Questions M2 (1391 questions)

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CAIE M2 2002 June Q1
1 One end of a light elastic string of natural length 1.6 m and modulus of elasticity 25 N is attached to a fixed point \(A\). A particle \(P\) of mass 0.15 kg is attached to the other end of the string. \(P\) is held at rest at a point 2 m vertically below \(A\) and is then released.
  1. For the motion from the instant of release until the string becomes slack, find the loss of elastic potential energy and the gain in gravitational potential energy.
  2. Hence find the speed of \(P\) at the instant the string becomes slack.
CAIE M2 2002 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_316_1065_712_541} Two identical uniform heavy triangular prisms, each of base width 10 cm , are arranged as shown at the ends of a smooth horizontal shelf of length 1 m . Some books, each of width 5 cm , are placed on the shelf between the prisms.
  1. Find how far the base of a prism can project beyond an end of the shelf without the prism toppling.
  2. Find the greatest number of books that can be stored on the shelf without either of the prisms toppling.
CAIE M2 2002 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_202_972_1619_584} A light elastic string has natural length 0.8 m and modulus of elasticity 12 N . The ends of the string are attached to fixed points \(A\) and \(B\), which are at the same horizontal level and 0.96 m apart. A particle of weight \(W \mathrm {~N}\) is attached to the mid-point of the string and hangs in equilibrium at a point 0.14 m below \(A B\) (see diagram). Find \(W\).
CAIE M2 2002 June Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_576_826_258_662} A hollow cone with semi-vertical angle \(45 ^ { \circ }\) is fixed with its axis vertical and its vertex \(O\) downwards. A particle \(P\) of mass 0.3 kg moves in a horizontal circle on the inner surface of the cone, which is smooth. \(P\) is attached to one end of a light inextensible string of length 1.2 m . The other end of the string is attached to the cone at \(O\) (see diagram). The string is taut and rotates at a constant angular speed of \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  1. Find the acceleration of \(P\).
  2. Find the tension in the string and the force exerted on \(P\) by the cone.
CAIE M2 2002 June Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-3_590_754_1425_699} A uniform lamina of weight 9 N has dimensions as shown in the diagram. The lamina is freely hinged to a fixed point at \(A\). A light inextensible string has one end attached to \(B\), and the other end attached to a fixed point \(C\), which is in the same vertical plane as the lamina. The lamina is in equilibrium with \(A B\) horizontal and angle \(A B C = 150 ^ { \circ }\).
  1. Show that the tension in the string is 12.2 N .
  2. Find the magnitude of the force acting on the lamina at \(A\).
CAIE M2 2002 June Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-4_182_844_264_653} A particle \(P\) of mass 0.4 kg travels on a horizontal surface along the line \(O A\) in the direction from \(O\) to \(A\). Air resistance of magnitude \(0.1 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\) after it passes through the fixed point \(O\) (see diagram). The speed of \(P\) at \(O\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Assume that the horizontal surface is smooth. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} x } = - \frac { 1 } { 4 }\), where \(x \mathrm {~m}\) is the distance of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance from \(O\) at which the speed of \(P\) is zero.
  2. Assume instead that the horizontal surface is not smooth and that the coefficient of friction between \(P\) and the surface is \(\frac { 3 } { 40 }\).
    (a) Show that \(4 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 3 )\).
    (b) Hence find the value of \(t\) for which the speed of \(P\) is zero.
CAIE M2 2002 June Q7
7 A ball is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an angle of \(30 ^ { \circ }\) above the horizontal. At time \(T\) s after projection, the ball passes through the point \(A\), whose horizontal and vertically upward displacements from \(O\) are 10 m and 2 m respectively.
  1. By using the equation of the trajectory, or otherwise, find the value of \(V\).
  2. Find the value of \(T\).
  3. Find the angle that the direction of motion of the ball at \(A\) makes with the horizontal.
CAIE M2 2003 June Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_533_497_269_824} A frame consists of a uniform circular ring of radius 25 cm and mass 1.5 kg , and a uniform rod of length 48 cm and mass 0.6 kg . The ends \(A\) and \(B\) of the rod are attached to points on the circumference of the ring, as shown in the diagram. Find the distance of the centre of mass of the frame from the centre of the ring.
CAIE M2 2003 June Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_439_608_1181_772} A uniform solid hemisphere, with centre \(O\) and radius 4 cm , is held so that a point \(P\) of its rim is in contact with a horizontal surface. The plane face of the hemisphere makes an angle of \(70 ^ { \circ }\) with the horizontal. \(Q\) is the point on the axis of symmetry of the hemisphere which is vertically above \(P\). The diagram shows the vertical cross-section of the hemisphere which contains \(O , P\) and \(Q\).
  1. Determine whether or not the centre of mass of the hemisphere is between \(O\) and \(Q\). The hemisphere is now released.
  2. State whether or not the hemisphere falls on to its plane face, giving a reason for your answer.
CAIE M2 2003 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-3_464_439_274_854} A uniform beam \(A B\) has length 6 m and mass 45 kg . One end of a light inextensible rope is attached to the beam at the point 2.5 m from \(A\). The other end of the rope is attached to a fixed point \(P\) on a vertical wall. The beam is in equilibrium with \(A\) in contact with the wall at a point 5 m below \(P\). The rope is taut and at right angles to \(A B\) (see diagram). Find
  1. the tension in the rope,
  2. the horizontal and vertical components of the force exerted by the wall on the beam at \(A\).
CAIE M2 2003 June Q4
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.