Questions — CAIE M2 (456 questions)

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CAIE M2 2004 November Q4
4 A particle is projected from a point \(O\) on horizontal ground with speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) to the horizontal. Given that the speed of the particle when it is at its highest point is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  1. show that \(\cos \theta = 0.8\),
  2. find, in either order,
    (a) the greatest height reached by the particle,
    (b) the distance from \(O\) at which the particle hits the ground.
CAIE M2 2004 November Q5
5 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 16 N is attached to a fixed point \(O\) of a horizontal table. A particle \(P\) of mass 0.8 kg is attached to the other end of the string. The particle \(P\) is released from rest on the table, at a point which is 0.5 m from \(O\). The coefficient of friction between the particle and the table is 0.2 . By considering work and energy, find the speed of \(P\) at the instant the string becomes slack.
CAIE M2 2004 November Q6
6 A horizontal turntable rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\) about its centre \(O\). A particle \(P\) of mass 0.08 kg is placed on the turntable. The particle moves with the turntable and no sliding takes place.
  1. It is given that \(\omega = 3\) and that the particle is about to slide on the turntable when \(O P = 0.5 \mathrm {~m}\). Find the coefficient of friction between the particle and the turntable.
  2. Given instead that the particle is about to slide when its speed is \(1.2 \mathrm {~ms} ^ { - 1 }\), find \(\omega\).
CAIE M2 2004 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-3_327_1006_1037_573} A light container has a vertical cross-section in the form of a trapezium. The container rests on a horizontal surface. Grain is poured into the container to a depth of \(y \mathrm {~m}\). As shown in the diagram, the cross-section \(A B C D\) of the grain is such that \(A B = 0.4 \mathrm {~m}\) and \(D C = ( 0.4 + 2 y ) \mathrm { m }\).
  1. When \(y = 0.3\), find the vertical height of the centre of mass of the grain above the base of the container.
  2. Find the value of \(y\) for which the container is about to topple.
CAIE M2 2005 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_552_604_264_772} A uniform solid cone has vertical height 28 cm and base radius 6 cm . The cone is held with a point of the circumference of its base in contact with a horizontal table, and with the base making an angle of \(\theta ^ { \circ }\) with the horizontal (see diagram). When the cone is released, it moves towards the equilibrium position in which its base is in contact with the table. Show that \(\theta < 40.6\), correct to 1 decimal place.
CAIE M2 2005 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_456_871_1228_635} An aircraft flies horizontally at a constant speed of \(220 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Initially it is flying due east. On reaching a point \(A\) it flies in a circular arc from \(A\) to \(B\), taking 50 s . At \(B\) the aircraft is flying due south (see diagram).
  1. Show that the radius of the arc is approximately 7000 m .
  2. Find the magnitude of the acceleration of the aircraft while it is flying between \(A\) and \(B\).
CAIE M2 2005 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_293_1045_267_550} A uniform lamina \(A B C D\) is in the form of a trapezium in which \(A B\) and \(D C\) are parallel and have lengths 2 m and 3 m respectively. \(B D\) is perpendicular to the parallel sides and has length 1 m (see diagram).
  1. Find the distance of the centre of mass of the lamina from \(B D\). The lamina has weight \(W \mathrm {~N}\) and is in equilibrium, suspended by a vertical string attached to the lamina at \(B\). The lamina rests on a vertical support at \(C\). The lamina is in a vertical plane with \(A B\) and \(D C\) horizontal.
  2. Find, in terms of \(W\), the tension in the string and the magnitude of the force exerted on the lamina at \(C\).
CAIE M2 2005 November Q4
4 A particle is projected from horizontal ground with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The greatest height reached by the particle is 10 m and the particle hits the ground at a distance of 40 m from the point of projection. In either order,
  1. find the values of \(u\) and \(\theta\),
  2. find the equation of the trajectory, in the form \(y = a x - b x ^ { 2 }\), where \(x \mathrm {~m}\) and \(y \mathrm {~m}\) are the horizontal and vertical displacements of the particle from the point of projection.
CAIE M2 2005 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_577_693_1740_724} A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light elastic string of natural length 5.5 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 6 m apart. \(P\) is held at rest at a point 1.25 m vertically above the mid-point of \(A B\) and then released. \(P\) travels a distance 5.25 m downwards before coming to instantaneous rest (see diagram). By considering the changes in gravitational potential energy and elastic potential energy as \(P\) travels downwards, find the value of \(\lambda\).
CAIE M2 2005 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-4_673_773_269_685} A horizontal circular disc of radius 4 m is free to rotate about a vertical axis through its centre \(O\). One end of a light inextensible rope of length 5 m is attached to a point \(A\) of the circumference of the disc, and an object \(P\) of mass 24 kg is attached to the other end of the rope. When the disc rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\), the rope makes an angle of \(\theta\) radians with the vertical and the tension in the rope is \(T \mathrm {~N}\) (see diagram). You may assume that the rope is always in the same vertical plane as the radius \(O A\) of the disc.
  1. Given that \(\cos \theta = \frac { 24 } { 25 }\), find the value of \(\omega\).
  2. Given instead that the speed of \(P\) is twice the speed of the point \(A\), find
    (a) the value of \(T\),
    (b) the speed of \(P\).
CAIE M2 2005 November Q7
7 A particle of mass 0.25 kg moves in a straight line on a smooth horizontal surface. A variable resisting force acts on the particle. At time \(t \mathrm {~s}\) the displacement of the particle from a point on the line is \(x \mathrm {~m}\), and its velocity is \(( 8 - 2 x ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It is given that \(x = 0\) when \(t = 0\).
  1. Find the acceleration of the particle in terms of \(x\), and hence find the magnitude of the resisting force when \(x = 1\).
  2. Find an expression for \(x\) in terms of \(t\).
  3. Show that the particle is always less than 4 m from its initial position. \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 University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2006 November Q1
1 A stone is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The horizontal and vertically upward displacements of the stone from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Find the equation of the stone's trajectory. The stone enters the sea at a horizontal distance of 24 m from the base of the cliff.
  2. Find the height above sea level of the top of the cliff.
CAIE M2 2006 November Q2
2 A horizontal turntable rotates with constant angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle of mass 0.06 kg is placed on the turntable at a point 0.25 m from its centre. The coefficient of friction between the particle and the turntable is \(\mu\). As the turntable rotates, the particle moves with the turntable and no sliding takes place.
  1. Find the vertical and horizontal components of the contact force exerted on the particle by the turntable.
  2. Show that \(\mu \geqslant 0.225\).
CAIE M2 2006 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-2_892_412_1217_865} A hollow cylinder of radius 0.35 m has a smooth inner surface. The cylinder is fixed with its axis vertical. One end of a light inextensible string of length 1.25 m is attached to a fixed point \(O\) on the axis of the cylinder. A particle \(P\) of mass 0.24 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle, in contact with the inner surface of the cylinder, and with the string taut (see diagram).
  1. Find the tension in the string.
  2. Given that the magnitude of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the force exerted on \(P\) by the cylinder.
CAIE M2 2006 November Q4
4 A stone is projected from a point on horizontal ground with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 4 } { 5 }\). At time 1.2 s after projection the stone passes through the point \(A\). Subsequently the stone passes through the point \(B\), which is at the same height above the ground as \(A\). Find the horizontal distance \(A B\).
CAIE M2 2006 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_383_1031_543_557} A non-uniform rod \(A B\) of length 2.5 m and mass 3 kg has its centre of mass at the point \(G\) of the rod, where \(A G = 1.5 \mathrm {~m}\). The rod hangs horizontally, in equilibrium, from strings attached at \(A\) and \(B\). The strings at \(A\) and \(B\) make angles with the vertical of \(\alpha ^ { \circ }\) and \(15 ^ { \circ }\) respectively. The tension in the string at \(B\) is \(T \mathrm {~N}\) (see diagram). Find
  1. the value of \(T\),
  2. the value of \(\alpha\).
CAIE M2 2006 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_597_690_1416_731} A large uniform lamina is in the shape of a right-angled triangle \(A B C\), with hypotenuse \(A C\), joined to a semicircle \(A D C\) with diameter \(A C\). The sides \(A B\) and \(B C\) have lengths 3 m and 4 m respectively, as shown in the diagram.
  1. Show that the distance from \(A B\) of the centre of mass of the semicircular part \(A D C\) of the lamina is \(\left( 2 + \frac { 2 } { \pi } \right) \mathrm { m }\).
  2. Show that the distance from \(A B\) of the centre of mass of the complete lamina is 2.14 m , correct to 3 significant figures.
CAIE M2 2006 November Q7
7 A cyclist starts from rest at a point \(O\) and travels along a straight path. At time \(t \mathrm {~s}\) after starting, the displacement of the cyclist from \(O\) is \(x \mathrm {~m}\), and the acceleration of the cyclist is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.6 x ^ { 0.2 }\).
  1. Find an expression for the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the cyclist in terms of \(x\).
  2. Show that \(t = 2.5 x ^ { 0.4 }\).
  3. Find the distance travelled by the cyclist in the first 10 s of the journey.
CAIE M2 2006 November Q8
8
\includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-4_933_275_689_934} The diagram shows a light elastic string of natural length 0.6 m and modulus of elasticity 5 N with one end attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at the point \(A\), which is 0.9 m vertically above \(O\). The particle is released from rest and travels vertically downwards through \(O\) to the point \(C\), where it starts to move upwards. \(B\) is the point of the line \(A C\) where the string first becomes slack.
  1. Find the speed of \(P\) at \(B\).
  2. The extension of the string when \(P\) is at \(C\) is \(x \mathrm {~m}\).
    (a) Show that \(x ^ { 2 } - 0.48 x - 0.81 = 0\).
    (b) Hence find the distance \(A C\).
CAIE M2 2007 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_711_398_269_877} Each of two identical light elastic strings has natural length 0.25 m and modulus of elasticity 4 N . A particle \(P\) of mass 0.6 kg is attached to one end of each of the strings. The other ends of the strings are attached to fixed points \(A\) and \(B\) which are 0.8 m apart on a smooth horizontal table. The particle is held at rest on the table, at a point 0.3 m from \(A B\) for which \(A P = B P\) (see diagram).
  1. Find the tension in the strings.
  2. The particle is released. Find its initial acceleration.
CAIE M2 2007 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_496_609_1535_769} One end of a light inextensible string of length 0.16 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) moves on the table in a horizontal circle, with the string taut and making an angle of \(30 ^ { \circ }\) with the downward vertical through \(A\) (see diagram). \(P\) moves with constant speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the tension in the string,
  2. the force exerted by the table on \(P\).
CAIE M2 2007 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-3_764_627_274_758} A uniform beam \(A B\) has length 2 m and mass 10 kg . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in a fixed position by a light inextensible string of length 2.4 m . One end of the string is attached to the beam at a point 0.7 m from \(A\). The other end of the string is attached to the wall at a point vertically above the hinge. The string is at right angles to \(A B\). The beam carries a load of weight 300 N at \(B\) (see diagram).
  1. Find the tension in the string. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally away from the wall and \(Y \mathrm {~N}\) vertically downwards.
  2. Find the values of \(X\) and \(Y\).
CAIE M2 2007 November Q4
4 A particle of mass 0.4 kg is released from rest and falls vertically. A resisting force of magnitude \(0.08 v \mathrm {~N}\) acts upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle at time \(t \mathrm {~s}\) after its release.
  1. Show that the acceleration of the particle is \(( 10 - 0.2 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find the velocity of the particle when \(t = 15\).
CAIE M2 2007 November Q5
5 Each of two light elastic strings, \(S _ { 1 }\) and \(S _ { 2 }\), has modulus of elasticity 16 N . The string \(S _ { 1 }\) has natural length 0.4 m and the string \(S _ { 2 }\) has natural length 0.5 m . One end of \(S _ { 1 }\) is attached to a fixed point \(A\) of a smooth horizontal table and the other end is attached to a particle \(P\) of mass 0.5 kg . One end of \(S _ { 2 }\) is attached to a fixed point \(B\) of the table and the other end is attached to \(P\). The distance \(A B\) is 1.5 m . The particle \(P\) is held at \(A\) and then released from rest.
  1. Find the speed of \(P\) at the instant that \(S _ { 2 }\) becomes slack.
  2. Find the greatest distance of \(P\) from \(A\) in the subsequent motion.
CAIE M2 2007 November Q6
6 marks
6 A particle is projected from a point \(O\) at an angle of \(35 ^ { \circ }\) above the horizontal. At time \(T\) s later the particle passes through a point \(A\) whose horizontal and vertically upward displacements from \(O\) are 8 m and 3 m respectively.
  1. By using the equation of the particle's trajectory, or otherwise, find (in either order) the speed of projection of the particle from \(O\) and the value of \(T\).
  2. Find the angle between the direction of motion of the particle at \(A\) and the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_476_895_269_625} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section of a uniform solid. The cross-section has the shape and dimensions shown. The centre of mass \(C\) of the solid lies in the plane of this cross-section. The distance of \(C\) from \(D E\) is \(y \mathrm {~cm}\).
  3. Find the value of \(y\). The solid is placed on a rough plane. The coefficient of friction between the solid and the plane is \(\mu\). The plane is tilted so that \(E F\) lies along a line of greatest slope.
  4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_375_431_1366_897} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The solid is placed so that \(F\) is higher up the plane than \(E\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 1 } { 2 }\). [3]
  5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_376_428_2069_900} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} The solid is now placed so that \(E\) is higher up the plane than \(F\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Show that \(\mu < \frac { 5 } { 6 }\). [3]