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CAIE M2 2015 November Q7
11 marks Challenging +1.2
7 A particle \(P\) of mass \(M \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 12.5 N . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically until it comes to instantaneous rest at the point \(B\). The greatest speed of \(P\) during its descent is \(4.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(e \mathrm {~m}\).
  1. Show that \(e = 0.64 M\).
  2. Find a second equation in \(e\) and \(M\), and hence find \(M\).
  3. Calculate the distance \(A B\).
CAIE M2 2016 November Q1
4 marks Standard +0.3
1 A particle \(P\) of mass 0.3 kg moves in a circle with centre \(O\) on a smooth horizontal surface. \(P\) is attached to \(O\) by a light elastic string of modulus of elasticity 12 N and natural length \(l \mathrm {~m}\). The speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the radius of the circle in which it moves is \(2 l \mathrm {~m}\). Calculate \(l\).
CAIE M2 2016 November Q2
7 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-2_318_495_484_824} A uniform wire is bent to form an object which has a semicircular arc with diameter \(A B\) of length 1.2 m , with a smaller semicircular arc with diameter \(B C\) of length 0.6 m . The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.
  1. Show that the distance of the centre of mass of the object from the line \(A C B\) is 0.191 m , correct to 3 significant figures. The object is freely suspended at \(A\) and hangs in equilibrium.
  2. Find the angle between \(A C B\) and the vertical.
CAIE M2 2016 November Q3
7 marks Standard +0.3
3 A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement is \(x \mathrm {~m}\) from \(O\). The force acting on \(B\) has magnitude \(\left( 2 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and is directed horizontally away from \(O\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 8\).
  2. Find the velocity of \(B\) when \(x = 1.5\). An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 6 - 3 x$$
  3. Find the magnitude of this extra force and state the direction in which it acts.
CAIE M2 2016 November Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-3_388_650_264_749} The diagram shows the cross-section \(A B C D\) through the centre of mass of a uniform solid prism. \(A B = 0.9 \mathrm {~m} , B C = 2 a \mathrm {~m} , A D = a \mathrm {~m}\) and angle \(A B C =\) angle \(B A D = 90 ^ { \circ }\).
  1. Calculate the distance of the centre of mass of the prism from \(A D\).
  2. Express the distance of the centre of mass of the prism from \(A B\) in terms of \(a\). The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(A D\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(B C\).
  3. Given that the prism is on the point of toppling, calculate \(a\).
CAIE M2 2016 November Q5
7 marks Standard +0.3
5 A small ball \(B\) of mass 0.4 kg moves in a horizontal circle with centre \(O\) and radius 0.6 m on a smooth horizontal surface. One end of a light inextensible string is attached to \(B\); the other end of the string is attached to a fixed point 0.45 m vertically above \(O\).
  1. Given that the tension in the string is 5 N , calculate the speed of \(B\).
  2. Find the greatest possible tension in the string for the motion, and the corresponding angular speed of \(B\).
CAIE M2 2016 November Q6
7 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-3_483_419_1800_863} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m . A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate
  1. the speed of \(P\) when it is at the same horizontal level as \(O\),
  2. the speed of \(P\) at the instant when the string becomes slack.
CAIE M2 2016 November Q7
11 marks Standard +0.8
7 A particle \(P\) is projected with speed \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a horizontal plane. In the subsequent motion, 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 $$y = k x - \frac { \left( 1 + k ^ { 2 } \right) x ^ { 2 } } { 245 }$$ where \(k\) is a constant. \(P\) passes through the points \(A ( 14 , a )\) and \(B ( 42,2 a )\), where \(a\) is a constant.
  1. Calculate the two possible values of \(k\) and hence show that the larger of the two possible angles of projection is \(63.435 ^ { \circ }\), correct to 3 decimal places. For the larger angle of projection, calculate
  2. the time after projection when \(P\) passes through \(A\),
  3. the speed and direction of motion of \(P\) when it passes through \(B\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2016 November Q1
3 marks Moderate -0.8
1 A stone \(S\) is thrown horizontally from the top \(T\) of a high tower. At the instant 1.6 s after \(S\) is thrown, the line \(S T\) makes an angle of \(30 ^ { \circ }\) below the horizontal. Find the speed with which \(S\) is thrown. [3]
CAIE M2 2016 November Q2
5 marks Standard +0.3
2 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string with modulus of elasticity 24 N and natural length 0.6 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) hangs in equilibrium vertically below \(A\).
  1. Find the distance \(A P\). The particle \(P\) is raised to \(A\) and released from rest.
  2. Calculate the greatest speed of \(P\) in the subsequent motion.
CAIE M2 2016 November Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-2_442_789_941_676} A non-uniform \(\operatorname { rod } A B\) of length 0.5 m is freely hinged to a fixed point at \(A\). The rod is in equilibrium at an angle of \(30 ^ { \circ }\) with the horizontal with \(B\) below the level of \(A\). Equilibrium is maintained by a force of magnitude \(F\) N applied at \(B\) acting at \(45 ^ { \circ }\) above the horizontal in the vertical plane containing \(A B\). The force exerted by the hinge on the rod has magnitude 10 N and acts at an angle of \(60 ^ { \circ }\) above the horizontal (see diagram).
  1. By resolving horizontally and vertically, calculate \(F\) and the weight of the rod.
  2. Find the distance of the centre of mass of the rod from \(A\).
CAIE M2 2016 November Q4
8 marks Standard +0.3
4 A particle \(P\) is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. \(P\) subsequently bounces when it first strikes the ground at the point \(A\).
  1. Find the time after projection when \(P\) first strikes the ground, and the distance \(O A\). When \(P\) bounces at \(A\) the horizontal component of the velocity of \(P\) is unchanged. The vertical component of velocity is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately after bouncing. \(P\) strikes the ground for the second time at \(B\) where it remains at rest.
  2. Calculate the first and last times after projection at which the speed of \(P\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M2 2016 November Q5
8 marks Standard +0.3
5 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A force of magnitude \(3 \mathrm { e } ^ { - t } \mathrm {~N}\) directed up a line of greatest slope acts on \(P\), where \(t \mathrm {~s}\) is the time after release.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 7.5 \mathrm { e } ^ { - t } - 5\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) up the plane at time \(t \mathrm {~s}\).
  2. Express \(v\) in terms of \(t\).
  3. Find the distance of \(P\) from \(O\) when \(v\) has its maximum value.
CAIE M2 2016 November Q6
9 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-3_656_757_781_694} The diagram shows the cross-section \(A B C D E F\) through the centre of mass of a uniform prism which rests with \(A B\) on rough horizontal ground. \(A B C D\) is a rectangle with \(A B = C D = 0.4 \mathrm {~m}\) and \(B C = A D = 1.8 \mathrm {~m}\). The other part of the cross-section is a semicircle with diameter \(D F\) and radius \(r \mathrm {~m}\).
  1. Given that the prism is on the point of toppling, show that \(r = 0.6\). A force of magnitude \(P \mathrm {~N}\) is applied to the prism, acting at \(60 ^ { \circ }\) to the upwards vertical along a tangent to the semicircle at a point between \(D\) and \(E\). The prism has weight 15 N and is in equilibrium on the point of toppling about \(B\).
  2. Show that \(P = 3.26\), correct to 3 significant figures.
  3. Find the smallest possible value of the coefficient of friction between the prism and the ground.
CAIE M2 2016 November Q7
10 marks Standard +0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-4_213_811_260_667} A small ball \(B\) of mass 0.5 kg moves in a horizontal circle with centre \(O\) and radius 0.4 m on the smooth inner surface of a hollow cone fixed with its vertex down. The axis of the cone is vertical and the semi-vertical angle is \(60 ^ { \circ }\) (see diagram).
  1. Show that the magnitude of the force exerted by the cone on \(B\) is 5.77 N , correct to 3 significant figures, and calculate the angular speed of \(B\). One end of a light elastic string of natural length 0.45 m and modulus of elasticity 36 N is attached to \(B\). The other end of the string is attached to the point on the axis 0.3 m above \(O\). The ball \(B\) again moves on the surface of the cone in the same horizontal circle as before.
  2. Calculate the speed of \(B\).
CAIE M2 2016 November Q2
7 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-2_318_488_484_826} A uniform wire is bent to form an object which has a semicircular arc with diameter \(A B\) of length 1.2 m , with a smaller semicircular arc with diameter \(B C\) of length 0.6 m . The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.
  1. Show that the distance of the centre of mass of the object from the line \(A C B\) is 0.191 m , correct to 3 significant figures. The object is freely suspended at \(A\) and hangs in equilibrium.
  2. Find the angle between \(A C B\) and the vertical.
CAIE M2 2016 November Q4
7 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-3_388_650_264_749} The diagram shows the cross-section \(A B C D\) through the centre of mass of a uniform solid prism. \(A B = 0.9 \mathrm {~m} , B C = 2 a \mathrm {~m} , A D = a \mathrm {~m}\) and angle \(A B C =\) angle \(B A D = 90 ^ { \circ }\).
  1. Calculate the distance of the centre of mass of the prism from \(A D\).
  2. Express the distance of the centre of mass of the prism from \(A B\) in terms of \(a\). The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(A D\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(B C\).
  3. Given that the prism is on the point of toppling, calculate \(a\).
CAIE M2 2016 November Q6
7 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{f9d76b90-9786-4a35-8f94-ffa7b18622b6-3_483_419_1800_863} The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre \(O\) and radius 0.9 m . A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point \(A\) on the inside of the tube. The other end of the string is attached to a particle \(P\) of mass 0.2 kg . The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate
  1. the speed of \(P\) when it is at the same horizontal level as \(O\),
  2. the speed of \(P\) at the instant when the string becomes slack.
CAIE M2 2017 November Q1
4 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{8a7016eb-4e76-4104-aa00-fbf09e1d739a-02_560_421_258_861} A hollow cylinder with a rough inner surface has radius 0.5 m . A particle \(P\) of mass 0.4 kg is in contact with the inner surface of the cylinder. The particle and cylinder rotate together with angular speed \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about the vertical axis of the cylinder, so that the particle moves in a horizontal circle (see diagram). Given that \(P\) is about to slip downwards, find the coefficient of friction between \(P\) and the surface of the cylinder.
CAIE M2 2017 November Q2
4 marks Standard +0.8
2 A small ball is projected from a point 1.5 m above horizontal ground. At a point 9 m above the ground the ball is travelling at \(45 ^ { \circ }\) above the horizontal and its velocity is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the angle of projection of the ball.
CAIE M2 2017 November Q3
7 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{8a7016eb-4e76-4104-aa00-fbf09e1d739a-04_305_510_264_813} One end of a light inextensible string of length 0.4 m is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass 0.6 kg is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the string taut and making an angle of \(60 ^ { \circ }\) with the horizontal (see diagram).
  1. Given that \(v = 0.5\), calculate the magnitude of the force that the surface exerts on \(P\).
  2. Find the greatest possible value of \(v\) for which \(P\) remains in contact with the surface.
CAIE M2 2017 November Q4
7 marks Moderate -0.3
4 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. 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.
  1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the trajectory of \(P\) is $$y = \frac { x } { \sqrt { 3 } } - \frac { 4 x ^ { 2 } } { 375 }$$
  2. Find the horizontal distance between the two points at which \(P\) is 5 m above the ground.
CAIE M2 2017 November Q5
8 marks Standard +0.8
5 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 24 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.3 \mathrm {~kg} . P\) is projected vertically upwards with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a position 1.2 m vertically below \(O\).
  1. Calculate the speed of the particle at the position where it is moving with zero acceleration. [5
  2. Show that the particle moves 1.2 m while moving upwards with constant deceleration.
CAIE M2 2017 November Q6
9 marks Standard +0.8
6 A solid object consists of a uniform hemisphere of radius 0.4 m attached to a uniform cylinder of radius 0.4 m so that the circumferences of their circular faces coincide. The hemisphere and cylinder each have weight 20 N . The centre of mass of the object lies at the centre \(O\) of their common circular face.
  1. Show that the height of the cylinder is 0.3 m .
    A new object is made by cutting the cylinder in half and removing the half not attached to the hemisphere. The cut is perpendicular to the axis of symmetry, so the new object consists of a hemisphere and a cylinder half the height of the original cylinder.
  2. Find the distance of the centre of mass of the new object from \(O\).
    The new object is placed with its hemispherical part on a rough horizontal surface. The new object is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting along its axis of symmetry, which is inclined at \(30 ^ { \circ }\) to the horizontal.
  3. Find \(P\).
CAIE M2 2017 November Q7
11 marks Challenging +1.2
7 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, and travels down a line of greatest slope. The coefficient of friction between \(P\) and the plane is 0.3 . A force of magnitude \(0.6 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \sqrt { } 3 - 1.5 - 3 x\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at a displacement \(x \mathrm {~m}\) from \(O\).
  2. Find the value of \(x\) for which \(P\) reaches its maximum velocity, and calculate this maximum velocity.
  3. Calculate the magnitude of the acceleration of \(P\) immediately after it has first come to instantaneous rest.