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CAIE M2 2011 June Q1
4 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_381_1079_255_534} A particle \(P\) of mass 0.4 kg is attached to a fixed point \(A\) by a light inextensible string. The string is inclined at \(60 ^ { \circ }\) to the vertical. \(P\) moves with constant speed in a horizontal circle of radius 0.2 m . The centre of the circle is vertically below \(A\) (see diagram).
  1. Show that the tension in the string is 8 N .
  2. Calculate the speed of the particle.
CAIE M2 2011 June Q2
6 marks Moderate -0.8
2 A stone is thrown with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally from the top of a vertical cliff 20 m above the sea. Calculate
  1. the distance from the foot of the cliff to the point where the stone enters the sea,
  2. the speed of the stone when it enters the sea.
CAIE M2 2011 June Q3
6 marks Challenging +1.8
3 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_786_1249_1455_447} A smooth hemispherical shell, with centre \(O\), weight 12 N and radius 0.4 m , rests on a horizontal plane. A particle of weight \(W \mathrm {~N}\) lies at rest on the inner surface of the hemisphere vertically below \(O\). A force of magnitude \(F \mathrm {~N}\) acting vertically upwards is applied to the highest point of the hemisphere, which is in equilibrium with its axis of symmetry inclined at \(20 ^ { \circ }\) to the horizontal (see diagram).
  1. Show, by taking moments about \(O\), that \(F = 16.48\) correct to 4 significant figures.
  2. Find the normal contact force exerted by the plane on the hemisphere in terms of \(W\). Hence find the least possible value of \(W\).
CAIE M2 2011 June Q4
8 marks Standard +0.8
4 The ends of a light elastic string of natural length 0.8 m and modulus of elasticity \(\lambda \mathrm { N }\) are attached to fixed points \(A\) and \(B\) which are 1.2 m apart at the same horizontal level. A particle of mass 0.3 kg is attached to the centre of the string, and released from rest at the mid-point of \(A B\). The particle descends 0.32 m vertically before coming to instantaneous rest.
  1. Calculate \(\lambda\).
  2. Calculate the speed of the particle when it is 0.25 m below \(A B\).
CAIE M2 2011 June Q5
8 marks Standard +0.3
5 One end of a light elastic string of natural length 0.3 m and modulus of elasticity 6 N is attached to a fixed point \(O\) on a smooth horizontal plane. The other end of the string is attached to a particle \(P\) of mass 0.2 kg , which moves on the plane in a circular path with centre \(O\). The angular speed of \(P\) is \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. For the case \(\omega = 5\), calculate the extension of the string.
  2. Express the extension of the string in terms of \(\omega\), and hence find the set of possible value of \(\omega\).
CAIE M2 2011 June Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-3_151_949_1206_598} \(O\) and \(A\) are fixed points on a horizontal surface, with \(O A = 0.5 \mathrm {~m}\). A particle \(P\) of mass 0.2 kg is projected horizontally with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in the direction \(O A\) and moves in a straight line (see diagram). At time \(t \mathrm {~s}\) after projection, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). The coefficient of friction between the surface and \(P\) is 0.5 , and a force of magnitude \(\frac { 0.4 } { x ^ { 2 } } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).
  1. Show that, while the particle is in motion, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 5 + \frac { 2 } { x ^ { 2 } } \right)\).
  2. Calculate the distance travelled by \(P\) before it comes to rest, and show that \(P\) does not subsequently move.
CAIE M2 2011 June Q7
9 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-4_597_1011_251_566} \(A B C D E\) is the cross-section through the centre of mass of a uniform prism resting in equilibrium with \(D E\) on a horizontal surface. The cross-section has the shape of a square \(O B C D\) with sides of length \(a \mathrm {~m}\), from which a quadrant \(O A E\) of a circle of radius 1 m has been removed (see diagram).
  1. Find the distance of the centre of mass of the prism from \(O\), giving the answer in terms of \(a , \pi\) and \(\sqrt { } 2\).
  2. Hence show that $$3 a ^ { 2 } ( 2 - a ) < \frac { 3 } { 2 } \pi - 2$$ and verify that this inequality is satisfied by \(a = 1.68\) but not by \(a = 1.67\).
CAIE M2 2012 June Q1
2 marks Easy -1.2
1 The end \(A\) of a \(\operatorname { rod } A B\) of length 1.2 m is freely pivoted at a fixed point. The rod rotates about \(A\) in a vertical plane. Calculate the angular speed of the rod at an instant when \(B\) has speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M2 2012 June Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-2_481_412_440_865} The diagram shows a circular object formed from a uniform semicircular lamina of weight 11 N and a uniform semicircular arc of weight 9 N . The lamina and the arc both have centre \(O\) and radius 0.7 m and are joined at the ends of their common diameter \(A B\).
  1. Show that the distance of the centre of mass of the object from \(O\) is 0.0371 m , correct to 3 significant figures. The object hangs in equilibrium, freely suspended at \(A\).
  2. Find the angle between \(A B\) and the vertical and state whether the lowest point of the object is on the lamina or on the arc.
CAIE M2 2012 June Q3
7 marks Standard +0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-2_268_652_1599_475} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-2_191_323_1653_1347} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a smooth hollow bowl whose axis is vertical and whose sloping side is inclined at \(60 ^ { \circ }\) to the horizontal. \(S\) moves with constant speed in a horizontal circle of radius 0.6 m (see Fig. 1). \(S\) is in contact with both the plane base and the sloping side of the bowl (see Fig. 2).
  1. Given that the magnitudes of the forces exerted on \(S\) by the base and sloping side of the bowl are equal, calculate the speed of \(S\).
  2. Given instead that \(S\) is on the point of losing contact with one of the surfaces, find the angular speed of \(S\).
CAIE M2 2012 June Q4
8 marks Challenging +1.2
4 A light elastic string has natural length 2.4 m and modulus of elasticity 21 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is projected vertically upwards with velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the mid-point of \(A B\). In the subsequent motion \(P\) is at instantaneous rest at a point 1.6 m above \(A B\).
  1. Find \(m\).
  2. Calculate the acceleration of \(P\) when it first passes through a point 0.5 m below \(A B\).
CAIE M2 2012 June Q5
9 marks Challenging +1.2
5 A particle \(P\) of mass 0.4 kg is released from rest at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The motion of \(P\) down the slope is opposed by a force of magnitude \(0.6 x \mathrm {~N}\), where \(x \mathrm {~m}\) is the distance \(P\) has travelled down the slope. \(P\) comes to rest before reaching the foot of the slope. Calculate
  1. the greatest speed of \(P\) during its motion,
  2. the distance travelled by \(P\) during its motion.
CAIE M2 2012 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-3_720_723_1165_712} The diagram shows the cross-section \(O A B C D E\) through the centre of mass of a uniform prism. The interior angles of the cross-section at \(O , A , C , D\) and \(E\) are all right angles. \(O A = 0.4 \mathrm {~m} , A B = 0.5 \mathrm {~m}\) and \(B C = C D = 1 \mathrm {~m}\).
  1. Calculate the distance of the centre of mass of the prism from \(O E\). The weight of the prism is 120 N . A force of magnitude \(F \mathrm {~N}\) acting along \(D E\) holds the prism in equilibrium when \(O A\) rests on a rough horizontal surface.
  2. Find the set of possible values of \(F\).
CAIE M2 2012 June Q7
9 marks Standard +0.3
7 A small ball \(B\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(41 ^ { \circ }\) above the horizontal from a point \(O\) which is 1.6 m above horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(B\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the trajectory of \(B\) is $$y = 0.869 x - 0.0390 x ^ { 2 }$$ where the coefficients are correct to 3 significant figures. A vertical fence is 1.5 m from \(O\) and perpendicular to the plane in which \(B\) moves. \(B\) just passes over the fence and subsequently strikes the ground at the point \(A\).
  2. Calculate the height of the fence, and the distance from the fence to \(A\).
CAIE M2 2012 June Q1
4 marks Standard +0.8
1 A particle \(P\) of mass 0.6 kg is projected horizontally with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a smooth horizontal surface. A horizontal force of magnitude \(0.3 x \mathrm {~N}\) acts on \(P\) in the direction \(O P\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\). Calculate the velocity of \(P\) when \(x = 8\).
CAIE M2 2012 June Q2
6 marks Standard +0.8
2 A uniform hemispherical shell of weight 8 N and a uniform solid hemisphere of weight 12 N are joined along their circumferences to form a non-uniform sphere of radius 0.2 m .
  1. Show that the distance between the centre of mass of the sphere and the centre of the sphere is 0.005 m . This sphere is placed on a horizontal surface with its axis of symmetry horizontal. The equilibrium of the sphere is maintained by a force of magnitude \(F \mathrm {~N}\) acting parallel to the axis of symmetry applied to the highest point of the sphere.
  2. Calculate \(F\).
CAIE M2 2012 June Q3
7 marks Challenging +1.2
3 A light elastic string has natural length 2.2 m and modulus of elasticity 14.3 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is released from rest at the mid-point of \(A B\). In the subsequent motion \(P\) has its greatest speed at a point 0.5 m below \(A B\).
  1. Find \(m\).
  2. Calculate the greatest speed of \(P\).
CAIE M2 2012 June Q4
7 marks Standard +0.3
4 A particle \(P\) of mass 0.25 kg moves in a straight line on a smooth horizontal surface. At time \(t \mathrm {~s}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A variable force of magnitude \(3 t \mathrm {~N}\) opposes the motion of \(P\).
  1. Given that \(P\) comes to rest when \(t = 3\), find \(v\) when \(t = 0\).
  2. Calculate the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 3\).
CAIE M2 2012 June Q5
7 marks Standard +0.3
5 A ball is projected with velocity \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. The ball subsequently bounces once on the ground at a point \(P\) before landing at a point \(Q\) where it remains at rest. The distance \(P Q\) is 17.1 m .
  1. Calculate the time taken by the ball to travel from \(O\) to \(P\) and the distance \(O P\).
  2. Given that the horizontal component of the velocity of the ball does not change at \(P\), calculate the speed of the ball when it leaves \(P\).
CAIE M2 2012 June Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{09971be0-73b6-4c73-8dfd-c89ff877950a-3_451_775_255_685} The diagram shows a uniform lamina \(A B C D E F\), formed from a semicircle with centre \(O\) and radius 1 m by removing a semicircular part with centre \(O\) and radius \(r \mathrm {~m}\).
  1. Show that the distance in metres of the centre of mass of the lamina from \(O\) is $$\frac { 4 \left( 1 + r + r ^ { 2 } \right) } { 3 \pi ( 1 + r ) } .$$ The centre of mass of the lamina lies on the \(\operatorname { arc } A B C\).
  2. Show that \(r = 0.494\), correct to 3 significant figures. The lamina is freely suspended at \(F\) and hangs in equilibrium.
  3. Find the angle between the diameter of the lamina and the vertical.
CAIE M2 2012 June Q7
10 marks Standard +0.3
7 Particles \(P\) and \(Q\), of masses 0.8 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string which passes through a small hole in a smooth horizontal table of negligible thickness. \(P\) moves with constant angular speed \(6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a circular path on the surface of the table.
  1. It is given that \(Q\) is stationary and that the part of string attached to \(Q\) is vertical. Calculate the radius of the path of \(P\), and find the speed of \(P\).
  2. It is given instead that the part of string attached to \(Q\) is inclined at \(60 ^ { \circ }\) to the vertical, and that \(Q\) moves in a horizontal circular path below the table, also with constant angular speed \(6.25 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Calculate the total length of the string.
    [0pt] [6]
CAIE M2 2012 June Q1
4 marks Moderate -0.8
1 A particle \(P\) 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. Calculate the distance \(O P\) at the instant 2 s after projection.
CAIE M2 2012 June Q2
4 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_448_547_434_799} The diagram shows a uniform object \(A B C\) of weight 3 N in the form of an arc of a circle with centre \(O\) and radius 0.7 m . The angle \(A O C\) is 2 radians. The object rests in equilibrium with \(A\) on a horizontal surface and \(C\) vertically above \(A\). Equilibrium is maintained by a horizontal force of magnitude \(F \mathrm {~N}\) applied at \(C\) in the plane of the object. Calculate \(F\).
CAIE M2 2012 June Q3
6 marks Standard +0.3
3 A particle \(P\) of mass 0.2 kg is projected horizontally from a fixed point \(O\), and moves in a straight line on a smooth horizontal surface. A force of magnitude \(0.4 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).
  1. Given that \(P\) comes to instantaneous rest when \(x = 2.5\), find the initial kinetic energy of \(P\).
  2. Find the value of \(x\) on the first occasion when the speed of \(P\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M2 2012 June Q4
8 marks Standard +0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-2_170_616_1649_767} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(S\) moves with constant speed in a horizontal circle of radius 0.4 m and is in contact with both the plane base and the curved surface of the cylinder (see diagram).
  1. Given that the horizontal and vertical forces exerted on \(S\) by the cylinder have equal magnitudes, calculate the speed of \(S\). \(S\) is now attached to the centre of the base of the cylinder by a horizontal light elastic string of natural length 0.25 m and modulus of elasticity 13 N . The sphere \(S\) is set in motion and moves in a horizontal circle with constant angular speed \(\omega \mathrm { rads } ^ { - 1 }\) and is in contact with both the plane base and the curved surface of the cylinder.
  2. It is given that the magnitudes of the horizontal and vertical forces exerted on \(S\) by the cylinder are equal if \(\omega = 8\). Calculate \(m\).
  3. For the value of \(m\) found in part (ii), find the least possible value of \(\omega\) for the motion.