Questions M2 (1391 questions)

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CAIE M2 2009 June Q3
3 A particle \(P\) starts from a fixed point \(O\) and moves in a straight line. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\), its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration is \(\frac { 1 } { x + 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Given that \(v = 2\) when \(x = 0\), use integration to show that \(v ^ { 2 } = 2 \ln \left( \frac { 1 } { 2 } x + 1 \right) + 4\).
  2. Find the value of \(v\) when the acceleration of \(P\) is \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M2 2009 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-3_437_444_269_849} A particle of mass 0.12 kg is moving on the smooth inside surface of a fixed hollow sphere of radius 0.5 m . The particle moves in a horizontal circle whose centre is 0.3 m below the centre of the sphere (see diagram).
  1. Show that the force exerted by the sphere on the particle has magnitude 2 N .
  2. Find the speed of the particle.
  3. Find the time taken for the particle to complete one revolution.
CAIE M2 2009 June Q5
5 A small stone is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) above the horizontal. Referred to horizontal and vertically upwards axes through \(O\), the equation of the stone's trajectory is \(y = 0.75 x - 0.02 x ^ { 2 }\), where \(x\) and \(y\) are in metres. Find
  1. the values of \(\theta\) and \(V\),
  2. the distance from \(O\) of the point where the stone hits the ground,
  3. the greatest height reached by the stone.
CAIE M2 2009 June Q6
5 marks
6
\includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-3_200_639_1754_753} A particle \(P\) of mass 1.6 kg is attached to one end of each of two light elastic strings. The other ends of the strings are attached to fixed points \(A\) and \(B\) which are 2 m apart on a smooth horizontal table. The string attached to \(A\) has natural length 0.25 m and modulus of elasticity 4 N , and the string attached to \(B\) has natural length 0.25 m and modulus of elasticity 8 N . The particle is held at the mid-point \(M\) of \(A B\) (see diagram).
  1. Find the tensions in the strings.
  2. Show that the total elastic potential energy in the two strings is 13.5 J .
    \(P\) is released from rest and in the subsequent motion both strings remain taut. The displacement of \(P\) from \(M\) is denoted by \(x \mathrm {~m}\). Find
  3. the initial acceleration of \(P\),
  4. the non-zero value of \(x\) at which the speed of \(P\) is zero. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_529_542_269_804} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A uniform solid body has a cross-section as shown in Fig. 1.
  5. Show that the centre of mass of the body is 2.5 cm from the plane face containing \(O B\) and 3.5 cm from the plane face containing \(O A\).
  6. The solid is placed on a rough plane which is initially horizontal. The coefficient of friction between the solid and the plane is \(\mu\).
    (a) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_332_469_1320_918} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The solid is placed with \(O A\) in contact with the plane, and then the plane is tilted so that \(O A\) lies along a line of greatest slope with \(A\) higher than \(O\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 5 } { 7 }\).
    [0pt] [5]
    (b)
    \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-4_291_465_1987_918} Instead, the solid is placed with \(O B\) in contact with the plane, and then the plane is tilted so that \(O B\) lies along a line of greatest slope with \(B\) higher than \(O\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Find another inequality for \(\mu\).
CAIE M2 2010 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-2_618_441_253_852} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg , and a uniform straight wire of length 40 cm and mass 0.9 kg . The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire.
CAIE M2 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-2_551_519_1231_813} A uniform solid cone has height 30 cm and base radius \(r \mathrm {~cm}\). The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35 ^ { \circ }\), when the cone topples. The diagram shows a cross-section of the cone.
  1. Find the value of \(r\).
  2. Show that the coefficient of friction between the cone and the plane is greater than 0.7 .
CAIE M2 2010 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_456_511_260_817} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T \mathrm {~N}\). The acceleration of the particle has magnitude \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(\tan \theta = 0.75\) and find the value of \(T\).
  2. Find the speed of the particle.
CAIE M2 2010 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-3_727_565_1256_790} A uniform lamina of weight 15 N is in the form of a trapezium \(A B C D\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(A B\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30 ^ { \circ }\) upwards from the horizontal (see diagram). Find the tension in the string.
CAIE M2 2010 June Q5
5 A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and direction upwards at an angle \(\theta\) to the horizontal. The particle passes through the point which is 7 m above the ground and 16 m horizontally from \(O\), and hits the ground at the point \(A\).
  1. Using the equation of the particle's trajectory and the identity \(\sec ^ { 2 } \theta = 1 + \tan ^ { 2 } \theta\), show that the possible values of \(\tan \theta\) are \(\frac { 3 } { 4 }\) and \(\frac { 17 } { 4 }\).
  2. Find the distance \(O A\) for each of the two possible values of \(\tan \theta\).
  3. Sketch in the same diagram the two possible trajectories.
CAIE M2 2010 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-4_324_1267_794_440} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(A B\) (see diagram).
  1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N .
    \(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
  2. Find the speed with which \(P\) passes through \(M\).
CAIE M2 2010 June Q7
7 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. \(P\) starts at the point \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves towards a fixed point \(A\) on the line. At time \(t \mathrm {~s}\) the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude (5-x) N acts on \(P\) in the direction towards \(O\).
  1. Form a differential equation in \(v\) and \(x\). By solving this differential equation, show that \(v = 10 - 2 x\).
  2. Find \(x\) in terms of \(t\), and hence show that the particle is always less than 5 m from \(O\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2010 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-2_618_441_253_852} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg , and a uniform straight wire of length 40 cm and mass 0.9 kg . The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire.
CAIE M2 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-2_551_519_1231_813} A uniform solid cone has height 30 cm and base radius \(r \mathrm {~cm}\). The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35 ^ { \circ }\), when the cone topples. The diagram shows a cross-section of the cone.
  1. Find the value of \(r\).
  2. Show that the coefficient of friction between the cone and the plane is greater than 0.7 .
CAIE M2 2010 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-3_456_511_260_817} A particle of mass 0.24 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle \(\theta\) with the vertical (see diagram), and the tension in the string is \(T \mathrm {~N}\). The acceleration of the particle has magnitude \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that \(\tan \theta = 0.75\) and find the value of \(T\).
  2. Find the speed of the particle.
CAIE M2 2010 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-3_727_565_1256_790} A uniform lamina of weight 15 N is in the form of a trapezium \(A B C D\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(A B\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30 ^ { \circ }\) upwards from the horizontal (see diagram). Find the tension in the string.
CAIE M2 2010 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{d3609757-50ae-4c68-8992-4304867a2d84-4_324_1267_794_440} A particle \(P\) of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points \(A\) and \(B\) which are 4.8 m apart at the same horizontal level. \(P\) hangs in equilibrium at a point 0.7 m vertically below the mid-point \(M\) of \(A B\) (see diagram).
  1. Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N .
    \(P\) is now held at rest at a point 1.8 m vertically below \(M\), and is then released.
  2. Find the speed with which \(P\) passes through \(M\).
CAIE M2 2010 June Q1
1 A particle is projected horizontally with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a high cliff. Find the direction of motion of the particle after 2 s .
CAIE M2 2010 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-2_319_908_438_616} A uniform solid cone has height 20 cm and base radius \(4 \mathrm {~cm} . P Q\) is a diameter of the base of the cone. The cone is held in equilibrium, with \(P\) in contact with a horizontal surface and \(P Q\) vertical, by a force applied at \(Q\). This force has magnitude 3 N and acts parallel to the axis of the cone (see diagram). Calculate the mass of the cone.
CAIE M2 2010 June Q3
3 Two particles \(P\) and \(Q\) are projected simultaneously with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a horizontal plane. Both particles subsequently pass at different times through the point \(A\) which has horizontal and vertically upward displacements from \(O\) of 40 m and 15 m respectively.
  1. By considering the equation of the trajectory of a projectile, show that each angle of projection satisfies the equation \(\tan ^ { 2 } \theta - 8 \tan \theta + 4 = 0\).
  2. Calculate the distance between the points at which \(P\) and \(Q\) strike the plane.
CAIE M2 2010 June Q4
4
\(A B\) is the diameter of a uniform semicircular lamina which has radius 0.3 m and mass 0.4 kg . The lamina is hinged to a vertical wall at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. One end of a light inextensible string is attached to the lamina at \(B\) and the other end of the string is attached to the wall vertically above \(A\). The lamina is in equilibrium in a vertical plane perpendicular to the wall with the string horizontal (see diagram).
  1. Show that the tension in the string is 2.00 N correct to 3 significant figures.
  2. Find the magnitude and direction of the force exerted on the lamina by the hinge.
    \includegraphics[max width=\textwidth, alt={}, center]{5a2248f6-3ef9-4e69-90cf-4d6a2351be14-3_956_540_258_804} A small ball \(B\) of mass 0.4 kg is attached to fixed points \(P\) and \(Q\) on a vertical axis by two light inextensible strings of equal length. Both strings are taut and each is inclined at \(30 ^ { \circ }\) to the vertical. The ball moves in a horizontal circle (see diagram).
CAIE M2 2010 June Q6
6 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. At time \(t \mathrm {~s}\), the displacement of \(P\) from a fixed point on the line is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that when \(t = 0 , x = 0\) and \(v = 9\). The motion of \(P\) is opposed by a force of magnitude \(3 \sqrt { } v \mathrm {~N}\).
  1. By solving an appropriate differential equation, show that \(v = ( 27 - 9 x ) ^ { \frac { 2 } { 3 } }\).
  2. Calculate the value of \(x\) when \(t = 0.5\).
CAIE M2 2010 June Q7
7 One end of a light elastic string of natural length 3 m and modulus of elasticity 24 N is attached to a fixed point \(O\). A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the extension of the string is \(x \mathrm {~m}\) the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } = 64 + 20 x - 20 x ^ { 2 }\).
  2. Find the greatest speed of the particle.
  3. Calculate the greatest tension in the string.
CAIE M2 2011 June Q1
1 A particle is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point on horizontal ground. Calculate the time taken for the particle to hit the ground.
CAIE M2 2011 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_406_483_431_829}
\(A O B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.6 m (see diagram).
  1. Calculate the distance of the centre of mass of the lamina from \(A\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
  2. Find the angle between the vertical and the side \(A O\) of the lamina.
CAIE M2 2011 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_247_839_1375_653} A light elastic string of natural length 1.2 m and modulus of elasticity 24 N is attached to fixed points \(A\) and \(B\) on a smooth horizontal surface, where \(A B = 1.2 \mathrm {~m}\). A particle \(P\) is attached to the mid-point of the string. \(P\) is projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the surface in a direction perpendicular to \(A B\) (see diagram). \(P\) comes to instantaneous rest at a distance 0.25 m from \(A B\).
  1. Show that the mass of \(P\) is 0.8 kg .
  2. Calculate the greatest deceleration of \(P\).