Questions M2 (1391 questions)

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CAIE M2 2012 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate
  1. the magnitude of the force applied at \(A\),
  2. the least possible value of the coefficient of friction at \(B\).
CAIE M2 2012 November Q3
3 A particle \(P\) of mass 0.2 kg is released from rest and falls vertically. At time \(t \mathrm {~s}\) after release \(P\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.8 v \mathrm {~N}\) acts on \(P\).
  1. Show that the acceleration of \(P\) is \(( 10 - 4 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find the value of \(v\) when \(t = 0.6\).
CAIE M2 2012 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_538_885_1809_628} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45 ^ { \circ }\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius 0.67 m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\).
CAIE M2 2012 November Q5
5 A particle \(P\) is projected with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. For the instant when the speed of \(P\) is \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and increasing,
  1. show that the vertical component of the velocity of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards,
  2. calculate the distance of \(P\) from \(O\).
CAIE M2 2012 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-3_582_862_577_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).
  1. Show that the centre of mass of the lamina lies on \(O B\).
  2. Calculate the distance of the centre of mass of the lamina from \(O\).
CAIE M2 2012 November Q7
7 A light elastic string has natural length 3 m and modulus of elasticity 45 N . A particle \(P\) of weight 6 N is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\) and \(A B = 4 \mathrm {~m}\). The particle \(P\) is released from rest at the point 1.5 m vertically below \(A\).
  1. Calculate the distance \(P\) moves after its release before first coming to instantaneous rest at a point vertically above \(B\). (You may assume that at this point the part of the string joining \(P\) to \(B\) is slack.)
  2. Show that the greatest speed of \(P\) occurs when it is 2.1 m below \(A\), and calculate this greatest speed.
  3. Calculate the greatest magnitude of the acceleration of \(P\).
CAIE M2 2012 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate
  1. the magnitude of the force applied at \(A\),
  2. the least possible value of the coefficient of friction at \(B\).
CAIE M2 2012 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-2_538_885_1809_628} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45 ^ { \circ }\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius 0.67 m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\).
CAIE M2 2012 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{576173a0-d932-45c3-94bc-d54105edc100-3_584_862_575_644} A uniform lamina \(O A B C D\) consists of a semicircle \(B C D\) with centre \(O\) and radius 0.6 m and an isosceles triangle \(O A B\), joined along \(O B\) (see diagram). The triangle has area \(0.36 \mathrm {~m} ^ { 2 }\) and \(A B = A O\).
  1. Show that the centre of mass of the lamina lies on \(O B\).
  2. Calculate the distance of the centre of mass of the lamina from \(O\).
CAIE M2 2012 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_426_531_258_808} A circular object is formed from a uniform semicircular lamina of weight 12 N and a uniform semicircular arc of weight 8 N . The lamina and the arc both have centre \(O\) and radius 0.6 m and are joined at the ends of their common diameter \(A B\). The object is freely pivoted to a fixed point at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. The object is in equilibrium acted on by a horizontal force of magnitude \(F\) N applied at the lowest point of the object, and acting in the plane of the object (see diagram).
  1. Show that the centre of mass of the object is at \(O\).
  2. Calculate \(F\).
CAIE M2 2012 November Q2
2 A light elastic string has natural length 4 m and modulus of elasticity 60 N . A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) at a distance of 6 m above \(B\). \(P\) is projected vertically upwards from the point 2 m vertically above \(B\). In the subsequent motion, \(P\) comes to instantaneous rest at a distance of 2 m below \(A\).
  1. Calculate the speed of projection of \(P\).
  2. Calculate the distance of \(P\) from \(A\) at an instant when \(P\) has its greatest kinetic energy, and calculate this kinetic energy.
CAIE M2 2012 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_397_1303_1790_422} The point \(O\) is 1.2 m below rough horizontal ground \(A B C\). A ball is projected from \(O\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the horizontal. The ball passes over the point \(A\) after travelling a horizontal distance of 2 m . The ball subsequently bounces once on the ground at \(B\). The ball leaves \(B\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and travels a further horizontal distance of 20 m before landing at \(C\) (see diagram).
  1. Calculate the height above the level of \(O\) of the ball when it is vertically above \(A\).
  2. Calculate the time after the instant of projection when the ball reaches \(B\).
  3. Find the angle which the trajectory of the ball makes with the horizontal immediately after it bounces at \(B\).
    \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-3_663_695_258_726} A cylinder of height 0.9 m and radius 0.9 m is placed symmetrically on top of a cylinder of height \(h \mathrm {~m}\) and radius \(r \mathrm {~m}\), where \(r < 0.9\), with plane faces in contact and axes in the same vertical line \(A B\), where \(A\) and \(B\) are centres of plane faces of the cylinders (see diagram). Both cylinders are uniform and made of the same material. The lower cylinder is gradually tilted and when the axis of symmetry is inclined at \(45 ^ { \circ }\) to the horizontal the upper cylinder is on the point of toppling without sliding.
CAIE M2 2012 November Q5
5 A small ball \(B\) of mass 0.2 kg is attached to fixed points \(P\) and \(Q\) by two light inextensible strings of equal length. \(P\) is vertically above \(Q\), the strings are taut and each is inclined at \(60 ^ { \circ }\) to the vertical. \(B\) moves with constant speed in a horizontal circle of radius 0.6 m .
  1. Given that the tension in the string \(P B\) is 7 N , calculate
    (a) the tension in string \(Q B\),
    (b) the speed of \(B\).
  2. Given instead that \(B\) is moving with angular speed \(7 \mathrm { rad } \mathrm { s } ^ { - 1 }\), calculate the tension in the string \(Q B\).
CAIE M2 2012 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-4_238_725_258_712} Two particles \(P\) and \(Q\), of masses 0.4 kg and 0.2 kg respectively, are attached to opposite ends of a light inextensible string. \(P\) is placed on a horizontal table and the string passes over a small smooth pulley at the edge of the table. The string is taut and the part of the string attached to \(Q\) is vertical (see diagram). The coefficient of friction between \(P\) and the table is 0.5 . \(Q\) is projected vertically downwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at time \(t \mathrm {~s}\) after the instant of projection the speed of the particles is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion of each particle is opposed by a resisting force of magnitude \(0.9 v \mathrm {~N}\). The particle \(P\) does not reach the pulley.
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 3 v\).
  2. Find the value of \(t\) when the particles have speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance that each particle has travelled in this time.
CAIE M2 2013 November Q1
1 A particle \(P\) of mass 0.3 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 \(O\) of a smooth horizontal plane. \(P\) moves on the plane at constant speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a circle with centre \(O\). Calculate the tension in the string.
CAIE M2 2013 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-2_359_686_484_731} A uniform frame consists of a semicircular arc \(A B C\) of radius 0.6 m together with its diameter \(A O C\), where \(O\) is the centre of the semicircle (see diagram).
  1. Calculate the distance of the centre of mass of the frame from \(O\). The frame is freely suspended at \(A\) and hangs in equilibrium.
  2. Calculate the angle between \(A C\) and the vertical.
CAIE M2 2013 November Q3
3 A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\). One force has magnitude \(4 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the positive \(x\)-direction. The other force has magnitude \(2.4 x ^ { 2 } \mathrm {~N}\) and acts in the negative \(x\)-direction.
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \mathrm { e } ^ { - x } - 3 x ^ { 2 }\).
  2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 2\).
CAIE M2 2013 November Q4
4 A small ball \(B\) is projected from a point \(O\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal.
  1. Calculate the speed and direction of motion of \(B\) for the instant 1.8 s after projection. The point \(O\) is 2 m above a horizontal plane.
  2. Calculate the time after projection when \(B\) reaches the plane.
CAIE M2 2013 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_540_537_255_804} A particle \(P\) of mass 0.2 kg is attached to a fixed point \(A\) by a light inextensible string of length 0.4 m . A second light inextensible string of length 0.3 m connects \(P\) to a fixed point \(B\) which is vertically below \(A\). The particle \(P\) moves in a horizontal circle, which has its centre on the line \(A B\), with the angle \(A P B = 90 ^ { \circ }\) (see diagram).
  1. Given that the tensions in the two strings are equal, calculate the speed of \(P\).
  2. It is given instead that \(P\) moves with its least possible angular speed for motion in this circle. Find this angular speed.
CAIE M2 2013 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_454_1029_1379_557}
\(A B C D\) is the cross-section through the centre of mass of a uniform rectangular block of weight 260 N . The lengths \(A B\) and \(B C\) are 1.5 m and 0.8 m respectively. The block rests in equilibrium with the point \(D\) on a rough horizontal floor. Equilibrium is maintained by a light rope attached to the point \(A\) on the block and the point \(E\) on the floor. The points \(E , A\) and \(B\) lie in a straight line inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).
  1. By taking moments about \(D\), show that the tension in the rope is 146 N , correct to 3 significant figures.
  2. Given that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the floor.
CAIE M2 2013 November Q7
7 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 32 N . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).
  1. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest. A horizontal plane is fixed at a distance 1 m below \(O\). The particle \(P\) is again released from rest at \(O\).
  2. Calculate the speed of \(P\) immediately before it collides with the plane.
  3. In the collision with the plane, \(P\) loses \(96 \%\) of its kinetic energy. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest above the plane, given that this occurs when the string is slack.
CAIE M2 2013 November Q2
2 A
  1. \(C\)
  2. \(C\)
CAIE M2 2013 November Q3
3
A
\(v\)
\(x\)
\includegraphics[max width=\textwidth, alt={}, center]{image-not-found}
\(v\)
\(O\)
\(v\)
\includegraphics[max width=\textwidth, alt={}, center]{2e61ea37-93e8-485d-a238-9c537ca233f0-2_342_609_488_767}
CAIE M2 2013 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{2e61ea37-93e8-485d-a238-9c537ca233f0-3_505_494_274_806}
A
\(x\)
\(x\)
\(v\)
A \(x\)
\(x\)
\(v\)
  1. \(v\)
  2. \(v\)
CAIE M2 2013 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{2e61ea37-93e8-485d-a238-9c537ca233f0-3_380_1023_1418_561}
E
\(v\)
  1. \(B\)
  2. \(v\)
    [0pt] [uestion is printed on the next page.] A
  3. \(C\) A \(x\)
  4. \(C\)

  5. \(C\)
    \(v \quad v\)