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CAIE M2 2017 June Q3
7 marks Challenging +1.2
3 An open box in the shape of a cube with edges of length 0.2 m is placed with its base horizontal and its four sides vertical. The four sides and base are uniform laminas, each with weight 3 N .
  1. Calculate the height of the centre of mass of the box above its base.
    The box is now fitted with a thin uniform square lid of weight 3 N and with edges of length 0.2 m . The lid is attached to the box by a hinge of length 0.2 m and weight 2 N . The lid of the box is held partly open.
  2. Find the angle which the lid makes with the horizontal when the centre of mass of the box (including the lid and hinge) is 0.12 m above the base of the box.
CAIE M2 2017 June Q4
8 marks Standard +0.8
4 A small object of mass 0.4 kg is released from rest at a point 8 m above the ground. The object descends vertically and when its downwards displacement from its initial position is \(x \mathrm {~m}\) the object has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). While the object is moving, a force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) opposes the motion.
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 0.5 v ^ { 2 }\).
  2. Express \(v\) in terms of \(x\).
  3. Find the increase in the value of \(v\) during the final 4 m of the descent of the object.
CAIE M2 2017 June Q5
8 marks Standard +0.3
5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Calculate the greatest speed of the particle during its descent.
  2. Find the greatest distance of the particle below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram).
  3. Find the distance of the centre of mass of the rod from \(A\).
    The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal.
  4. Calculate \(P\).
    In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium.
  5. Find the coefficient of friction at \(A\). \(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\).
  6. Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
    When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\).
  7. Calculate \(V\) and \(a\).
  8. Find the direction of motion of \(P\) when it passes through \(A\).
CAIE M2 2017 June Q1
4 marks Standard +0.3
1 A particle is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle is descending at an angle of \(40 ^ { \circ }\) below the horizontal.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{37a752de-04aa-4b65-bc44-a5f28d769902-04_376_713_260_715} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\) which hangs vertically below \(A\). The particle is also attached to one end of a light elastic string of natural length 0.25 m . The other end of this string is attached to a point \(B\) which is 0.6 m from \(P\) and on the same horizontal level as \(P\). Equilibrium is maintained by a horizontal force of magnitude 7 N applied to \(P\) (see Fig. 1).
  1. Calculate the modulus of elasticity of the elastic string.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{37a752de-04aa-4b65-bc44-a5f28d769902-05_348_488_262_826} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \(P\) is released from rest by removing the 7 N force. In its subsequent motion \(P\) first comes to instantaneous rest at a point where \(B P = 0.3 \mathrm {~m}\) and the elastic string makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2).
  2. Find the value of \(m\).
CAIE M2 2017 June Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{37a752de-04aa-4b65-bc44-a5f28d769902-06_351_607_269_769} An object is made from a uniform solid hemisphere of radius 0.56 m and centre \(O\) by removing a hemisphere of radius 0.28 m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.
  1. Calculate the distance of the centre of mass of the object from \(O\).
    [0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
    The object has weight 24 N . A uniform hemisphere \(H\) of radius 0.28 m is placed in the hollow part of the object to create a non-uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is 0.15 m from \(O\).
  2. Calculate the weight of \(H\).
CAIE M2 2017 June Q4
8 marks Moderate -0.3
4 A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the trajectory of the particle. \includegraphics[max width=\textwidth, alt={}, center]{37a752de-04aa-4b65-bc44-a5f28d769902-08_63_1569_488_328}
    The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d \mathrm {~m}\) from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.
  2. Show that \(d\) is less than 30 .
  3. Find the value of \(x\) when the particle is 14 m below the level of \(O\). \includegraphics[max width=\textwidth, alt={}, center]{37a752de-04aa-4b65-bc44-a5f28d769902-10_501_614_258_762} A uniform semicircular lamina of radius 0.7 m and weight 14 N has diameter \(A B\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(A B\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(A B\) and the horizontal is \(30 ^ { \circ }\) and \(A P = 0.9 \mathrm {~m}\) (see diagram).
  4. Show that the magnitude of the force exerted by the peg on the lamina is 7.12 N , correct to 3 significant figures.
  5. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\).
CAIE M2 2018 June Q1
4 marks Moderate -0.5
1 A small ball \(B\) is projected from a point \(O\) on horizontal ground. The initial velocity of \(B\) has horizontal and vertically upwards components of \(18 \mathrm {~ms} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. For the instant 4 s after projection, find the speed and direction of motion of \(B\).
A non-uniform rod \(A B\) of length 0.5 m and weight 8 N is freely hinged to a fixed point at \(A\). The rod makes an angle of \(30 ^ { \circ }\) with the horizontal with \(B\) above the level of \(A\). The rod is held in equilibrium by a force of magnitude 12 N acting in the vertical plane containing the rod at an angle of \(30 ^ { \circ }\) to \(A B\) applied at \(B\) (see diagram). Find the distance of the centre of mass of the rod from \(A\).
CAIE M2 2018 June Q3
7 marks Standard +0.3
3 A particle \(P\) of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point \(O\). At time \(t \mathrm {~s}\) after projection the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(0.8 t \mathrm {~N}\) directed away from \(O\) acts on \(P\) and a force of magnitude \(2 \mathrm { e } ^ { - t } \mathrm {~N}\) opposes the motion of \(P\).
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 2 t - 5 \mathrm { e } ^ { - t }\).
  2. Given that \(v = 8\) when \(t = 1\), express \(v\) in terms of \(t\).
  3. Find the speed of projection of \(P\).
CAIE M2 2018 June Q4
9 marks Standard +0.3
4 A small object is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the path.
    The object passes through the point with coordinates \(( 24,18 )\).
  2. Find \(V\).
  3. The object passes through two points which are 22.5 m above the level of \(O\). Find the values of \(x\) for these points.
CAIE M2 2018 June Q5
8 marks Standard +0.8
5 A particle \(P\) of mass 0.7 kg is attached by a light elastic string to a fixed point \(O\) on a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The natural length of the string is 0.5 m and the modulus of elasticity is 20 N . The particle \(P\) is projected up the line of greatest slope through \(O\) from a point \(A\) below the level of \(O\). The initial kinetic energy of \(P\) is 1.8 J and the initial elastic potential energy in the string is also 1.8 J .
  1. Find the distance \(O A\).
    ....................................................................................................................................
  2. Find the greatest speed of \(P\) in the motion.
CAIE M2 2018 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{c403a227-586d-4c1f-a392-e475234fc0a0-10_262_732_264_705} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle \(Q\) of mass 0.3 kg . The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins \(Q\) to a fixed point \(A\) which is 0.4 m vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).
  1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended.
  2. Find the distance \(H P\) given that the angular speed of \(P\) is \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
CAIE M2 2018 June Q7
10 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{c403a227-586d-4c1f-a392-e475234fc0a0-12_732_581_260_774} A uniform solid cone has height 1.2 m and base radius 0.5 m . A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).
  1. Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is \(0.0352 \pi \mathrm {~m} ^ { 3 }\).
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  2. Find the distance of the centre of mass of the object from its base.
    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 2018 June Q2
6 marks Standard +0.8
2 One end of a light elastic string is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass 0.4 kg . The string has natural length 0.6 m and modulus of elasticity 24 N . The particle is released from rest at \(O\). Find the two possible values of the distance \(O P\) for which the particle has speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M2 2018 June Q3
5 marks Standard +0.8
3 \(A B C\) is an object made from a uniform wire consisting of two straight portions \(A B\) and \(B C\), in which \(A B = a , B C = x\) and angle \(A B C = 90 ^ { \circ }\). When the object is freely suspended from \(A\) and in equilibrium, the angle between \(A B\) and the horizontal is \(\theta\) (see diagram).
  1. Show that \(x ^ { 2 } \tan \theta - 2 a x - a ^ { 2 } = 0\).
  2. Given that \(\tan \theta = 1.25\), calculate the length of the wire in terms of \(a\).
CAIE M2 2018 June Q4
7 marks Moderate -0.3
4 A particle \(P\) is projected from a point \(O\) on horizontal ground with initial speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angle of projection \(30 ^ { \circ }\). At the instant \(t s\) after projection, the horizontal and vertically upwards 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 find the equation of the trajectory of \(P\). \(P\) is at the same height above the ground at two points which are a horizontal distance apart of 15 m .
  2. Calculate this height. \includegraphics[max width=\textwidth, alt={}, center]{874622ab-4c75-4a32-bae5-eef780ed0cc0-08_607_1022_255_573} A uniform object is made by joining a solid cone of height 0.8 m and base radius 0.6 m and a cylinder. The cylinder has length 0.4 m and radius 0.5 m . The cylinder has a cylindrical hole of length 0.4 m and radius \(x \mathrm {~m}\) drilled through it along the axis of symmetry. A plane face of the cylinder is attached to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is placed with points on the base of the cone and the base of the cylinder in contact with a horizontal surface (see diagram). The object is on the point of toppling.
  3. Show that the centre of mass of the object is 0.15 m from the base of the cone.
  4. Find \(x\).
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
CAIE M2 2018 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{874622ab-4c75-4a32-bae5-eef780ed0cc0-10_757_464_258_836} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) is also attached to one end of a second light inextensible string of length 0.6 m , the other end of which is attached to a fixed point \(B\) vertically below \(A\). The particle moves in a horizontal circle of radius 0.3 m , which has its centre at the mid-point of \(A B\), with both strings straight (see diagram).
  1. Calculate the least possible angular speed of \(P\).
    The string \(A P\) will break if its tension exceeds 8 N . The string \(B P\) will break if its tension exceeds 5 N .
  2. Find the greatest possible speed of \(P\). \(7 \quad\) A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) above horizontal ground. At time \(t \mathrm {~s}\) after its release the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards. A vertically downwards force of magnitude \(0.6 t \mathrm {~N}\) acts on \(P\). A vertically upwards force of magnitude \(k \mathrm { e } ^ { - t } \mathrm {~N}\), where \(k\) is a constant, also acts on \(P\).
  3. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 5 k \mathrm { e } ^ { - t } + 3 t\).
  4. Find the greatest value of \(k\) for which \(P\) does not initially move upwards.
  5. Given that \(k = 1\), and that \(P\) strikes the ground when \(t = 2\), find the height of \(O\) above the ground. [5]
    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 2018 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-10_262_732_264_705} A particle \(P\) of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle \(Q\) of mass 0.3 kg . The string passes through a small hole \(H\) in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins \(Q\) to a fixed point \(A\) which is 0.4 m vertically below \(H\). The particle \(P\) moves on the surface in a horizontal circle with centre \(H\) (see diagram).
  1. Calculate the greatest possible speed of \(P\) for which the elastic string is not extended.
  2. Find the distance \(H P\) given that the angular speed of \(P\) is \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
CAIE M2 2018 June Q7
10 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-12_732_581_260_774} A uniform solid cone has height 1.2 m and base radius 0.5 m . A uniform object is made by drilling a cylindrical hole of radius 0.2 m through the cone along the axis of symmetry (see diagram).
  1. Show that the height of the object is 0.72 m and that the volume of the cone removed by the drilling is \(0.0352 \pi \mathrm {~m} ^ { 3 }\).
    [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  2. Find the distance of the centre of mass of the object from its base.
    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 Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-03_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 Q2
5 marks Standard +0.3
2 A particle is projected with speed \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. At the instant 4 s after projection the speed of the particle is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its direction of motion is \(30 ^ { \circ }\) above the horizontal. Find \(V\) and \(\theta\).
CAIE M2 2019 June Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{bba68fb2-88c6-4883-931b-f738cda2dce3-05_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 2019 June Q4
8 marks Moderate -0.3
4 A small ball is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal 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.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence find the equation of the trajectory of the ball.
  2. Find \(x\) for the position of the ball when its path makes an angle of \(15 ^ { \circ }\) below the horizontal. [4]
CAIE M2 2019 June Q5
8 marks Challenging +1.2
5 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point \(( 0.5 + x ) \mathrm { m }\) vertically below \(O\). The particle \(P\) comes to instantaneous rest at \(O\).
  1. Find \(x\).
  2. Find the greatest speed of \(P\).
CAIE M2 2019 June Q6
7 marks Standard +0.3
6 \(A B C\) is a uniform lamina in the form of a triangle with \(A B = 0.3 \mathrm {~m} , B C = 0.6 \mathrm {~m}\) and a right angle at \(B\) (see diagram).
  1. State the distances of the centre of mass of the lamina from \(A B\) and from \(B C\). Distance from \(A B\) Distance from \(B C\) \(\_\_\_\_\) The lamina is freely suspended at \(B\) and hangs in equilibrium.
  2. Find the angle between \(A B\) and the horizontal.
    A force of magnitude 12 N is applied along the edge \(A C\) of the lamina in the direction from \(A\) towards \(C\). The lamina, still suspended at \(B\), is now in equilibrium with \(A B\) vertical.
  3. Calculate the weight of the lamina.
CAIE M2 2019 June Q7
12 marks Challenging +1.8
7 A particle \(P\) of mass 0.5 kg is attached to a fixed point \(O\) by a light elastic string of natural length 1 m and modulus of elasticity 16 N . The particle \(P\) is projected vertically upwards from \(O\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 x ^ { 2 } \mathrm {~N}\) acts on \(P\) when \(P\) has displacement \(x \mathrm {~m}\) above \(O\). After projection the upwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that, before the string becomes taut, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 10 - 0.2 x ^ { 2 }\).
  2. Find the velocity of \(P\) at the instant the string becomes taut.
  3. Find an expression for the acceleration of \(P\) while it is moving upwards after the string becomes taut.
  4. Verify that \(P\) comes to instantaneous rest before the extension of the string is 0.5 m .
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.