Questions M2 (1391 questions)

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CAIE M2 2018 June Q6
5 marks
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
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
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
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
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
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
4 marks
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
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
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
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.
CAIE M2 2019 June Q1
1 A small ball is projected from a point \(O\) on horizontal ground at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(t \mathrm {~s}\) after projection the vertically upwards displacement of the ball from \(O\) is \(\left( 14 t - k t ^ { 2 } \right) \mathrm { m }\), where \(k\) is a constant.
  1. State the value of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-03_56_1563_495_331}
  2. Show that the initial speed of the ball is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the horizontal displacement of the ball from \(O\) when \(t = 3\).
CAIE M2 2019 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-04_442_554_260_794} A uniform lamina \(A B C E F G\) is formed from a square \(A B D G\) by removing a smaller square \(C D F E\) from one corner. \(A B = 0.7 \mathrm {~m}\) and \(D F = 0.3 \mathrm {~m}\) (see diagram). Find the distance of the centre of mass of the lamina from \(A\).
CAIE M2 2019 June Q3
3 A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.5 m . The point \(A\) is 0.3 m above a smooth horizontal surface. The particle \(P\) moves in a horizontal circle on the surface with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Calculate the tension in the string.
    \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-05_67_1569_486_328}
  2. Find the magnitude of the force exerted by the surface on \(P\).
CAIE M2 2019 June Q4
4 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 16 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.8 m vertically below \(O\). When the extension of the string is \(x \mathrm {~m}\), the downwards velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and a force of magnitude \(25 x ^ { 2 } \mathrm {~N}\) opposes the motion of \(P\).
  1. Show that, when \(P\) is moving downwards, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 40 x - 50 x ^ { 2 }\).
  2. For the instant when \(P\) has its greatest downwards speed, find the kinetic energy of \(P\) and the elastic potential energy stored in the string.
CAIE M2 2019 June Q5
5 A light elastic string has natural length \(a \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\). When the length of the string is 1.6 m the tension is 4 N . When the length of the string is 2 m the tension is 6 N .
  1. Find the values of \(a\) and \(\lambda\).
    One end of the string is attached to a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle \(P\) moves with constant speed on the surface in a circle with centre \(O\) and radius 1.9 m .
  2. Find the speed of \(P\).
CAIE M2 2019 June Q6
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
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
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
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
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
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
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
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
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
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
    (a) calculate \(R\) when \(\omega = 2\),
    (b) find the greatest possible value of \(T\) and the corresponding speed of \(P\).