Questions — OCR MEI M3 (71 questions)

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OCR MEI M3 Q2
2
  1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
  2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e0e5580a-e1f0-46f8-9304-2a96533af186-03_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. Express \(v ^ { 2 }\) in terms of \(\theta\).
    2. Show that \(R = 5.88 \sin \theta - 3.43\).
    3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
    4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.
OCR MEI M3 2006 January Q1
1
    1. Write down the dimensions of force. The period, \(t\), of a vibrating wire depends on its tension, \(F\), its length, \(l\), and its mass per unit length, \(\sigma\).
    2. Assuming that the relationship is of the form \(t = k F ^ { \alpha } l ^ { \beta } \sigma ^ { \gamma }\), where \(k\) is a dimensionless constant, use dimensional analysis to determine the values of \(\alpha , \beta\) and \(\gamma\). Two lengths are cut from a reel of uniform wire. The first has length 1.2 m , and it vibrates under a tension of 90 N . The second has length 2.0 m , and it vibrates with the same period as the first wire.
    3. Find the tension in the second wire. (You may assume that changing the tension does not significantly change the mass per unit length.)
  2. The midpoint M of a vibrating wire is moving in simple harmonic motion in a straight line, with amplitude 0.018 m and period 0.01 s .
    1. Find the maximum speed of M .
    2. Find the distance of M from the centre of the motion when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR MEI M3 2006 January Q2
2
  1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
  2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-3_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. Express \(v ^ { 2 }\) in terms of \(\theta\).
    2. Show that \(R = 5.88 \sin \theta - 3.43\).
    3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
    4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.
OCR MEI M3 2006 January Q3
3 A light elastic rope has natural length 15 m . One end of the rope is attached to a fixed point O and the other end is attached to a small rock of mass 12 kg . When the rock is hanging in equilibrium vertically below O , the length of the rope is 15.8 m .
  1. Show that the modulus of elasticity of the rope is 2205 N . The rock is pulled down to the point 20 m vertically below O , and is released from rest in this position. It moves upwards, and comes to rest instantaneously, with the rope slack, at the point A .
  2. Find the acceleration of the rock immediately after it is released.
  3. Use an energy method to find the distance OA. At time \(t\) seconds after release, the rope is still taut and the displacement of the rock below the equilibrium position is \(x\) metres.
  4. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 12.25 x\).
  5. Write down an expression for \(x\) in terms of \(t\), and hence find the time between releasing the rock and the rope becoming slack.
OCR MEI M3 2006 January Q4
4 The region between the curve \(y = 4 - x ^ { 2 }\) and the \(x\)-axis, from \(x = 0\) to \(x = 2\), is occupied by a uniform lamina. The units of the axes are metres.
  1. Show that the coordinates of the centre of mass of this lamina are \(( 0.75,1.6 )\). This lamina and another exactly like it are attached to a uniform rod PQ , of mass 12 kg and length 8 m , to form a rigid body as shown in Fig. 4. Each lamina has mass 6.5 kg . The ends of the rod are at \(\mathrm { P } ( - 4,0 )\) and \(\mathrm { Q } ( 4,0 )\). The rigid body lies entirely in the \(( x , y )\) plane. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-4_511_956_1836_557} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
  2. Find the coordinates of the centre of mass of the rigid body. The rigid body is freely suspended from the point \(\mathrm { A } ( 2,4 )\) and hangs in equilibrium.
  3. Find the angle that PQ makes with the horizontal.
OCR MEI M3 2007 January Q1
1
  1. Write down the dimensions of velocity, acceleration and force. The force \(F\) of gravitational attraction between two objects with masses \(m _ { 1 }\) and \(m _ { 2 }\), at a distance \(r\) apart, is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { r ^ { 2 } }$$ where \(G\) is the universal constant of gravitation.
  2. Show that the dimensions of \(G\) are \(\mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 }\).
  3. In SI units (based on the kilogram, metre and second) the value of \(G\) is \(6.67 \times 10 ^ { - 11 }\). Find the value of \(G\) in imperial units based on the pound \(( 0.4536 \mathrm {~kg} )\), foot \(( 0.3048 \mathrm {~m} )\) and second.
  4. For a planet of mass \(m\) and radius \(r\), the escape velocity \(v\) from the planet's surface is given by $$v = \sqrt { \frac { 2 G m } { r } }$$ Show that this formula is dimensionally consistent.
  5. For a planet in circular orbit of radius \(R\) round a star of mass \(M\), the time \(t\) taken to complete one orbit is given by $$t = k G ^ { \alpha } M ^ { \beta } R ^ { \gamma }$$ where \(k\) is a dimensionless constant.
    Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
OCR MEI M3 2007 January Q2
2
  1. A light inextensible string has length 1.8 m . One end of the string is attached to a fixed point O , and the other end is attached to a particle of mass 5 kg . The particle moves in a complete vertical circle with centre O , so that the string remains taut throughout the motion. Air resistance may be neglected.
    1. Show that, at the highest point of the circle, the speed of the particle is at least \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    2. Find the least possible tension in the string when the particle is at the lowest point of the circle.
  2. Fig. 2 shows a hollow cone mounted with its axis of symmetry vertical and its vertex V pointing downwards. The cone rotates about its axis with a constant angular speed of \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle P of mass 0.02 kg is in contact with the rough inside surface of the cone, and does not slip. The particle P moves in a horizontal circle of radius 0.32 m . The angle between VP and the vertical is \(\theta\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b209dbe7-769c-4301-a2f3-108c27c8cefb-3_588_510_1046_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} In the case when \(\omega = 8.75\), there is no frictional force acting on P .
    1. Show that \(\tan \theta = 0.4\). Now consider the case when \(\omega\) takes a constant value greater than 8.75.
    2. Draw a diagram showing the forces acting on P .
    3. You are given that the coefficient of friction between P and the surface is 0.11 . Find the maximum possible value of \(\omega\) for which the particle does not slip.
OCR MEI M3 2007 January Q3
3 Ben has mass 60 kg and he is considering doing a bungee jump using an elastic rope with natural length 32 m . One end of the rope is attached to a fixed point O , and the other end is attached to Ben. When Ben is supported in equilibrium by the rope, the length of the rope is 32.8 m . To predict what will happen, Ben is modelled as a particle B, the rope is assumed to be light, and air resistance is neglected. B is released from rest at O and falls vertically. When the rope becomes stretched, \(x \mathrm {~m}\) denotes the extension of the rope.
  1. Find the stiffness of the rope.
  2. Use an energy argument to show that, when B comes to rest instantaneously with the rope stretched, $$x ^ { 2 } - 1.6 x - 51.2 = 0$$ Hence find the length of the rope when B is at its lowest point.
  3. Show that, while the rope is stretched, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12.25 x = 9.8$$ where \(t\) is the time measured in seconds.
  4. Find the time taken for B to travel between the equilibrium position \(( x = 0.8 )\) and the lowest point.
  5. Find the acceleration of \(\mathbf { B }\) when it is at the lowest point, and comment on the implications for Ben.
OCR MEI M3 2007 January Q4
4 In this question, \(a\) is a constant with \(a > 1\).
Fig. 4 shows the region bounded by the curve \(y = \frac { 1 } { x ^ { 2 } }\) for \(1 \leqslant x \leqslant a\), the \(x\)-axis, and the lines \(x = 1\) and \(x = a\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b209dbe7-769c-4301-a2f3-108c27c8cefb-5_447_848_543_612} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} This region is occupied by a uniform lamina ABCD , where A is \(( 1,1 ) , \mathrm { B }\) is \(( 1,0 ) , \mathrm { C }\) is \(( a , 0 )\) and D is \(\left( a , \frac { 1 } { a ^ { 2 } } \right)\). The centre of mass of this lamina is \(( \bar { x } , \bar { y } )\).
  1. Find \(\bar { x }\) in terms of \(a\), and show that \(\bar { y } = \frac { a ^ { 3 } - 1 } { 6 \left( a ^ { 3 } - a ^ { 2 } \right) }\).
  2. In the case \(a = 2\), the lamina is freely suspended from the point A , and hangs in equilibrium. Find the angle which AB makes with the vertical. The region shown in Fig. 4 is now rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.
  3. Find the \(x\)-coordinate of the centre of mass of this solid of revolution, in terms of \(a\), and show that it is less than 1.5.
OCR MEI M3 2008 January Q1
1
    1. Write down the dimensions of force and the dimensions of density. When a wire, with natural length \(l _ { 0 }\) and cross-sectional area \(A\), is stretched to a length \(l\), the tension \(F\) in the wire is given by $$F = \frac { E A \left( l - l _ { 0 } \right) } { l _ { 0 } }$$ where \(E\) is Young's modulus for the material from which the wire is made.
    2. Find the dimensions of Young's modulus \(E\). A uniform sphere of radius \(r\) is made from material with density \(\rho\) and Young's modulus \(E\). When the sphere is struck, it vibrates with periodic time \(t\) given by $$t = k r ^ { \alpha } \rho ^ { \beta } E ^ { \gamma }$$ where \(k\) is a dimensionless constant.
    3. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
  2. Fig. 1 shows a fixed point A that is 1.5 m vertically above a point B on a rough horizontal surface. A particle P of mass 5 kg is at rest on the surface at a distance 0.8 m from B , and is connected to A by a light elastic string with natural length 1.5 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-2_405_538_1338_845} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} The coefficient of friction between P and the surface is 0.4 , and P is on the point of sliding. Find the stiffness of the string.
OCR MEI M3 2008 January Q2
2
  1. A small ball of mass 0.01 kg is moving in a vertical circle of radius 0.55 m on the smooth inside surface of a fixed sphere also of radius 0.55 m . When the ball is at the highest point of the circle, the normal reaction between the surface and the ball is 0.1 N . Modelling the ball as a particle and neglecting air resistance, find
    1. the speed of the ball when it is at the highest point of the circle,
    2. the normal reaction between the surface and the ball when the vertical height of the ball above the lowest point of the circle is 0.15 m .
  2. A small object Q of mass 0.8 kg moves in a circular path, with centre O and radius \(r\) metres, on a smooth horizontal surface. A light elastic string, with natural length 2 m and modulus of elasticity 160 N , has one end attached to Q and the other end attached to O . The object Q has a constant angular speed of \(\omega\) rad s \(^ { - 1 }\).
    1. Show that \(\omega ^ { 2 } = \frac { 100 ( r - 2 ) } { r }\) and deduce that \(\omega < 10\).
    2. Find expressions, in terms of \(r\) only, for the elastic energy stored in the string, and for the kinetic energy of Q . Show that the kinetic energy of Q is greater than the elastic energy stored in the string.
    3. Given that the angular speed of Q is \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find the tension in the string.
OCR MEI M3 2008 January Q3
3 A particle is oscillating in a vertical line. At time \(t\) seconds, its displacement above the centre of the oscillations is \(x\) metres, where \(x = A \sin \omega t + B \cos \omega t\) (and \(A , B\) and \(\omega\) are constants).
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \omega ^ { 2 } x\). When \(t = 0\), the particle is 2 m above the centre of the oscillations, the velocity is \(1.44 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, and the acceleration is \(0.18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards.
  2. Find \(A , B\) and \(\omega\).
  3. Show that the period of oscillation is 20.9 s (correct to 3 significant figures), and find the amplitude.
  4. Find the total distance travelled by the particle between \(t = 12\) and \(t = 24\).
OCR MEI M3 2008 January Q4
4 Fig. 4.1 shows the region \(R\) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 8\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-4_597_1018_411_566} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure}
  1. Find the \(x\)-coordinate of the centre of mass of a uniform solid of revolution obtained by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
  2. Find the coordinates of the centre of mass of a uniform lamina in the shape of the region \(R\).
  3. Using your answer to part (ii), or otherwise, find the coordinates of the centre of mass of a uniform lamina in the shape of the region (shown shaded in Fig. 4.2) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the line \(y = \frac { 1 } { 2 }\) and the line \(x = 1\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-4_595_1015_1610_607} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure}
OCR MEI M3 2009 January Q2
2
  1. Fig. 2 shows a light inextensible string of length 3.3 m passing through a small smooth ring R of mass 0.27 kg . The ends of the string are attached to fixed points A and B , where A is vertically above \(B\). The ring \(R\) is moving with constant speed in a horizontal circle of radius \(1.2 \mathrm {~m} , \mathrm { AR } = 2.0 \mathrm {~m}\) and \(\mathrm { BR } = 1.3 \mathrm {~m}\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-3_570_659_493_781} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. Show that the tension in the string is 6.37 N .
    2. Find the speed of R .
  2. One end of a light inextensible string of length 1.25 m is attached to a fixed point O . The other end is attached to a particle P of mass 0.2 kg . The particle P is moving in a vertical circle with centre O and radius 1.25 m , and when P is at the highest point of the circle there is no tension in the string.
    1. Show that when P is at the highest point its speed is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the instant when the string OP makes an angle of \(60 ^ { \circ }\) with the upward vertical, find
    2. the radial and tangential components of the acceleration of P ,
    3. the tension in the string.
OCR MEI M3 2009 January Q3
3 An elastic rope has natural length 25 m and modulus of elasticity 980 N . One end of the rope is attached to a fixed point O , and a rock of mass 5 kg is attached to the other end; the rock is always vertically below O.
  1. Find the extension of the rope when the rock is hanging in equilibrium. When the rock is moving with the rope stretched, its displacement is \(x\) metres below the equilibrium position at time \(t\) seconds.
  2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 7.84 x\). The rock is released from a position where the rope is slack, and when the rope just becomes taut the speed of the rock is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the distance below the equilibrium position at which the rock first comes instantaneously to rest.
  4. Find the maximum speed of the rock.
  5. Find the time between the rope becoming taut and the rock first coming to rest.
  6. State three modelling assumptions you have made in answering this question.
OCR MEI M3 2009 January Q4
4
  1. The region bounded by the \(x\)-axis and the semicircle \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(- a \leqslant x \leqslant a\) is occupied by a uniform lamina with area \(\frac { 1 } { 2 } \pi a ^ { 2 }\). Show by integration that the \(y\)-coordinate of the centre of mass of this lamina is \(\frac { 4 a } { 3 \pi }\).
  2. A uniform solid cone is formed by rotating the region between the \(x\)-axis and the line \(y = m x\), for \(0 \leqslant x \leqslant h\), through \(2 \pi\) radians about the \(x\)-axis.
    1. Show that the \(x\)-coordinate of the centre of mass of this cone is \(\frac { 3 } { 4 } h\).
      [0pt] [You may use the formula \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]
      From such a uniform solid cone with radius 0.7 m and height 2.4 m , a cone of material is removed. The cone removed has radius 0.4 m and height 1.1 m ; the centre of its base coincides with the centre of the base of the original cone, and its axis of symmetry is also the axis of symmetry of the original cone. Fig. 4 shows the resulting object; the vertex of the original cone is V, and A is a point on the circumference of its base. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-5_716_1228_1027_497} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
    2. Find the distance of the centre of mass of this object from V . This object is suspended by a string attached to a point Q on the line VA, and hangs in equilibrium with VA horizontal.
    3. Find the distance VQ.
OCR MEI M3 2010 January Q1
1
    1. Write down the dimensions of density, kinetic energy and power. A sphere of radius \(r\) is moved at constant velocity \(v\) through a fluid.
    2. In a viscous fluid, the power required is \(6 \pi \eta r v ^ { 2 }\), where \(\eta\) is the viscosity of the fluid. Find the dimensions of viscosity.
    3. In a non-viscous fluid, the power required is \(k \rho ^ { \alpha } r ^ { \beta } v ^ { \gamma }\), where \(\rho\) is the density of the fluid and \(k\) is a dimensionless constant. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
  2. A rock of mass 5.5 kg is connected to a fixed point O by a light elastic rope with natural length 1.2 m . The rock is released from rest in a position 2 m vertically below O , and it next comes to instantaneous rest when it is 1.5 m vertically above O . Find the stiffness of the rope.
OCR MEI M3 2010 January Q2
2
  1. A uniform solid hemisphere of volume \(\frac { 2 } { 3 } \pi a ^ { 3 }\) is formed by rotating the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), through \(2 \pi\) radians about the \(x\)-axis. Show that the \(x\)-coordinate of the centre of mass of the hemisphere is \(\frac { 3 } { 8 } a\).
  2. A uniform lamina is bounded by the \(x\)-axis, the line \(x = 1\), and the curve \(y = 2 - \sqrt { x }\) for \(1 \leqslant x \leqslant 4\). Its corners are \(\mathrm { A } ( 1,1 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 4,0 )\).
    1. Find the coordinates of the centre of mass of the lamina. The lamina is suspended with AB vertical and BC horizontal by light vertical strings attached to A and C , as shown in Fig. 2. The weight of the lamina is \(W\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-2_346_684_1672_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure}
    2. Find the tensions in the two strings in terms of \(W\).
OCR MEI M3 2010 January Q3
3 A particle P of mass 0.6 kg is connected to a fixed point O by a light inextensible string of length 1.25 m . When it is 1.25 m vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with horizontal velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves in part of a vertical circle with centre O and radius 1.25 m . When OP makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 3.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-3_602_627_484_758} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Show that \(v ^ { 2 } = 11.5 + 24.5 \cos \theta\).
  2. Find the tension in the string in terms of \(\theta\).
  3. Find the speed of P at the instant when the string becomes slack. A second light inextensible string, of length 0.35 m , is attached to P , and the other end of this string is attached to a point C which is 1.2 m vertically below O . The particle P now moves in a horizontal circle with centre C and radius 0.35 m , as shown in Fig. 3.2. The speed of P is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-3_518_488_1701_826} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  4. Find the tension in the string OP and the tension in the string CP.
OCR MEI M3 2010 January Q4
4 Fig. 4 shows a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Two fixed points A and B on the plane are 4.55 m apart with B higher than A on a line of greatest slope. A particle P of mass 0.25 kg is in contact with the plane and is connected to A and to B by two light elastic strings. The string AP has natural length 1.5 m and modulus of elasticity 7.35 N ; the string BP has natural length 2.5 m and modulus of elasticity 7.35 N . The particle P moves along part of the line AB , with both strings taut throughout the motion. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-4_598_1006_568_571} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Show that, when \(\mathrm { AP } = 1.55 \mathrm {~m}\), the acceleration of P is zero.
  2. Taking \(\mathrm { AP } = ( 1.55 + x ) \mathrm { m }\), write down the tension in the string AP , in terms of \(x\), and show that the tension in the string BP is \(( 1.47 - 2.94 x ) \mathrm { N }\).
  3. Show that the motion of P is simple harmonic, and find its period. The particle P is released from rest with \(\mathrm { AP } = 1.5 \mathrm {~m}\).
  4. Find the time after release when P is first moving down the plane with speed \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
OCR MEI M3 2011 January Q2
2
  1. A particle P , of mass 48 kg , is moving in a horizontal circle of radius 8.4 m at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), in contact with a smooth horizontal surface. A light inextensible rope of length 30 m connects P to a fixed point A which is vertically above the centre C of the circle, as shown in Fig. 2.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_526_490_482_870} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
    \end{figure}
    1. Given that \(V = 3.5\), find the tension in the rope and the normal reaction of the surface on P .
    2. Calculate the value of \(V\) for which the normal reaction is zero.
  2. The particle P , of mass 48 kg , is now placed on the highest point of a fixed solid sphere with centre O and radius 2.5 m . The surface of the sphere is smooth. The particle P is given an initial horizontal velocity of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it then moves in part of a vertical circle with centre O and radius 2.5 m . When OP makes an angle \(\theta\) with the upward vertical and P is still in contact with the surface of the sphere, P has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction of the sphere on P is \(R \mathrm {~N}\), as shown in Fig. 2.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-3_590_617_1706_804} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure}
    1. Show that \(v ^ { 2 } = u ^ { 2 } + 49 - 49 \cos \theta\).
    2. Find an expression for \(R\) in terms of \(u\) and \(v\).
    3. Given that P loses contact with the surface of the sphere at the instant when its speed is \(4.15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(u\).
OCR MEI M3 2011 January Q3
3 A block of mass 200 kg is connected to a horizontal ceiling by four identical light elastic ropes, each having natural length 7 m and stiffness \(180 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). It is also connected to the floor by a single light elastic rope having stiffness \(80 \mathrm { Nm } ^ { - 1 }\). Throughout this question you may assume that all five ropes are stretched and vertical, and you may neglect air resistance. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-4_665_623_482_760} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Fig. 3 shows the block resting in equilibrium, with each of the top ropes having length 10 m and the bottom rope having length 8 m .
  1. Find the tension in one of the top ropes.
  2. Find the natural length of the bottom rope. The block now moves vertically up and down. At time \(t\) seconds, the block is \(x\) metres below its equilibrium position.
  3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\). The motion is started by pulling the block down 2.2 m below its equilibrium position and releasing it from rest. The block then executes simple harmonic motion with amplitude 2.2 m .
  4. Find the maximum magnitude of the acceleration of the block.
  5. Find the speed of the block when it has travelled 3.8 m from its starting point.
  6. Find the distance travelled by the block in the first 5 s .
OCR MEI M3 2011 January Q4
4
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_705_501_260_863} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure} The region \(R\), shown in Fig. 4.1, is bounded by the curve \(x ^ { 2 } - y ^ { 2 } = k ^ { 2 }\) for \(k \leqslant x \leqslant 4 k\) and the line \(x = 4 k\), where \(k\) is a positive constant. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when \(R\) is rotated about the \(x\)-axis.
  2. A uniform lamina occupies the region bounded by the curve \(y = \frac { x ^ { 3 } } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant 2 a\), the \(x\)-axis and the line \(x = 2 a\), where \(a\) is a positive constant. The vertices of the lamina are \(\mathrm { O } ( 0,0 ) , \mathrm { A } ( 2 a , 8 a )\) and \(\mathrm { B } ( 2 a , 0 )\), as shown in Fig. 4.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_714_509_1546_858} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure}
    1. Find the coordinates of the centre of mass of the lamina.
    2. The lamina is freely suspended from the point A and hangs in equilibrium. Find the angle that AB makes with the vertical.
OCR MEI M3 2012 January Q1
1 The surface tension of a liquid enables a metal needle to be at rest on the surface of the liquid. The greatest mass \(m\) of a needle of length \(a\) which can be supported in this way by a liquid of surface tension \(S\) is given by $$m = \frac { 2 S a } { g }$$ where \(g\) is the acceleration due to gravity.
  1. Show that the dimensions of surface tension are \(\mathrm { MT } ^ { - 2 }\). The surface tension of water is 0.073 when expressed in SI units (based on kilograms, metres and seconds).
  2. Find the surface tension of water when expressed in a system of units based on grams, centimetres and minutes. Liquid will rise up a capillary tube to a height \(h\) given by \(h = \frac { 2 S } { \rho g r }\), where \(\rho\) is the density of the liquid and
    \(r\) is the radius of the capillary tube. \(r\) is the radius of the capillary tube.
  3. Show that the equation \(h = \frac { 2 S } { \rho g r }\) is dimensionally consistent.
  4. Find the radius of a capillary tube in which water will rise to a height of 25 cm . (The density of water is 1000 in SI units.) When liquid is poured onto a horizontal surface, it forms puddles of depth \(d\). You are given that \(d = k S ^ { \alpha } \rho ^ { \beta } g ^ { \gamma }\) where \(k\) is a dimensionless constant.
  5. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\). Water forms puddles of depth 0.44 cm . Mercury has surface tension 0.487 and density 13500 in SI units.
  6. Find the depth of puddles formed by mercury on a horizontal surface.
OCR MEI M3 2012 January Q2
2 A light inextensible string of length 5 m has one end attached to a fixed point A and the other end attached to a particle P of mass 0.72 kg . At first, P is moving in a vertical circle with centre A and radius 5 m . When P is at the highest point of the circle it has speed \(10 \mathrm {~ms} ^ { - 1 }\).
  1. Find the tension in the string when the speed of P is \(15 \mathrm {~ms} ^ { - 1 }\). The particle P now moves at constant speed in a horizontal circle with radius 1.4 m and centre at the point C which is 4.8 m vertically below A .
  2. Find the tension in the string.
  3. Find the time taken for P to make one complete revolution. Another light inextensible string, also of length 5 m , now has one end attached to P and the other end attached to the fixed point B which is 9.6 m vertically below A . The particle P then moves with constant speed \(7 \mathrm {~ms} ^ { - 1 }\) in the circle with centre C and radius 1.4 m , as shown in Fig. 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-3_693_465_1078_817} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  4. Find the tension in the string PA and the tension in the string PB .