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CAIE M2 2003 June Q4
7 marks Standard +0.8
4 A particle of mass 0.2 kg moves in a straight line on a smooth horizontal surface. When its displacement from a fixed point on the surface is \(x \mathrm {~m}\), its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion is opposed by a force of magnitude \(\frac { 1 } { 3 v } \mathrm {~N}\).
  1. Show that \(3 v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 5\).
  2. Find the value of \(v\) when \(x = 7.4\), given that \(v = 4\) when \(x = 0\).
CAIE M2 2003 June Q5
7 marks Moderate -0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-3_305_634_1747_758} A toy aircraft of mass 0.5 kg is attached to one end of a light inextensible string of length 9 m . The other end of the string is attached to a fixed point \(O\). The aircraft moves with constant speed in a horizontal circle. The string is taut, and makes an angle of \(60 ^ { \circ }\) with the upward vertical at \(O\) (see diagram). In a simplified model of the motion, the aircraft is treated as a particle and the force of the air on the aircraft is taken to act vertically upwards with magnitude 8 N . Find
  1. the tension in the string,
  2. the speed of the aircraft.
CAIE M2 2003 June Q6
9 marks Moderate -0.3
6 A particle is projected with speed \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on horizontal ground. The angle of projection is \(\alpha ^ { \circ }\) above the horizontal. The particle reaches the ground again after 10 s .
  1. Find the value of \(\alpha\).
  2. Find the greatest height reached by the particle.
  3. At time \(T\) s after the instant of projection the direction of motion of the particle is at an angle of \(45 ^ { \circ }\) above the horizontal. Find the value of \(T\).
CAIE M2 2003 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-4_232_905_762_621} A light elastic string has natural length 10 m and modulus of elasticity 130 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. A small stone is attached to the mid-point of the string and hangs in equilibrium at a point 2.5 m below \(A B\), as shown in the diagram. With the stone in this position the length of the string is 13 m .
  1. Find the tension in the string.
  2. Show that the mass of the stone is 3 kg . The stone is now held at rest at a point 8 m vertically below the mid-point of \(A B\).
  3. Find the elastic potential energy of the string in this position.
  4. The stone is now released. Find the speed with which it passes through the mid-point of \(A B\).
CAIE M2 2004 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_182_843_264_651} A uniform rigid plank has mass 10 kg and length 4 m . The plank has 0.9 m of its length in contact with a horizontal platform. A man \(M\) of mass 75 kg stands on the end of the plank which is in contact with the platform. A child \(C\) of mass 25 kg walks on to the overhanging part of the plank (see diagram). Find the distance between the man and the child when the plank is on the point of tilting.
CAIE M2 2004 June Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_291_732_822_708} A uniform lamina \(A B C D E\) consists of a rectangular part with sides 5 cm and 10 cm , and a part in the form of a quarter of a circle of radius 5 cm , as shown in the diagram.
  1. Show that the distance of the centre of mass of the part \(C D E\) of the lamina is \(\frac { 20 } { 3 \pi } \mathrm {~cm}\) from \(C E\).
  2. Find the distance of the centre of mass of the lamina \(A B C D E\) from the edge \(A B\).
CAIE M2 2004 June Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-2_145_792_1656_680} A particle \(P\) of mass 0.6 kg moves in a straight line on a smooth horizontal surface. A force of magnitude \(\frac { 3 } { x ^ { 3 } }\) newtons acts on the particle in the direction from \(P\) to \(O\), where \(O\) is a fixed point of the surface and \(x \mathrm {~m}\) is the distance \(O P\) (see diagram). The particle \(P\) is released from rest at the point where \(x = 10\). Find the speed of \(P\) when \(x = 2.5\).
CAIE M2 2004 June Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-3_737_700_264_721} A uniform beam has length 2.4 m and weight 68 N . The beam is hinged at a fixed point of a vertical wall, and held in a horizontal position by a light rod of length 2.5 m . One end of the rod is attached to the beam at a point 0.7 m from the wall, and the other end of the rod is attached to the wall at a point vertically below the hinge. The beam carries a load of 750 N at its end (see diagram).
  1. Find the force in the rod. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally towards the wall and \(Y \mathrm {~N}\) vertically downwards.
  2. Find the values of \(X\) and \(Y\).
CAIE M2 2004 June Q5
7 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{835616aa-0b2b-4e8c-bbbf-60b72dc5ea3e-3_321_698_1692_726} One end of a light elastic string of natural length 4 m and modulus of elasticity 200 N is attached to a fixed point \(A\). The other end is attached to the end \(C\) of a uniform rod \(C D\) of mass 10 kg . One end of another light elastic string, which is identical to the first, is attached to a fixed point \(B\) and the other end is attached to \(D\), as shown in the diagram. The distance \(A B\) is equal to the length of the rod, and \(A B\) is horizontal. The rod is released from rest with \(C\) at \(A\) and \(D\) at \(B\). While the strings are taut, the speed of the rod is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the rod is at a distance of \(( 4 + x ) \mathrm { m }\) below \(A B\).
  1. Show that \(v ^ { 2 } = 10 \left( 8 + 2 x - x ^ { 2 } \right)\).
  2. Hence find the value of \(x\) when the rod is at its lowest point.
CAIE M2 2004 June Q6
9 marks Standard +0.3
6 A particle is projected from a point \(O\) on horizontal ground. The velocity of projection has magnitude \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and direction upwards at \(35 ^ { \circ }\) to the horizontal. The particle passes through the point \(M\) at time \(T\) seconds after the instant of projection. The point \(M\) is 2 m above the ground and at a horizontal distance of 25 m from \(O\).
  1. Find the values of \(V\) and \(T\).
  2. Find the speed of the particle as it passes through \(M\) and determine whether it is moving upwards or downwards.
CAIE M2 2004 June Q7
10 marks Standard +0.3
7 One end of a light inextensible string of length 0.15 m is attached to a fixed point which is above a smooth horizontal surface. A particle of mass 0.5 kg is attached to the other end of the string. The particle moves with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, with the string taut and making an angle of \(\theta ^ { \circ }\) with the downward vertical.
  1. Given that \(\theta = 60\) and that the particle is not in contact with the surface, find \(v\).
  2. Given instead that \(\theta = 45\) and \(v = 0.9\), and that the particle is in contact with the surface, find
    (a) the tension in the string,
    (b) the force exerted by the surface on the particle.
CAIE M2 2005 June Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_643_218_264_959} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of a light elastic string of natural length 0.8 m and modulus of elasticity 8 N . One end of the string is attached to a fixed point \(A\) and the other end is attached to a fixed point \(B\) which is 2 m vertically below \(A\). When the particle is in equilibrium the distance \(A P\) is 1.1 m (see diagram). Find the value of \(m\).
CAIE M2 2005 June Q2
6 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-2_561_570_1274_790} A particle of mass 0.15 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The magnitude of the acceleration of the particle is \(7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The string makes an angle of \(\theta ^ { \circ }\) with the downward vertical, as shown in the diagram. Find
  1. the value of \(\theta\) to the nearest whole number,
  2. the tension in the string,
  3. the speed of the particle.
CAIE M2 2005 June Q3
6 marks Standard +0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-3_426_429_264_858} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \(A B C D E F\) is the L -shaped cross-section of a uniform solid. This cross-section passes through the centre of mass of the solid and has dimensions as shown in Fig. 1.
  1. Find the distance of the centre of mass of the solid from the edge \(A B\) of the cross-section. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-3_588_1020_1087_561} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The solid rests in equilibrium with the face containing the edge \(A F\) of the cross-section in contact with a horizontal table. The weight of the solid is \(W\) N. A horizontal force of magnitude \(P\) N is applied to the solid at the point \(B\), in the direction of \(B C\) (see Fig. 2). The table is sufficiently rough to prevent sliding.
  2. Find \(P\) in terms of \(W\), given that the equilibrium of the solid is about to be broken.
CAIE M2 2005 June Q4
7 marks Standard +0.8
4 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. \(P\) is released from rest at a point on the table 3.5 m from \(O\). The speed of \(P\) at the instant the string becomes slack is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the work done against friction during the period from the release of \(P\) until the string becomes slack,
  2. the coefficient of friction between \(P\) and the table.
CAIE M2 2005 June Q5
7 marks Standard +0.8
5 The acceleration of a particle moving in a straight line is \(( x - 2.4 ) \mathrm { m } \mathrm { s } ^ { - 2 }\) when its displacement from a fixed point \(O\) of the line is \(x \mathrm {~m}\). The velocity of the particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and it is given that \(v = 2.5\) when \(x = 0\). Find
  1. an expression for \(v\) in terms of \(x\),
  2. the minimum value of \(v\).
CAIE M2 2005 June Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-4_620_899_644_623} A rigid rod consists of two parts. The part \(B C\) is in the form of an arc of a circle of radius 2 m and centre \(O\), with angle \(B O C = \frac { 1 } { 4 } \pi\) radians. \(B C\) is uniform and has weight 3 N . The part \(A B\) is straight and of length 2 m ; it is uniform and has weight 4 N . The part \(A B\) of the rod is a tangent to the arc \(B C\) at \(B\). The end \(A\) of the rod is freely hinged to a fixed point of a vertical wall. The rod is held in equilibrium, with the straight part \(A B\) making an angle of \(\frac { 1 } { 4 } \pi\) radians with the wall, by means of a horizontal string attached to \(C\). The string is in the same vertical plane as the rod, and the tension in the string is \(T \mathrm {~N}\) (see diagram).
  1. Show that the centre of mass \(G\) of the part \(B C\) of the rod is at a distance of 2.083 m from the wall, correct to 4 significant figures.
  2. Find the value of \(T\).
  3. State the magnitude of the horizontal component and the magnitude of the vertical component of the force exerted on the rod by the hinge. \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-5_579_1118_264_516} A particle \(A\) is released from rest at time \(t = 0\), at a point \(P\) which is 7 m above horizontal ground. At the same instant as \(A\) is released, a particle \(B\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(P\) is 24 m . Particle \(B\) moves in the vertical plane containing \(O\) and \(P\), with initial speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle of \(\theta\) above the horizontal (see diagram). Write down
  4. an expression for the height of \(A\) above the ground at time \(t \mathrm {~s}\),
  5. an expression in terms of \(V , \theta\) and \(t\) for
    (a) the horizontal distance of \(B\) from \(O\),
    (b) the height of \(B\) above the ground. At time \(t = T\) the particles \(A\) and \(B\) collide at a point above the ground.
  6. Show that \(\tan \theta = \frac { 7 } { 24 }\) and that \(V T = 25\).
  7. Deduce that \(7 V ^ { 2 } > 3125\).
CAIE M2 2006 June Q1
5 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_248_1267_276_440} A light elastic string has natural length 0.6 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.63 m apart. A particle \(P\) of mass 0.064 kg is attached to the mid-point of the string and hangs in equilibrium at a point 0.08 m below \(A B\) (see diagram). Find
  1. the tension in the string,
  2. the value of \(\lambda\).
CAIE M2 2006 June Q2
5 marks Standard +0.3
2 A uniform solid cone has height 38 cm .
  1. Write down the distance of the centre of mass of the cone from its base. \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_497_547_1224_840} The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted, and the cone remains in equilibrium until the angle of inclination of the plane reaches \(31 ^ { \circ }\) (see diagram), when the cone topples.
  2. Find the radius of the cone.
  3. Show that \(\mu \geqslant 0.601\), correct to 3 significant figures, where \(\mu\) is the coefficient of friction between the cone and the plane.
CAIE M2 2006 June Q3
6 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-3_437_567_269_788} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) moves with constant speed in a horizontal circle, with the string taut and inclined at \(35 ^ { \circ }\) to the vertical. \(O P\) rotates with angular speed \(2.2 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about the vertical axis through \(O\) (see diagram). Find
  1. the value of \(L\),
  2. the speed of \(P\) in \(\mathrm { m } \mathrm { s } ^ { - 1 }\).
CAIE M2 2006 June Q4
7 marks Standard +0.3
4 An object of mass 0.4 kg is projected vertically upwards from the ground, with an initial speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.1 v\) newtons acts on the object during its ascent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the object at time \(t \mathrm {~s}\) after it starts to move.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.25 ( v + 40 )\).
  2. Find the value of \(t\) at the instant that the object reaches its maximum height.
CAIE M2 2006 June Q5
7 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-3_531_791_1633_678} A uniform lamina of weight 15 N has dimensions as shown in the diagram.
  1. Show that the distance of the centre of mass of the lamina from \(A B\) is 0.22 m . The lamina is freely hinged at \(B\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(C\). The string passes over a fixed smooth pulley and a particle of mass 1.1 kg is attached to the other end of the string. The lamina is in equilibrium with \(B C\) horizontal. The string is taut and makes an angle of \(\theta ^ { \circ }\) with the horizontal at \(C\), and the particle hangs freely below the pulley (see diagram).
  2. Find the value of \(\theta\).
CAIE M2 2006 June Q6
9 marks Standard +0.8
6 A light elastic string has natural length 2 m and modulus of elasticity 0.8 N . One end of the string is attached to a fixed point \(O\) of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 12 } { 13 }\). A particle \(P\) of mass 0.052 kg is attached to the other end of the string. The coefficient of friction between the particle and the plane is 0.4 . \(P\) is released from rest at \(O\).
  1. When \(P\) has moved \(d\) metres down the plane from \(O\), where \(d > 2\), find expressions in terms of \(d\) for
    (a) the loss in gravitational potential energy of \(P\),
    (b) the gain in elastic potential energy of the string,
    (c) the work done by the frictional force acting on \(P\).
  2. Show that \(d ^ { 2 } - 6 d + 4 = 0\) when \(P\) is at its lowest point, and hence find the value of \(d\) in this case.
CAIE M2 2006 June Q7
11 marks Standard +0.3
7 A stone is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The stone is at its highest point when it has travelled a horizontal distance of 19.2 m .
  1. Find the value of \(V\). After passing through its highest point the stone strikes a vertical wall at a point 4 m above the ground.
  2. Find the horizontal distance between \(O\) and the wall. At the instant when the stone hits the wall the horizontal component of the stone's velocity is halved in magnitude and reversed in direction. The vertical component of the stone's velocity does not change as a result of the stone hitting the wall.
  3. Find the distance from the wall of the point where the stone reaches the ground.
CAIE M2 2007 June Q1
4 marks Moderate -0.8
1 \includegraphics[max width=\textwidth, alt={}, center]{57f7ca89-f028-447a-9ac9-55f931201e6b-2_467_645_274_749} A uniform semicircular lamina has radius 5 m . The lamina rotates in a horizontal plane about a vertical axis through \(O\), the mid-point of its diameter. The angular speed of the lamina is \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Find
  1. the distance of the centre of mass of the lamina from \(O\),
  2. the speed with which the centre of mass of the lamina is moving.