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CAIE M2 2011 November Q2
7 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_525_913_1123_616} A particle \(P\) is projected from a point \(O\) at an angle of \(60 ^ { \circ }\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45 ^ { \circ }\) (see diagram).
  1. Show that the speed of projection of \(P\) is \(8.20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Calculate the time after projection when the direction of motion of \(P\) is \(45 ^ { \circ }\) above the horizontal.
CAIE M2 2011 November Q3
8 marks Standard +0.3
3 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.25 \mathrm {~kg} . P\) hangs in equilibrium below \(O\).
  1. Calculate the distance \(O P\). The particle \(P\) is raised, and is released from rest at \(O\).
  2. Calculate the speed of \(P\) when it passes through the equilibrium position.
  3. Calculate the greatest value of the distance \(O P\) in the subsequent motion.
CAIE M2 2011 November Q4
9 marks Challenging +1.2
4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.
  1. Find the value of \(\theta\) at which the cone is about to topple.
  2. Given that the cylinder does not topple, find the greatest possible value of \(h\). The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-3_476_1211_836_466}
  3. Given that the solid immediately topples, find the least possible value of \(h\).
CAIE M2 2011 November Q5
10 marks Standard +0.3
5 A ball of mass 0.05 kg is released from rest at a height \(h \mathrm {~m}\) above the ground. At time \(t \mathrm {~s}\) after its release, the downward velocity of the ball is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance opposes the motion of the ball with a force of magnitude \(0.01 \nu \mathrm {~N}\).
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 0.2 v\). Hence find \(v\) in terms of \(t\).
  2. Given that the ball reaches the ground when \(t = 2\), calculate \(h\).
CAIE M2 2011 November Q6
11 marks Challenging +1.2
6 A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m . One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is \(T \mathrm {~N}\), and the bead rotates with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle about the vertical axis through \(A\) and \(C\).
  1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m , calculate \(\omega\), and hence find the kinetic energy of \(B\).
  2. Given instead that angle \(A B C = 90 ^ { \circ }\), and that \(A B\) makes an angle \(\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\) with the vertical, calculate \(T\) and \(\omega\).
CAIE M2 2011 November Q1
5 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-2_334_679_258_731} A non-uniform \(\operatorname { rod } A B\), of length 0.6 m and weight 9 N , has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\) (see diagram).
  1. Calculate \(T\).
  2. Find the least possible value of the coefficient of friction at \(A\).
CAIE M2 2011 November Q2
7 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-2_525_913_1123_616} A particle \(P\) is projected from a point \(O\) at an angle of \(60 ^ { \circ }\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45 ^ { \circ }\) (see diagram).
  1. Show that the speed of projection of \(P\) is \(8.20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Calculate the time after projection when the direction of motion of \(P\) is \(45 ^ { \circ }\) above the horizontal.
CAIE M2 2011 November Q4
9 marks Challenging +1.2
4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.
  1. Find the value of \(\theta\) at which the cone is about to topple.
  2. Given that the cylinder does not topple, find the greatest possible value of \(h\). The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-3_476_1211_836_466}
  3. Given that the solid immediately topples, find the least possible value of \(h\).
CAIE M2 2011 November Q1
3 marks Moderate -0.8
1 A particle is projected with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal from a point on horizontal ground. Calculate the speed of the particle 2 s after the instant of projection.
CAIE M2 2011 November Q2
5 marks Standard +0.3
2 An object is made from two identical uniform rods \(A B\) and \(B C\) each of length 0.6 m and weight 7 N . The rods are rigidly joined to each other at \(B\) and angle \(A B C = 90 ^ { \circ }\).
  1. Calculate the distance of the centre of mass of the object from \(B\). The object is freely suspended at \(A\) and a force of magnitude \(F \mathrm {~N}\) is applied to the rod \(B C\) at \(C\). The object is in equilibrium with \(A B\) inclined at \(45 ^ { \circ }\) to the horizontal.
  2. (a) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_401_314_799_995} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Calculate \(F\) given that the force acts horizontally as shown in Fig. 1.
    (b) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_503_273_1446_1014} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Calculate \(F\) given instead that the force acts perpendicular to the rod as shown in Fig. 2.
CAIE M2 2011 November Q3
6 marks Standard +0.3
3 A particle \(P\) is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the path of \(P\) is \(y = x - 0.016 x ^ { 2 }\).
  2. Calculate the horizontal distance between the two positions at which \(P\) is 2.4 m above the ground.
CAIE M2 2011 November Q4
8 marks Standard +0.3
4 A particle \(P\) of mass 0.4 kg is projected horizontally with velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a smooth horizontal surface. The motion of \(P\) is opposed by a resisting force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after projection.
  1. Show that \(v = \frac { 8 } { 1 + 4 t }\).
  2. Calculate the distance \(O P\) when \(t = 1.5\).
CAIE M2 2011 November Q5
9 marks Standard +0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-3_927_1022_689_559} One end of a light inextensible string of length 1.2 m is attached to a fixed point \(O\) on a smooth horizontal surface. Particles \(P , Q\) and \(R\) are attached to the string so that \(O P = P Q = Q R = 0.4 \mathrm {~m}\). The particles rotate in horizontal circles about \(O\) with constant angular speed \(\omega \operatorname { rads } ^ { - 1 }\) and with \(O , P\), \(Q\) and \(R\) in a straight line (see diagram). \(R\) has mass 0.2 kg , and the tensions in the parts of the string attached to \(Q\) are 6 N and 10 N .
  1. Show that \(\omega = 5\).
  2. Calculate the mass of \(Q\).
  3. Given that the kinetic energy of \(P\) is equal to the kinetic energy of \(R\), calculate the tension in the part of the string attached to \(O\).
CAIE M2 2011 November Q6
9 marks Challenging +1.2
6 A uniform solid consists of a hemisphere with centre \(O\) and radius 0.6 m joined to a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere coincides with one of the plane faces of the cylinder.
  1. Calculate the distance of the centre of mass of the solid from \(O\).
    [0pt] [The volume of a hemisphere of radius \(r\) is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
  2. \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-4_547_631_593_797} A cylindrical hole, of length 0.48 m , starting at the plane face of the solid, is made along the axis of symmetry (see diagram). The resulting solid has its centre of mass at \(O\). Show that the area of the cross-section of the hole is \(\frac { 3 } { 16 } \pi \mathrm {~m} ^ { 2 }\).
  3. It is possible to increase the length of the cylindrical hole so that the solid still has its centre of mass at \(O\). State the increase in the length of the hole.
CAIE M2 2011 November Q7
10 marks Challenging +1.2
7 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a particle \(P\) of mass 0.8 kg . The other end of the string is attached to a fixed point \(O\) at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The particle rests in equilibrium on the plane.
  1. Calculate the extension of the string. \(P\) is projected from its equilibrium position up the plane along a line of greatest slope. In the subsequent motion \(P\) just reaches \(O\), and later just reaches the foot of the plane. Calculate
  2. the speed of projection of \(P\),
  3. the length of the line of greatest slope of the plane.
CAIE M2 2012 November Q2
7 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate
  1. the magnitude of the force applied at \(A\),
  2. the least possible value of the coefficient of friction at \(B\).
CAIE M2 2012 November Q3
7 marks Standard +0.8
3 A particle \(P\) of mass 0.2 kg is released from rest and falls vertically. At time \(t \mathrm {~s}\) after release \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.8 v \mathrm {~N}\) acts on \(P\).
  1. Show that the acceleration of \(P\) is \(( 10 - 4 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find the value of \(v\) when \(t = 0.6\).
CAIE M2 2012 November Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_538_885_1809_628} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45 ^ { \circ }\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius 0.67 m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\).
CAIE M2 2012 November Q5
7 marks Standard +0.3
5 A particle \(P\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. For the instant when the speed of \(P\) is \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and increasing,
  1. show that the vertical component of the velocity of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards,
  2. calculate the distance of \(P\) from \(O\).
CAIE M2 2012 November Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-3_582_862_577_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).
  1. Show that the centre of mass of the lamina lies on \(O B\).
  2. Calculate the distance of the centre of mass of the lamina from \(O\).
CAIE M2 2012 November Q7
12 marks Challenging +1.8
7 A light elastic string has natural length 3 m and modulus of elasticity 45 N . A particle \(P\) of weight 6 N is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\) and \(A B = 4 \mathrm {~m}\). The particle \(P\) is released from rest at the point 1.5 m vertically below \(A\).
  1. Calculate the distance \(P\) moves after its release before first coming to instantaneous rest at a point vertically above \(B\). (You may assume that at this point the part of the string joining \(P\) to \(B\) is slack.)
  2. Show that the greatest speed of \(P\) occurs when it is 2.1 m below \(A\), and calculate this greatest speed.
  3. Calculate the greatest magnitude of the acceleration of \(P\).
CAIE M2 2012 November Q2
7 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate
  1. the magnitude of the force applied at \(A\),
  2. the least possible value of the coefficient of friction at \(B\).
CAIE M2 2012 November Q4
6 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-2_538_885_1809_628} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45 ^ { \circ }\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius 0.67 m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\).
CAIE M2 2012 November Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-3_584_862_575_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).
  1. Show that the centre of mass of the lamina lies on \(O B\).
  2. Calculate the distance of the centre of mass of the lamina from \(O\).
CAIE M2 2012 November Q1
6 marks Standard +0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_426_531_258_808} A circular object is formed from a uniform semicircular lamina of weight 12 N and a uniform semicircular arc of weight 8 N . The lamina and the arc both have centre \(O\) and radius 0.6 m and are joined at the ends of their common diameter \(A B\). The object is freely pivoted to a fixed point at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. The object is in equilibrium acted on by a horizontal force of magnitude \(F\) N applied at the lowest point of the object, and acting in the plane of the object (see diagram).
  1. Show that the centre of mass of the object is at \(O\).
  2. Calculate \(F\).