Questions M2 (1537 questions)

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CAIE M2 2019 June Q6
8 marks Standard +0.3
6 A particle is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find \(\theta\).
  2. Show that at the instant 4 s after projection the particle is 33.75 m below the level of the point of projection and find the direction of motion at this instant.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-12_259_609_255_769} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows an object made from a uniform wire of length 0.8 m . The object consists of a straight part \(A B\), and a semicircular part \(B C\) such that \(A , B\) and \(C\) lie in the same straight line. The radius of the semicircle is \(r \mathrm {~m}\) and the centre of mass of the object is 0.1 m from line \(A B C\).
  3. Show that \(r = 0.2\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-13_615_383_260_881} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The object is freely suspended at \(A\) and a horizontal force of magnitude 7 N is applied to the object at \(C\) so that the object is in equilibrium with \(A B C\) vertical (see Fig. 2).
  4. Calculate the weight of the object.
    The 7 N force is removed and the object hangs in equilibrium with \(A B C\) at an angle of \(\theta ^ { \circ }\) with the vertical.
  5. Find \(\theta\).
    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 2019 June Q1
5 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{111bcbf6-daaf-4d8d-9299-d591ac7369f1-03_231_970_258_591} A particle \(P\) of mass 0.3 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.8 m . The fixed point \(O\) is 0.15 m vertically below \(A\). The particle \(P\) moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle with centre \(O\) (see diagram).
  1. Show that the tension in the string is 16 N .
  2. Find the value of \(v\).
CAIE M2 2019 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{111bcbf6-daaf-4d8d-9299-d591ac7369f1-05_448_802_258_676} The diagram shows the cross-section through the centre of mass of a uniform solid object. The object is a cylinder of radius 0.2 m and length 0.7 m , from which a hemisphere of radius 0.2 m has been removed at one end. The point \(A\) is the centre of the plane face at the other end of the object. Find the distance of the centre of mass of the object from \(A\).
[0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
CAIE M2 2016 March Q1
5 marks Standard +0.3
1 A particle is projected from a point on horizontal ground. At the instant 2 s after projection, the particle has travelled a horizontal distance of 30 m and is at its greatest height above the ground. Find the initial speed and the angle of projection of the particle.
CAIE M2 2016 March Q2
5 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_295_805_484_671} A uniform solid hemisphere of weight 60 N and radius 0.8 m rests in limiting equilibrium with its curved surface on a rough horizontal plane. The axis of symmetry of the hemisphere is inclined at an angle of \(\theta\) to the horizontal, where \(\cos \theta = 0.28\). Equilibrium is maintained by a horizontal force of magnitude \(P\) N applied to the lowest point of the circular rim of the hemisphere (see diagram).
  1. Show that \(P = 8.75\).
  2. Find the coefficient of friction between the hemisphere and the plane.
CAIE M2 2016 March Q3
5 marks Standard +0.8
3 A stone is thrown with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point on horizontal ground. Find the distance between the two points at which the path of the stone makes an angle of \(45 ^ { \circ }\) with the horizontal.
CAIE M2 2016 March Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_549_579_1505_781} A uniform lamina is made by joining a rectangle \(A B C D\), in which \(A B = C D = 0.56 \mathrm {~m}\) and \(B C = A D = 2 \mathrm {~m}\), and a square \(E F G A\) of side 1.2 m . The vertex \(E\) of the square lies on the edge \(A D\) of the rectangle (see diagram). The centre of mass of the lamina is a distance \(h \mathrm {~m}\) from \(B C\) and a distance \(v \mathrm {~m}\) from \(B A G\).
  1. Find the value of \(h\) and show that \(v = h\). The lamina is freely suspended at the point \(B\) and hangs in equilibrium.
  2. State the angle which the edge \(B C\) makes with the horizontal. Instead, the lamina is now freely suspended at the point \(F\) and hangs in equilibrium.
  3. Calculate the angle between \(F G\) and the vertical.
CAIE M2 2016 March Q5
9 marks Standard +0.3
5 A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(A\), and \(P\) hangs in equilibrium.
  1. Calculate the extension of the string. \(P\) is projected vertically downwards from the equilibrium position with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the distance \(A P\) when the speed of \(P\) is \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(P\) is below the equilibrium position.
  3. Calculate the speed of \(P\) when it is 0.5 m above the equilibrium position.
CAIE M2 2016 March Q6
9 marks Challenging +1.8
6 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a plane inclined at \(30 ^ { \circ }\) to the horizontal. At time \(t \mathrm {~s}\) after its release, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement \(x \mathrm {~m}\) down the plane from \(O\). The coefficient of friction between \(P\) and the plane increases as \(P\) moves down the plane, and equals \(0.1 x ^ { 2 }\).
  1. Show that \(2 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - ( \sqrt { } 3 ) x ^ { 2 }\).
  2. Calculate the maximum speed of \(P\).
  3. Find the value of \(x\) at the point where \(P\) comes to rest.
CAIE M2 2016 March Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-3_451_432_1434_852} One end of a light inextensible string is attached to the highest point \(A\) of a solid fixed sphere with centre \(O\) and radius 0.6 m . The other end of the string is attached to a particle \(P\) of mass 0.2 kg which rests in contact with the smooth surface of the sphere. The angle \(A O P = 60 ^ { \circ }\) (see diagram). The sphere exerts a contact force of magnitude \(R \mathrm {~N}\) on \(P\) and the tension in the string is \(T \mathrm {~N}\).
  1. By resolving vertically, show that \(R + ( \sqrt { } 3 ) T = 4\). \(P\) is now set in motion, and moves with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on the surface of the sphere.
  2. Find an equation involving \(R , T\) and \(\omega\).
  3. Hence
    1. calculate \(R\) when \(\omega = 2\),
    2. find the greatest possible value of \(T\) and the corresponding speed of \(P\).
CAIE M2 2019 March Q1
5 marks Moderate -0.5
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
6 marks Standard +0.3
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
6 marks Moderate -0.8
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 Standard +0.3
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
8 marks Standard +0.3
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
11 marks Challenging +1.2
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 Moderate -0.5
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
5 marks Standard +0.3
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
6 marks Moderate -0.3
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
6 marks Standard +0.3
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
9 marks Standard +0.8
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
10 marks Standard +0.3
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
3 marks Moderate -0.8
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
6 marks Standard +0.3
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
8 marks Standard +0.3
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.