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CAIE M2 2012 June Q5
9 marks Challenging +1.8
5 A light elastic string has natural length 3 m and modulus of elasticity 45 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 on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4 m , and \(A\) is higher than \(B\).
  1. Calculate the distance \(A P\) when \(P\) rests on the slope in equilibrium. \(P\) is released from rest at the point between \(A\) and \(B\) where \(A P = 2.5 \mathrm {~m}\).
  2. Find the maximum speed of \(P\).
  3. Show that \(P\) is at rest when \(A P = 1.6 \mathrm {~m}\).
CAIE M2 2012 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{98bbefd8-b3dd-49f1-8591-e939282cb05c-3_341_791_886_678} A uniform lamina \(A B C D E\) consists of a rectangle \(B C D E\) and an isosceles triangle \(A B E\) joined along their common edge \(B E\). For the triangle, \(A B = A E , B E = a \mathrm {~m}\) and the perpendicular height is \(h \mathrm {~m}\). For the rectangle, \(B C = D E = 0.5 \mathrm {~m}\) and \(C D = B E = a \mathrm {~m}\) (see diagram).
  1. Show that the distance in metres of the centre of mass of the lamina from \(B E\) towards \(C D\) is $$\frac { 3 - 4 h ^ { 2 } } { 12 + 12 h }$$ The lamina is freely suspended at \(E\) and hangs in equilibrium.
  2. Given that \(D E\) is horizontal, calculate \(h\).
  3. Given instead that \(h = 0.5\) and \(A E\) is horizontal, calculate \(a\).
CAIE M2 2012 June Q7
10 marks Standard +0.3
7 The equation of the trajectory of a projectile is \(y = 0.6 x - 0.017 x ^ { 2 }\), referred to horizontal and vertically upward axes through the point of projection.
  1. Find the angle of projection of the projectile, and show that the initial speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the speed and direction of motion of the projectile when it is at a height of 5.2 m above the level of the point of projection for the second time.
CAIE M2 2013 June Q1
5 marks Moderate -0.8
1 A small ball is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) 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 \(t\).
  2. Show that the equation of the trajectory of the ball is \(y = x - \frac { 1 } { 40 } x ^ { 2 }\).
  3. State the distance from \(O\) of the point at which the ball first strikes the ground.
CAIE M2 2013 June Q2
7 marks Standard +0.3
2 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 19.2 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) is released from rest at the point 2.7 m vertically above \(A\). Calculate
  1. the initial acceleration of \(P\),
  2. the speed of \(P\) when it reaches \(A\).
CAIE M2 2013 June Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-2_373_759_1119_694} A uniform object \(A B C\) is formed from two rods \(A B\) and \(B C\) joined rigidly at right angles at \(B\). The rod \(A B\) has length 0.3 m and the rod \(B C\) has length 0.2 m . The object rests with the end \(A\) on a rough horizontal surface and the \(\operatorname { rod } A B\) vertical. The object is held in equilibrium by a horizontal force of magnitude 4 N applied at \(B\) and acting in the direction \(C B\) (see diagram).
  1. Find the distance of the centre of mass of the object from \(A B\).
  2. Calculate the weight of the object.
  3. Find the least possible value of the coefficient of friction between the surface and the object.
CAIE M2 2013 June Q4
6 marks Standard +0.8
4 A particle of mass 0.2 kg is projected vertically downwards with initial speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resisting force of magnitude \(0.09 v \mathrm {~N}\) acts vertically upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the downwards velocity of the particle at time \(t \mathrm {~s}\) after being set in motion.
  1. Show that the acceleration of the particle is \(( 10 - 0.45 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find \(v\) when \(t = 1.5\).
CAIE M2 2013 June Q5
7 marks Moderate -0.8
5 A particle \(P\) is projected with speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point \(O\). For the instant 2.5 s after projection, calculate
  1. the speed of \(P\),
  2. the angle between \(O P\) and the horizontal.
CAIE M2 2013 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_259_890_584_630} One end of a light inextensible string of length 0.2 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.3 kg is attached to the other end of the string. \(P\) moves on the table in a horizontal circle, with the string taut and making an angle of \(60 ^ { \circ }\) with the downward vertical (see diagram).
  1. Calculate the tension in the string if the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. For the motion as described, show that the angular speed of \(P\) cannot exceed \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and hence find the greatest possible value for the kinetic energy of \(P\).
CAIE M2 2013 June Q7
9 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_519_860_1430_641} \(O A B C\) is the cross-section through the centre of mass of a uniform prism of weight 20 N . The crosssection is in the shape of a sector of a circle with centre \(O\), radius \(O A = r \mathrm {~m}\) and angle \(A O C = \frac { 2 } { 3 } \pi\) radians. The prism lies on a plane inclined at an angle \(\theta\) radians to the horizontal, where \(\theta < \frac { 1 } { 3 } \pi\). OC lies along a line of greatest slope with \(O\) higher than \(C\). The prism is freely hinged to the plane at \(O\). A force of magnitude 15 N acts at \(A\), in a direction towards to the plane and at right angles to it (see diagram). Given that the prism remains in equilibrium, find the set of possible values of \(\theta\).
CAIE M2 2013 June Q1
4 marks Standard +0.3
1 A small sphere of mass 0.4 kg moves with constant speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle inside a smooth fixed hollow cylinder of diameter 0.6 m . The axis of the cylinder is vertical, and the sphere is in contact with both the horizontal base and the vertical curved surface of the cylinder.
  1. Calculate the magnitude of the force exerted on the sphere by the vertical curved surface of the cylinder.
  2. Hence show that the magnitude of the total force exerted on the sphere by the cylinder is 5 N .
CAIE M2 2013 June Q2
6 marks Standard +0.3
2 A uniform semicircular lamina of radius 0.25 m has diameter \(A B\). It is freely suspended at \(A\) from a fixed point and hangs in equilibrium.
  1. Find the distance of the centre of mass of the lamina from the diameter \(A B\).
  2. Calculate the angle which the diameter \(A B\) makes with the vertical. The lamina is now held in equilibrium with the diameter \(A B\) vertical by means of a force applied at \(B\). This force has magnitude 6 N and acts at \(45 ^ { \circ }\) to the upward vertical in the plane of the lamina.
  3. Calculate the weight of the lamina.
CAIE M2 2013 June Q3
7 marks Challenging +1.2
3 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 1.6 m and modulus of elasticity 18 N . The other end of the string is attached to a fixed point \(O\) which is 1.6 m above a smooth horizontal surface. \(P\) is placed on the surface vertically below \(O\) and then projected horizontally. \(P\) moves with initial speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on the surface. Show that, when \(O P = 1.8 \mathrm {~m}\),
  1. \(P\) is at instantaneous rest,
  2. \(P\) is on the point of losing contact with the surface.
CAIE M2 2013 June Q4
6 marks Moderate -0.8
4 A ball \(B\) is projected from a point \(O\) on horizontal ground at an angle of \(40 ^ { \circ }\) above the horizontal. \(B\) hits the ground 1.8 s after the instant of projection. Calculate
  1. the speed of projection of \(B\),
  2. the greatest height of \(B\),
  3. the distance from \(O\) of the point at which \(B\) hits the ground.
CAIE M2 2013 June Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808} A block \(B\) of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle \(P\) of mass 0.3 kg . \(B\) is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. \(P\) is released from rest at a point next to the pulley and falls vertically. At the first instant when \(P\) is 0.8 m below the pulley and descending, \(B\) is in limiting equilibrium with the part of the string attached to \(B\) horizontal (see diagram).
  1. Calculate the speed of \(P\) when \(B\) is first in limiting equilibrium.
  2. Find the coefficient of friction between \(B\) and the table.
CAIE M2 2013 June Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_291_993_1238_575} A uniform solid cone of height 0.6 m and mass 0.5 kg has its axis of symmetry vertical and its vertex \(V\) uppermost. The semi-vertical angle of the cone is \(60 ^ { \circ }\) and the surface is smooth. The cone is fixed to a horizontal surface. A particle \(P\) of mass 0.2 kg is connected to \(V\) by a light inextensible string of length 0.4 m (see diagram).
  1. Calculate the height, above the horizontal surface, of the centre of mass of the cone with the particle. \(P\) is set in motion, and moves with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a circular path on the surface of the cone.
  2. Show that the tension in the string is 1.96 N , and calculate the magnitude of the force exerted on \(P\) by the cone.
  3. Find the speed of \(P\).
CAIE M2 2013 June Q7
10 marks Standard +0.8
7 A particle \(P\) of mass 0.5 kg moves in a straight line on a smooth horizontal surface. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\). A single horizontal force of magnitude \(0.16 \mathrm { e } ^ { x } \mathrm {~N}\) acts on \(P\) in the direction \(O P\). The velocity of \(P\) when it is at \(O\) is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v = 0.8 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
  2. Find the time taken by \(P\) to travel 1.4 m from \(O\).
CAIE M2 2013 June Q4
Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{3607beed-0b98-46aa-8d95-403b05446a63-2_622_496_1121_804} A ll l \(l \quad x \quad x v l \quad O \quad l\) xl \(l \quad x\) \(l\) \(k v l\) \section*{\(x\)} \(l\) \(l\)
  2. l l \(l\)
  3. Cll \(x\) \(l v\)
CAIE M2 2013 June Q5
Standard +0.3
5 Oll \(x\) vll \(l\)
  1. \(l x l\) vl \(l\) vll \(l\)
  2. \(x l\) (a)
    (b)
    (c) \(x l\)
  3. Cll
CAIE M2 2013 June Q6
Easy -4.0
6
6 \(\begin{array} { l l l l } A & l & & v l \\ \text { lll } & l & l \end{array}\) l l l \(l\) l ll l
  1. Cll \(\operatorname { All } _ { x } l _ { x }\)
  2. \(B\) (a)
    (b) \(v\)
  3. \(v\)
  4. \(v\)
  5. \(v\)
  6. v
  7. v $$l v \quad l$$ l ll
    \multirow[t]{3}{*}{A ll ll \(l\)}\multirow{3}{*}{}\multirow[t]{2}{*}{\(k v\)}x ll\multirow{3}{*}{}
    ll l\(v l\)
    -
    \multirow[t]{5}{*}{(i)}\multirow[b]{5}{*}{
    }
    (ii)\(x\) vl Cll\(v\)\(l l\)\(l\)
    CAIE M2 2014 June Q2
    Moderate -0.3
    2 \includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-2_85_86_228_255} \includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-2_560_483_434_831} \(\theta \quad \theta \quad \theta\)
    6. \(\theta\) \(\theta\)
    7. \(\theta\) \(\theta\)
    CAIE M2 2014 June Q5
    Standard +0.3
    5 \includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-3_350_611_276_767} \(\theta\) \(\theta\)
    1. \(\theta\)
    CAIE M2 2014 June Q6
    Moderate -0.5
    6
    1. \(\theta\) \(\theta\) \(\theta\)

    2. [0pt] [uestion 7 is printed on the next page.]
    CAIE M2 2014 June Q7
    Standard +0.3
    7 \includegraphics[max width=\textwidth, alt={}, center]{edb816ee-741a-4d2d-8d60-f2a801005b1e-4_342_732_278_683}
    4. \multirow[t]{3}{*}{P m}mmv
      Cmv\multirow{5}{*}{
      m
      n
      }      		\multirow{5}{*}{}
      m mm
      C mmC mAmC mAm\multirow{3}{*}{}\multirow[t]{3}{*}{}\multirow[t]{2}{*}{C m}
    CAIE M2 2014 June Q2
    Standard +0.3
    2 \includegraphics[max width=\textwidth, alt={}, center]{3788852a-cd1a-49ae-be79-f365894bfa71-2_85_86_228_255} \includegraphics[max width=\textwidth, alt={}, center]{3788852a-cd1a-49ae-be79-f365894bfa71-2_560_483_434_831} \(\theta \quad \theta \quad \theta\)
    6. \(\theta\) \(\theta\)