Questions — CAIE M2 (456 questions)

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CAIE M2 2017 March Q1
1 A small ball is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. Find the distance from the point of projection of the ball at the instant when it is travelling horizontally.
CAIE M2 2017 March Q2
2 A cylindrical container is open at the top. The curved surface and the circular base of the container are both made from the same thin uniform material. The container has radius 0.2 m and height 0.9 m .
  1. Show that the centre of mass of the container is 0.405 m from the base.
    The container is placed with its base on a rough inclined plane. The container is in equilibrium on the point of slipping down the plane and also on the point of toppling.
  2. Find the coefficient of friction between the container and the plane.
CAIE M2 2017 March Q3
4 marks
3 A particle \(P\) is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) below the horizontal, from a point \(O\) which is 30 m above horizontal ground.
  1. Calculate the time taken by \(P\) to reach the ground.
  2. Calculate the speed and direction of motion of \(P\) immediately before it reaches the ground. [4]
CAIE M2 2017 March Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fef8f0eb-dfed-4d2b-8a58-ca3c85b28686-06_474_631_260_756} The diagram shows a uniform lamina \(A B C D\) with \(A B = 0.75 \mathrm {~m} , A D = 0.6 \mathrm {~m}\) and \(B C = 0.9 \mathrm {~m}\). Angle \(B A D =\) angle \(A B C = 90 ^ { \circ }\).
  1. Show that the distance of the centre of mass of the lamina from \(A B\) is 0.38 m , and find the distance of the centre of mass from \(B C\).
    The lamina is freely suspended at \(B\) and hangs in equilibrium.
  2. Find the angle between \(B C\) and the vertical.
    \includegraphics[max width=\textwidth, alt={}, center]{fef8f0eb-dfed-4d2b-8a58-ca3c85b28686-08_428_455_260_845} Two particles \(P\) and \(Q\) have masses 0.4 kg and \(m \mathrm {~kg}\) respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string of length 0.5 m which is inclined at an angle of \(60 ^ { \circ }\) to the vertical. \(P\) and \(Q\) are joined to each other by a light inextensible vertical string. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string. The string \(B Q\) is taut and horizontal. The particles rotate in horizontal circles about an axis through \(A\) and \(B\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram). The tension in the string joining \(P\) and \(Q\) is 1.5 N .
CAIE M2 2017 March Q7
7 One end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.4 kg which hangs in equilibrium vertically below \(O\).
  1. Calculate the extension of the string.
    \(P\) is projected vertically downwards from the equilibrium position with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Calculate the distance \(P\) travels before it is first at instantaneous rest.
    When \(P\) is first at instantaneous rest a stationary particle of mass 0.4 kg becomes attached to \(P\).
  3. Find the greatest speed of the combined particle in the subsequent motion.
CAIE M2 2019 March Q1
1 A particle is projected with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. Find the speed and direction of motion of the particle at the instant 4 s after projection.
CAIE M2 2019 March Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-04_606_376_260_881} A uniform object is made by joining together three solid cubes with edges \(3 \mathrm {~m} , 2 \mathrm {~m}\) and 1 m . The object has an axis of symmetry, with the cubes stacked vertically and the cube of edge 2 m between the other two cubes (see diagram).
  1. Calculate the distance of the centre of mass of the object above the base of the largest cube.
    The smallest cube is now removed from the object. It is replaced by a heavier uniform cube with 1 m edges which is made of a different material. The centre of mass of the object is now at the base of the 2 m cube.
  2. Find the ratio of the masses of the two cubes of edge 1 m .
CAIE M2 2019 March Q3
3 A small ball is projected from a point \(O\) on horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of the ball from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively, where \(x = 4 t\) and \(y = 6 t - 5 t ^ { 2 }\).
  1. Find the equation of the trajectory of the ball.
  2. Hence or otherwise calculate the angle of projection of the ball and its initial speed.
CAIE M2 2019 March Q4
6 marks
4
\includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_337_526_262_726}
\includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-07_111_116_486_1308} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The particle \(P\) moves in a horizontal circle which has centre \(O\). It is given that \(A O\) is vertical and that angle \(O A P\) is \(60 ^ { \circ }\) (see diagram). Calculate the speed of \(P\). [6]
CAIE M2 2019 March Q5
5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.4 m vertically below \(O\).
  1. Find the greatest speed of \(P\).
  2. Calculate the greatest distance of \(P\) below \(O\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-10_608_611_258_767} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section of a solid cylinder through which a cylindrical hole has been drilled to make a uniform prism. The radius of the cylinder is \(5 r\) and the radius of the hole is \(r\). The centre of the hole is a distance \(2 r\) from the centre of the cylinder.
CAIE M2 2019 March Q7
7 A particle \(P\) is projected horizontally from a point \(O\) on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.2 . A horizontal force of magnitude \(0.06 t \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(t \mathrm {~s}\) is the time after projection. \(P\) comes to rest when \(t = 4\).
  1. The particle begins to move again when \(t = 8\). Show that the mass of \(P\) is 0.24 kg .
  2. Show that, for \(0 \leqslant t \leqslant 4 , \frac { \mathrm {~d} v } { \mathrm {~d} t } = 0.25 t - 2\), and find the speed of projection of \(P\).
  3. Find the distance from \(O\) at which \(P\) comes to rest.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE M2 2002 November Q1
3 marks
1
\includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_373_552_260_799} A uniform isosceles triangular lamina \(A B C\) is right-angled at \(B\). The length of \(A C\) is 24 cm . The lamina rotates in a horizontal plane, about a vertical axis through the mid-point of \(A C\), with angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Find the speed with which the centre of mass of the lamina is moving.
[0pt] [3]
CAIE M2 2002 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_319_874_968_639} A uniform rod \(A B\), of length 2 m and mass 10 kg , is freely hinged to a fixed point at the end \(B\). A light elastic string, of modulus of elasticity 200 N , has one end attached to the end \(A\) of the rod and the other end attached to a fixed point \(O\), which is in the same vertical plane as the rod. The rod is horizontal and in equilibrium, with \(O A = 3 \mathrm {~m}\) and angle \(O A B = 150 ^ { \circ }\) (see diagram). Find
  1. the tension in the string,
  2. the natural length of the string.
CAIE M2 2002 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-2_502_789_1742_680} A stone is projected horizontally, with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from the top of a vertical cliff of height 45 m above sea level (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of the stone from the top of the cliff are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the stone's trajectory.
  2. Find the angle the trajectory makes with the horizontal at the point where the stone reaches sea level.
CAIE M2 2002 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-3_604_490_258_831} A small ball \(B\) of mass 0.5 kg is attached to points \(P\) and \(Q\) on a fixed vertical axis by two light inextensible strings of equal length. Both of the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical, as shown in the diagram. The ball moves with constant speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.8 m . Find the tension in the string \(P B\).
CAIE M2 2002 November Q5
5 A light elastic string has natural length 2 m and modulus of elasticity 1.5 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.075 \mathrm {~kg} . P\) is released from rest at \(O\). Find
  1. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
  2. the acceleration with which \(P\) starts to move up the plane immediately after it has reached its lowest point.
CAIE M2 2002 November Q6
3 marks
6
\includegraphics[max width=\textwidth, alt={}, center]{fcf239a6-6558-43ec-b404-70aa349af6a9-3_177_880_1658_635} A particle \(P\) of mass \(\frac { 1 } { 10 } \mathrm {~kg}\) travels in a straight line on a smooth horizontal surface. It passes through the fixed point \(O\) with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\). After \(t\) seconds its displacement from \(O\) is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 } . P\) is subject to a single force of magnitude \(\frac { v } { 200 } \mathrm {~N}\) which acts in a direction opposite to the motion (see diagram).
  1. Find an expression for \(v\) in terms of \(x\).
  2. Find an expression for \(x\) in terms of \(t\).
  3. Show that the value of \(x\) is less than 100 for all values of \(t\).
  4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_477_684_264_774} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross section through the centre of mass \(C\) of a uniform L-shaped prism. \(C\) is \(x \mathrm {~cm}\) from \(O Y\) and \(y \mathrm {~cm}\) from \(O X\). Find the values of \(x\) and \(y\).
  5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_257_428_1064_902} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism is placed on a rough plane with \(O X\) in contact with the plane. The plane is tilted from the horizontal so that \(O X\) lies along a line of greatest slope, as shown in Fig. 2. When the angle of inclination of the plane is sufficiently great the prism starts to slide (without toppling). Show that the coefficient of friction between the prism and the plane is less than \(\frac { 7 } { 5 }\).
  6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fcf239a6-6558-43ec-b404-70aa349af6a9-4_303_414_1710_909} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} The prism is now placed on a rough plane with \(O Y\) in contact with the plane. The plane is tilted from the horizontal so that \(O Y\) lies along a line of greatest slope, as shown in Fig. 3. When the angle of inclination of the plane is sufficiently great the prism starts to topple (without sliding). Find the least possible value of the coefficient of friction between the prism and the plane. [3]
CAIE M2 2003 November Q1
1 A railway engine of mass 50000 kg travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal circular track of radius 1250 m . Find the magnitude of the horizontal force on the engine.
CAIE M2 2003 November Q2
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-2_376_569_559_466} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-2_485_456_450_1226} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A uniform solid cone has height 20 cm and base radius 10 cm . It is placed with its axis vertical on a rough horizontal plane (see Fig. 1). The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(\theta ^ { \circ }\), when the cone begins to topple without sliding (see Fig. 2).
  1. Find the value of \(\theta\).
  2. What can you say about the value of the coefficient of friction between the cone and the plane?
CAIE M2 2003 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{be83d46f-bf5b-4382-b424-bb5067626adc-2_433_446_1635_854} One end of a light elastic spring, of natural length 0.4 m and modulus of elasticity 88 N , is attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the spring and is held, with the spring compressed, at a point 0.3 m vertically above \(O\), as shown in the diagram. \(P\) is now released from rest and moves vertically upwards.
  1. Find the initial acceleration of \(P\).
  2. Find the initial elastic potential energy of the spring.
  3. Find the speed of \(P\) when the distance \(O P\) is 0.4 m . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-3_362_657_269_744} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows a uniform lamina \(A B C D\) with dimensions \(A B = 15.5 \mathrm {~cm} , B C = 8 \mathrm {~cm}\) and \(C D = 9.5 \mathrm {~cm}\). Angles \(A B C\) and \(B C D\) are right angles.
CAIE M2 2003 November Q5
5 A stone is projected from a point on horizontal ground with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\alpha ^ { \circ }\) above the horizontal. The stone is moving horizontally when it hits a vertical wall at a point 7.2 m above the ground.
  1. Find the value of \(\alpha\). After rebounding at right angles from the wall the speed of the stone is halved. Find
  2. the distance from the wall of the point at which the stone hits the ground,
  3. the angle which the direction of motion of the stone makes with the horizontal, immediately before the stone hits the ground.
CAIE M2 2003 November Q6
6 A cyclist and his machine have a total mass of 80 kg . The cyclist starts from rest and rides from the bottom to the top of a straight slope inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). The cyclist exerts a constant force of magnitude 120 N . There is a resisting force of magnitude \(8 v \mathrm {~N}\) acting on the cyclist, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after the start.
  1. Show that \(\left( \frac { 1 } { 5 - v } \right) \frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1 } { 10 }\).
  2. Solve this differential equation and hence show that \(v = 5 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 10 } t } \right)\).
  3. Given that the cyclist takes 20 s to reach the top of the slope, find the length of the slope.
CAIE M2 2004 November Q1
1 A light elastic string has natural length 1.5 m and modulus of elasticity 60 N . The string is stretched between two fixed points \(A\) and \(B\), which are at the same horizontal level and 2 m apart.
  1. Find the tension in the string. A particle of weight \(W \mathrm {~N}\) is now attached to the mid-point of the string and the particle is in equilibrium at a point 0.75 m vertically below the mid-point of \(A B\).
  2. Find the value of \(W\).
CAIE M2 2004 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{81411376-b926-4857-bc9b-ac85d7957f3d-2_333_737_762_705} A uniform rod \(A B\) of length 1.2 m and weight 30 N is in equilibrium with the end \(A\) in contact with a vertical wall. \(A B\) is held at right angles to the wall by a light inextensible string. The string has one end attached to the rod at \(B\) and the other end attached to a point \(C\) of the wall. The point \(C\) is 0.5 m vertically above \(A\) (see diagram). Find
  1. the tension in the string,
  2. the horizontal and vertical components of the force exerted on the rod by the wall at \(A\).
CAIE M2 2004 November Q3
3 A car of mass 1000 kg is moving on a straight horizontal road. The driving force of the car is \(\frac { 28000 } { v } \mathrm {~N}\) and the resistance to motion is \(4 \nu \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car \(t\) seconds after it passes a fixed point on the road.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 7000 - v ^ { 2 } } { 250 v }\). The car passes points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.
  2. Find the time taken for the car to travel from \(A\) to \(B\).