniform lamina occupying this region.
OCR MEI M3 2014 June Q1
18 marks Standard +0.8
1
  1. The speed \(v\) of sound in a solid material is given by \(v = \sqrt { \frac { E } { \rho } }\), where \(E\) is Young's modulus for the material and \(\rho\) is its density.
    1. Find the dimensions of Young's modulus. The density of steel is \(7800 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\) and the speed of sound in steel is \(6100 \mathrm {~ms} ^ { - 1 }\).
    2. Find Young's modulus for steel, stating the units in which your answer is measured. A tuning fork has cylindrical prongs of radius \(r\) and length \(l\). The frequency \(f\) at which the tuning fork vibrates is given by \(f = k c ^ { \alpha } E ^ { \beta } \rho ^ { \gamma }\), where \(c = \frac { l ^ { 2 } } { r }\) and \(k\) is a dimensionless constant.
    3. Find \(\alpha , \beta\) and \(\gamma\).
  2. A particle P is performing simple harmonic motion along a straight line, and the centre of the oscillations is O . The points X and Y on the line are on the same side of O , at distances 3.9 m and 6.0 m from O respectively. The speed of P is \(1.04 \mathrm {~ms} ^ { - 1 }\) when it passes through X and \(0.5 \mathrm {~ms} ^ { - 1 }\) when it passes through Y.
    1. Find the amplitude and the period of the oscillations.
    2. Find the time taken for P to travel directly from X to Y .
OCR MEI M3 2014 June Q2
19 marks Standard +0.3
2
  1. The fixed point A is vertically above the fixed point B . A light inextensible string of length 5.4 m has one end attached to A and the other end attached to B. The string passes through a small smooth ring R of mass 0.24 kg , and R is moving at constant angular speed in a horizontal circle. The circle has radius 1.6 m , and \(\mathrm { AR } = 3.4 \mathrm {~m} , \mathrm { RB } = 2.0 \mathrm {~m}\), as shown in Fig. 2 . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-3_565_504_447_753} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. Find the tension in the string.
    2. Find the angular speed of R .
  2. A particle P of mass 0.3 kg is joined to a fixed point O by a light inextensible string of length 1.8 m . The particle P moves without resistance in part of a vertical circle with centre O and radius 1.8 m . When OP makes an angle of \(25 ^ { \circ }\) with the downward vertical, the tension in the string is 15 N .
    1. Find the speed of P when OP makes an angle of \(25 ^ { \circ }\) with the downward vertical.
    2. Find the tension in the string when OP makes an angle of \(60 ^ { \circ }\) with the upward vertical.
    3. Find the speed of P at the instant when the string becomes slack.
OCR MEI M3 2014 June Q3
17 marks Standard +0.8
3 The fixed points A and B lie on a line of greatest slope of a smooth inclined plane, with B higher than A . The horizontal distance from A to B is 2.4 m and the vertical distance is 0.7 m . The fixed point C is 2.5 m vertically above B . A light elastic string of natural length 2.2 m has one end attached to C and the other end attached to a small block of mass 9 kg which is in contact with the plane. The block is in equilibrium when it is at A, as shown in Fig. 3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-4_712_641_488_687} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that the modulus of elasticity of the string is 37.73 N . The block starts at A and is at rest. A constant force of 18 N , acting in the direction AB , is then applied to the block so that it slides along the line AB .
  2. Find the magnitude and direction of the acceleration of the block
    (A) when it leaves the point A ,
    (B) when it reaches the point B .
  3. Find the speed of the block when it reaches the point B .
OCR MEI M3 2014 June Q4
18 marks Challenging +1.3
4 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { - x }\) and the line \(x = k\), where \(k\) is a positive constant.
  1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid, and show that it can be written in the form $$\frac { 1 } { 2 } - \frac { k } { \mathrm { e } ^ { 2 k } - 1 } .$$
  2. The solid in part (i) is placed with its larger circular face in contact with a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4, and you are given that no slipping occurs. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5a0df44f-f8f0-44d4-b2f6-70a5314706f9-5_508_483_712_790} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} Show that, whatever the value of \(k\), the solid will not topple.
  3. A uniform lamina occupies the region \(R\). Find, in terms of \(k\), the coordinates of the centre of mass of this lamina. \section*{END OF QUESTION PAPER}
OCR MEI M3 2015 June Q2
18 marks Challenging +1.2
2
  1. A particle P of mass \(m\) is attached to a fixed point O by a light inextensible string of length \(a\). P is moving without resistance in a complete vertical circle with centre O and radius \(a\). When P is at the highest point of the circle, the tension in the string is \(T _ { 1 }\). When OP makes an angle \(\theta\) with the upward vertical, the tension in the string is \(T _ { 2 }\). Show that $$T _ { 2 } = T _ { 1 } + 3 m g ( 1 - \cos \theta ) .$$
  2. The fixed point A is 1.2 m vertically above the fixed point C . A particle Q of mass 0.9 kg is joined to A , to C , and to a particle R of mass 1.5 kg , by three light inextensible strings of lengths \(1.3 \mathrm {~m} , 0.5 \mathrm {~m}\) and 1.8 m respectively. The particle Q moves in a horizontal circle with centre C , and R moves in a horizontal circle at the same constant angular speed as Q , in such a way that \(\mathrm { A } , \mathrm { C } , \mathrm { Q }\) and R are always coplanar. The string QR makes an angle of \(60 ^ { \circ }\) with the downward vertical. This situation is shown in Fig. 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-3_579_1191_881_406} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. Find the tensions in the strings QR and AQ .
    2. Find the angular speed of the system.
    3. Find the tension in the string CQ .
OCR MEI M3 2015 June Q3
18 marks Challenging +1.2
3 Fig. 3 shows the fixed points A and F which are 9.5 m apart on a smooth horizontal surface and points B and D on the line AF such that \(\mathrm { AB } = \mathrm { DF } = 3.0 \mathrm {~m}\). A small block of mass 10.5 kg is joined to A by a light elastic string of natural length 3.0 m and stiffness \(12 \mathrm { Nm } ^ { - 1 }\); the block is joined to F by a light elastic string of natural length 3.0 m and stiffness \(30 \mathrm { Nm } ^ { - 1 }\). The block is released from rest at B and then slides along part of the line AF . The block has zero acceleration when it is at a point C , and it comes to instantaneous rest at a point E . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-4_221_1082_536_502} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Find the distance BC . At time \(t \mathrm {~s}\) the displacement of the block from C is \(x \mathrm {~m}\), measured in the direction AF .
  2. Show that, when the block is between B and \(\mathrm { D } , \frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\).
  3. Find the maximum speed of the block.
  4. Find the distance of the block from C when its speed is \(4.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  5. Find the time taken for the block to travel from B to D.
  6. Find the distance DE .
OCR MEI M3 2015 June Q4
18 marks Challenging +1.2
4
  1. A uniform lamina occupies the region bounded by the \(x\)-axis and the curve \(y = \frac { x ^ { 2 } ( a - x ) } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant a\). Find the coordinates of the centre of mass of this lamina.
  2. The region \(A\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(x = 3\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(y = 5\). These regions are shown in Fig. 4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-5_604_460_605_792} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
    1. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(A\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
    2. Using your answer to part (i), or otherwise, find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. \section*{END OF QUESTION PAPER}
OCR MEI M3 2016 June Q1
19 marks Standard +0.3
1
  1. In an investigation, small spheres are dropped into a long column of a viscous liquid and their terminal speeds measured. It is thought that the terminal speed \(V\) of a sphere depends on a product of powers of its radius \(r\), its weight \(m g\) and the viscosity \(\eta\) of the liquid, and is given by $$V = k r ^ { \alpha } ( m g ) ^ { \beta } \eta ^ { \gamma } ,$$ where \(k\) is a dimensionless constant.
    1. Given that the dimensions of viscosity are \(\mathrm { ML } ^ { - 1 } \mathrm {~T} ^ { - 1 }\) find \(\alpha , \beta\) and \(\gamma\). A sphere of mass 0.03 grams and radius 0.2 cm has a terminal speed of \(6 \mathrm {~ms} ^ { - 1 }\) when falling through a liquid with viscosity \(\eta\). A second sphere of radius 0.25 cm falling through the same liquid has a terminal speed of \(8 \mathrm {~ms} ^ { - 1 }\).
    2. Find the mass of the second sphere.
  2. A manufacturer is testing different types of light elastic ropes to be used in bungee jumping. You may assume that air resistance is negligible. A bungee jumper of mass 80 kg is connected to a fixed point A by one of these elastic ropes. The natural length of this rope is 25 m and its modulus of elasticity is 1600 N . At one instant, the jumper is 30 m directly below A and he is moving vertically upwards at \(15 \mathrm {~ms} ^ { - 1 }\). He comes to instantaneous rest at a point B , with the rope slack.
    1. Find the distance AB . The same bungee jumper now tests a second rope, also of natural length 25 m . He falls from rest at A . It is found that he first comes instantaneously to rest at a distance 54 m directly below A .
    2. Find the modulus of elasticity of this second rope. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_559_705_262_680} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
      \end{figure} The region R shown in Fig. 2.1 is bounded by the curve \(y = k ^ { 2 } - x ^ { 2 }\), for \(0 \leqslant x \leqslant k\), and the coordinate axes. The \(x\)-coordinate of the centre of mass of a uniform lamina occupying the region R is 0.75 .
    3. Show that \(k = 2\). A uniform solid S is formed by rotating the region R through \(2 \pi\) radians about the \(x\)-axis.
    4. Show that the centre of mass of S is at \(( 0.625,0 )\). Fig. 2.2 shows a solid T made by attaching the solid S to the base of a uniform solid circular cone C . The cone \(C\) is made of the same material as \(S\) and has height 8 cm and base radius 4 cm . \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-3_455_794_1521_639} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
      \end{figure}
    5. Show that the centre of mass of T is at a distance of 6.75 cm from the vertex of the cone. [You may quote the standard results that the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) and its centre of mass is \(\frac { 3 } { 4 } h\) from its vertex.]
    6. The solid T is suspended from a point P on the circumference of the base of C . Find the acute angle between the axis of symmetry of T and the vertical. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-4_668_262_255_904} \captionsetup{labelformat=empty} \caption{Fig. 3}
      \end{figure} One end of a light elastic string, of natural length 2.7 m and modulus of elasticity 54 N , is attached to a fixed point L . The other end of the string is attached to a particle P of mass 2.5 kg . One end of a second light elastic string, of natural length 1.7 m and modulus of elasticity 8.5 N , is attached to P . The other end of this second string is attached to a fixed point M , which is 6 m vertically below L . This situation is shown in Fig. 3. The particle P is released from rest when it is 4.2 m below L . Both strings remain taut throughout the subsequent motion. At time \(t \mathrm {~s}\) after P is released from rest, its displacement below L is \(x \mathrm {~m}\).
    7. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 10 ( x - 4 )\).
    8. Write down the value of \(x\) when P is at the centre of its motion.
    9. Find the amplitude and the period of the oscillations.
    10. Find the velocity of P when \(t = 1.2\).
OCR MEI M3 2016 June Q4
18 marks Challenging +1.2
4 A particle P of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point O . Particle P is projected so that it moves in complete vertical circles with centre O ; there is no air resistance. A and B are two points on the circle, situated on opposite sides of the vertical through O . The lines OA and OB make angles \(\alpha\) and \(\beta\) with the upward vertical as shown in Fig. 4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{68cbb8bb-2898-4812-a221-6ea5363b0812-5_414_399_434_833} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} The speed of P at A is \(\sqrt { \frac { 17 a g } { 3 } }\). The speed of P at B is \(\sqrt { 5 a g }\) and \(\cos \beta = \frac { 2 } { 3 }\).
  1. Show that \(\cos \alpha = \frac { 1 } { 3 }\). On one occasion, when P is at its lowest point and moving in a clockwise direction, it collides with a stationary particle Q . The two particles coalesce and the combined particle continues to move in the same vertical circle. When this combined particle reaches the point A , the string becomes slack.
  2. Show that when the string becomes slack, the speed of the combined particle is \(\sqrt { \frac { a g } { 3 } }\). The mass of the particle Q is \(k m\).
  3. Find the value of \(k\).
  4. Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.
OCR MEI M4 Q2
12 marks Challenging +1.8
2 A rigid circular hoop of radius \(a\) is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length \(\frac { 5 } { 2 } a\) is passed over the pulley. A particle of mass \(m\) is attached to the string at \(\mathrm { B } . \mathrm { PB }\) is vertical and angle \(\mathrm { APB } = \theta\). A small smooth ring of mass \(m\) is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5bed3ad4-0e20-4458-a37f-655faf84c31a-02_568_549_1306_758} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Show that \(\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta\) and hence show that the potential energy of the system relative to P is \(V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)\).
  2. Hence find the positions of equilibrium and investigate their stability.
OCR MEI M4 2006 June Q1
12 marks Challenging +1.8
1 A spherical raindrop falls through a stationary cloud. Water condenses on the raindrop and it gains mass at a rate proportional to its surface area. At time \(t\) the radius of the raindrop is \(r\). Initially the raindrop is at rest and \(r = r _ { 0 }\). The density of the water is \(\rho\).
  1. Show that \(\frac { \mathrm { d } r } { \mathrm {~d} t } = k\), where \(k\) is a constant. Hence find the mass of the raindrop in terms of \(r _ { 0 } , \rho , k\) and \(t\).
  2. Assuming that air resistance is negligible, find the velocity of the raindrop in terms of \(r _ { 0 } , k\) and \(t\).
OCR MEI M4 2006 June Q2
12 marks Challenging +1.2
2 A rigid circular hoop of radius \(a\) is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length \(\frac { 5 } { 2 } a\) is passed over the pulley. A particle of mass \(m\) is attached to the string at \(\mathrm { B } . \mathrm { PB }\) is vertical and angle \(\mathrm { APB } = \theta\). A small smooth ring of mass \(m\) is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c97056a9-4156-4ecd-a80e-1a82c81ab824-2_568_549_1306_758} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure}
  1. Show that \(\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta\) and hence show that the potential energy of the system relative to P is \(V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)\).
  2. Hence find the positions of equilibrium and investigate their stability.
OCR MEI M4 2006 June Q3
24 marks Challenging +1.8
3 An aeroplane is taking off from a runway. It starts from rest. The resultant force in the direction of motion has power, \(P\) watts, modelled by $$P = 0.0004 m \left( 10000 v + v ^ { 3 } \right) ,$$ where \(m \mathrm {~kg}\) is the mass of the aeroplane and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity at time \(t\) seconds. The displacement of the aeroplane from its starting point is \(x \mathrm {~m}\). To take off successfully the aeroplane must reach a speed of \(80 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before it has travelled 900 m .
  1. Formulate and solve a differential equation for \(v\) in terms of \(x\). Hence show that the aeroplane takes off successfully.
  2. Formulate a differential equation for \(v\) in terms of \(t\). Solve the differential equation to show that \(v = 100 \tan ( 0.04 t )\). What feature of this result casts doubt on the validity of the model?
  3. In fact the model is only valid for \(0 \leqslant t \leqslant 11\), after which the power remains constant at the value attained at \(t = 11\). Will the aeroplane take off successfully?
OCR MEI M4 2006 June Q4
24 marks Challenging +1.8
4 A flagpole AB of length \(2 a\) is modelled as a thin rigid rod of variable mass per unit length given by $$\rho = \frac { M } { 8 a ^ { 2 } } ( 5 a - x ) ,$$ where \(x\) is the distance from A and \(M\) is the mass of the flagpole.
  1. Show that the moment of inertia of the flagpole about an axis through A and perpendicular to the flagpole is \(\frac { 7 } { 6 } M a ^ { 2 }\). Show also that the centre of mass of the flagpole is at a distance \(\frac { 11 } { 12 } a\) from A . The flagpole is hinged to a wall at A and can rotate freely in a vertical plane. A light inextensible rope of length \(2 \sqrt { 2 } a\) is attached to the end B and the other end is attached to a point on the wall a distance \(2 a\) vertically above A, as shown in Fig. 4. The flagpole is initially at rest when lying vertically against the wall, and then is displaced slightly so that it falls to a horizontal position, at which point the rope becomes taut and the flagpole comes to rest. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c97056a9-4156-4ecd-a80e-1a82c81ab824-4_403_365_1174_849} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
  2. Find an expression for the angular velocity of the flagpole when it has turned through an angle \(\theta\).
  3. Show that the vertical component of the impulse in the rope when it becomes taut is \(\frac { 1 } { 12 } M \sqrt { 77 a g }\). Hence write down the horizontal component.
  4. Find the horizontal and vertical components of the impulse that the hinge exerts on the flagpole when the rope becomes taut. Hence find the angle that this impulse makes with the horizontal.
OCR MEI M4 2007 June Q1
12 marks Challenging +1.2
1 A light elastic string has one end fixed to a vertical pole at A . The string passes round a smooth horizontal peg, P , at a distance \(a\) from the pole and has a smooth ring of mass \(m\) attached at its other end B . The ring is threaded onto the pole below A . The ring is at a distance \(y\) below the horizontal level of the peg. This situation is shown in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-2_462_275_557_897} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} The string has stiffness \(k\) and natural length equal to the distance AP .
  1. Express the extension of the string in terms of \(y\) and \(a\). Hence find the potential energy of the system relative to the level of P .
  2. Use the potential energy to find the equilibrium position of the system, and show that it is stable.
  3. Calculate the normal reaction exerted by the pole on the ring in the equilibrium position.
OCR MEI M4 2007 June Q2
12 marks Challenging +1.2
2 A railway truck of mass \(m _ { 0 }\) travels along a horizontal track. There is no driving force and the resistances to motion are negligible. The truck is being filled with coal which falls vertically into it at a mass rate \(k\). The process starts as the truck passes a point O with speed \(u\). After time \(t\), the truck has velocity \(v\) and the displacement from O is \(x\).
  1. Show that \(v = \frac { m _ { 0 } u } { m _ { 0 } + k t }\) and find \(x\) in terms of \(m _ { 0 } , u , k\) and \(t\).
  2. Find the distance that the truck has travelled when its speed has been halved.
OCR MEI M4 2007 June Q3
24 marks Challenging +1.3
3
  1. Show, by integration, that the moment of inertia of a uniform rod of mass \(m\) and length \(2 a\) about an axis through its centre and perpendicular to the rod is \(\frac { 1 } { 3 } m a ^ { 2 }\). A pendulum of length 1 m is made by attaching a uniform sphere of mass 2 kg and radius 0.1 m to the end of a uniform rod AB of mass 1.2 kg and length 0.8 m , as shown in Fig. 3. The centre of the sphere is collinear with A and B . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8aab7e54-a204-481b-8f09-4bf4ca4e115d-3_442_291_717_886} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure}
  2. Find the moment of inertia of the pendulum about an axis through A perpendicular to the rod. The pendulum can swing freely in a vertical plane about a fixed horizontal axis through A .
  3. The pendulum is held with AB at an angle \(\alpha\) to the downward vertical and released from rest. At time \(t , \mathrm { AB }\) is at an angle \(\theta\) to the vertical. Find an expression for \(\dot { \theta } ^ { 2 }\) in terms of \(\theta\) and \(\alpha\).
  4. Hence, or otherwise, show that, provided that \(\alpha\) is small, the pendulum performs simple harmonic motion. Calculate the period.
OCR MEI M4 2007 June Q4
24 marks Challenging +1.8
4 A particle of mass 2 kg starts from rest at a point O and moves in a horizontal line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) under the action of a force \(F \mathrm {~N}\), where \(F = 2 - 8 v ^ { 2 }\). The displacement of the particle from O at time \(t\) seconds is \(x \mathrm {~m}\).
  1. Formulate and solve a differential equation to show that \(v ^ { 2 } = \frac { 1 } { 4 } \left( 1 - \mathrm { e } ^ { - 8 x } \right)\).
  2. Hence express \(F\) in terms of \(x\) and find, by integration, the work done in the first 2 m of the motion.
  3. Formulate and solve a differential equation to show that \(v = \frac { 1 } { 2 } \left( \frac { 1 - \mathrm { e } ^ { - 4 t } } { 1 + \mathrm { e } ^ { - 4 t } } \right)\).
  4. Calculate \(v\) when \(t = 1\) and when \(t = 2\), giving your answers to four significant figures. Hence find the impulse of the force \(F\) over the interval \(1 \leqslant t \leqslant 2\).
OCR MEI M4 2008 June Q1
12 marks Challenging +1.2
1 A rocket in deep space starts from rest and moves in a straight line. The initial mass of the rocket is \(m _ { 0 }\) and the propulsion system ejects matter at a constant mass rate \(k\) with constant speed \(u\) relative to the rocket. At time \(t\) the speed of the rocket is \(v\).
  1. Show that while mass is being ejected from the rocket, \(\left( m _ { 0 } - k t \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = u k\).
  2. Hence find an expression for \(v\) in terms of \(t\).
  3. Find the speed of the rocket when its mass is \(\frac { 1 } { 3 } m _ { 0 }\).
OCR MEI M4 2008 June Q2
12 marks Challenging +1.2
2 A car of mass \(m \mathrm {~kg}\) starts from rest at a point O and moves along a straight horizontal road. The resultant force in the direction of motion has power \(P\) watts, given by \(P = m \left( k ^ { 2 } - v ^ { 2 } \right)\), where \(v \mathrm {~ms} ^ { - 1 }\) is the velocity of the car and \(k\) is a positive constant. The displacement from O in the direction of motion is \(x \mathrm {~m}\).
  1. Show that \(\left( \frac { k ^ { 2 } } { k ^ { 2 } - v ^ { 2 } } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\), and hence find \(x\) in terms of \(v\) and \(k\).
  2. How far does the car travel before reaching \(90 \%\) of its terminal velocity?