Questions M2 (1537 questions)

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CAIE M2 2007 November Q2
6 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_496_609_1535_769} One end of a light inextensible string of length 0.16 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.4 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 \(30 ^ { \circ }\) with the downward vertical through \(A\) (see diagram). \(P\) moves with constant speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the tension in the string,
  2. the force exerted by the table on \(P\).
CAIE M2 2007 November Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-3_764_627_274_758} A uniform beam \(A B\) has length 2 m and mass 10 kg . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in a fixed position by a light inextensible string of length 2.4 m . One end of the string is attached to the beam at a point 0.7 m from \(A\). The other end of the string is attached to the wall at a point vertically above the hinge. The string is at right angles to \(A B\). The beam carries a load of weight 300 N at \(B\) (see diagram).
  1. Find the tension in the string. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally away from the wall and \(Y \mathrm {~N}\) vertically downwards.
  2. Find the values of \(X\) and \(Y\).
CAIE M2 2007 November Q4
7 marks Standard +0.3
4 A particle of mass 0.4 kg is released from rest and falls vertically. A resisting force of magnitude \(0.08 v \mathrm {~N}\) acts upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle at time \(t \mathrm {~s}\) after its release.
  1. Show that the acceleration of the particle is \(( 10 - 0.2 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find the velocity of the particle when \(t = 15\).
CAIE M2 2007 November Q5
7 marks Challenging +1.2
5 Each of two light elastic strings, \(S _ { 1 }\) and \(S _ { 2 }\), has modulus of elasticity 16 N . The string \(S _ { 1 }\) has natural length 0.4 m and the string \(S _ { 2 }\) has natural length 0.5 m . One end of \(S _ { 1 }\) is attached to a fixed point \(A\) of a smooth horizontal table and the other end is attached to a particle \(P\) of mass 0.5 kg . One end of \(S _ { 2 }\) is attached to a fixed point \(B\) of the table and the other end is attached to \(P\). The distance \(A B\) is 1.5 m . The particle \(P\) is held at \(A\) and then released from rest.
  1. Find the speed of \(P\) at the instant that \(S _ { 2 }\) becomes slack.
  2. Find the greatest distance of \(P\) from \(A\) in the subsequent motion.
CAIE M2 2007 November Q6
9 marks Standard +0.2
6 A particle is projected from a point \(O\) at an angle of \(35 ^ { \circ }\) above the horizontal. At time \(T\) s later the particle passes through a point \(A\) whose horizontal and vertically upward displacements from \(O\) are 8 m and 3 m respectively.
  1. By using the equation of the particle's trajectory, or otherwise, find (in either order) the speed of projection of the particle from \(O\) and the value of \(T\).
  2. Find the angle between the direction of motion of the particle at \(A\) and the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_476_895_269_625} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section of a uniform solid. The cross-section has the shape and dimensions shown. The centre of mass \(C\) of the solid lies in the plane of this cross-section. The distance of \(C\) from \(D E\) is \(y \mathrm {~cm}\).
  3. Find the value of \(y\). The solid is placed on a rough plane. The coefficient of friction between the solid and the plane is \(\mu\). The plane is tilted so that \(E F\) lies along a line of greatest slope.
  4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_375_431_1366_897} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The solid is placed so that \(F\) is higher up the plane than \(E\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 1 } { 2 }\). [3]
  5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_376_428_2069_900} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} The solid is now placed so that \(E\) is higher up the plane than \(F\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Show that \(\mu < \frac { 5 } { 6 }\). [3]
CAIE M2 2008 November Q1
4 marks Standard +0.3
1 One end of a light elastic rope of natural length 2.5 m and modulus of elasticity 80 N is attached to a fixed point \(A\). A stone \(S\) of mass 8 kg is attached to the other end of the rope. \(S\) is held at a point 6 m vertically below \(A\) and then released. Find the initial acceleration of \(S\).
CAIE M2 2008 November Q2
4 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-2_485_863_495_641} A uniform solid cylinder has height 24 cm and radius \(r \mathrm {~cm}\). A uniform solid cone has base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder and the cone are both placed with their axes vertical on a rough horizontal plane (see diagram, which shows cross-sections of the solids). The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches \(\alpha ^ { \circ }\), when both solids topple simultaneously.
  1. Find the value of \(h\).
  2. Given that \(r = 10\), find the value of \(\alpha\).
CAIE M2 2008 November Q3
7 marks Moderate -0.3
3 A particle \(P\) of mass 0.5 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 \(\left( 1 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and acts in the positive \(x\)-direction, and the other force has magnitude \(8 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the negative \(x\)-direction.
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2 + 0.6 x ^ { 2 } - 16 \mathrm { e } ^ { - x }\).
  2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 3\).
  3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_259_745_278_740} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A small sphere \(A\) of mass 0.15 kg is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(A\) moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.35 m , and is continuously in contact with both the plane base and the curved surface of the cylinder. Fig. 1 shows a vertical cross-section of the cylinder through its axis. Find the magnitude of the force exerted on \(A\) by
CAIE M2 2008 November Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-4_495_1405_264_370} \(A B C D\) is a central cross-section of a uniform rectangular block of mass 35 kg . The lengths of \(A B\) and \(B C\) are 1.2 m and 0.8 m respectively. The block is held in equilibrium by a rope, one end of which is attached to the point \(E\) of a rough horizontal floor. The other end of the rope is attached to the block at \(A\). The rope is in the same vertical plane as \(A B C D\), and \(E A B\) is a straight line making an angle of \(20 ^ { \circ }\) with the horizontal (see diagram).
  1. Show that the tension in the rope is 187 N , correct to the nearest whole number.
  2. The block is on the point of slipping. Find the coefficient of friction between the block and the floor.
CAIE M2 2008 November Q6
9 marks Standard +0.3
6 A light elastic string has natural length 4 m and modulus of elasticity 2 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.1 \mathrm {~kg} . P\) is held at rest at \(O\) and then released. The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(x \mathrm {~m}\).
  1. Show that \(v ^ { 2 } = 45 - 5 ( x - 1 ) ^ { 2 }\). Hence find
  2. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
  3. the maximum speed of \(P\).
CAIE M2 2008 November Q7
10 marks Standard +0.3
7 A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and direction \(60 ^ { \circ }\) upwards from the horizontal. At time \(t \mathrm {~s}\) later the horizontal and vertical displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Write down expressions for \(x\) and \(y\) in terms of \(V\) and \(t\) and hence show that the equation of the trajectory of \(P\) is $$y = ( \sqrt { } 3 ) x - \frac { 20 x ^ { 2 } } { V ^ { 2 } }$$ \(P\) passes through the point \(A\) at which \(x = 70\) and \(y = 10\). Find
  2. the value of \(V\),
  3. the direction of motion of \(P\) at the instant it passes through \(A\).
CAIE M2 2009 November Q1
3 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_113_787_264_680} A light elastic spring of natural length 0.25 m and modulus of elasticity 100 N is held horizontally between two parallel plates. The axis of the spring is at right angles to each of the plates. The horizontal force exerted on the spring by each of the plates is 20 N (see diagram). Find the amount by which the spring is compressed and hence write down the distance between the plates.
CAIE M2 2009 November Q2
5 marks Standard +0.3
2 A particle of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 4 N . The other end of the string is attached to a fixed point \(O\). The particle is held at a point which is \(( 0.6 + x ) \mathrm { m }\) vertically below \(O\). The particle is released from rest. In the subsequent motion the speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the string becomes slack. By considering energy, find the value of \(x\).
CAIE M2 2009 November Q3
6 marks Standard +0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_408_291_1027_927} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform solid cylinder has mass 8 kg and height 16 cm . A uniform solid cone, whose base radius is the same as the radius of the cylinder, has mass 2 kg and height 12 cm . A composite solid is formed by joining the cylinder and cone so that the base of the cone coincides with one end of the cylinder (see Fig. 1).
  1. Show that the centre of mass of the composite solid is 10.2 cm from its base. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_401_444_1877_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The composite solid is held with a point on the circumference of its base in contact with a horizontal table. The base makes an angle \(\theta ^ { \circ }\) with the table (see Fig. 2, which shows a cross-section). When the cone is released it moves towards the equilibrium position in which its base is in contact with the table.
  2. Given that the radius of the base is 4 cm , find the greatest possible value of \(\theta\), correct to 1 decimal place.
CAIE M2 2009 November Q4
7 marks Standard +0.3
4 A particle is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. After 0.3 s the particle is moving with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\tan ^ { - 1 } \left( \frac { 7 } { 24 } \right)\) above the horizontal.
  1. Show that \(V \cos \theta = 24\).
  2. Find the value of \(V \sin \theta\), and hence find \(V\) and \(\theta\).
CAIE M2 2009 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_405_545_630_799} A uniform lamina \(A O B\) is in the shape of a sector of a circle with centre \(O\) and radius 0.5 m , and has angle \(A O B = \frac { 1 } { 3 } \pi\) radians and weight 3 N . The lamina is freely hinged at \(O\) to a fixed point and is held in equilibrium with \(A O\) vertical by a force of magnitude \(F \mathrm {~N}\) acting at \(B\). The direction of this force is at right angles to \(O B\) (see diagram). Find
  1. the value of \(F\),
  2. the magnitude of the force acting on the lamina at \(O\).
CAIE M2 2009 November Q6
11 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-3_504_878_1557_632} One end of a light inextensible string of length 0.7 m is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass 0.25 kg . The particle \(P\) moves in a circle on a smooth horizontal table with constant speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string is taut and makes an angle of \(40 ^ { \circ }\) with the vertical (see diagram). Find
  1. the tension in the string,
  2. the force exerted on \(P\) by the table. \(P\) now moves in the same horizontal circle with constant angular speed \(\omega \operatorname { rad~s } ^ { - 1 }\).
  3. Find the maximum value of \(\omega\) for which \(P\) remains on the table.
CAIE M2 2009 November Q7
10 marks Standard +0.3
7 A particle \(P\) of mass 0.1 kg is projected vertically upwards from a point \(O\) with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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 projection.
  1. Show that, while \(P\) is moving upwards, \(\frac { 1 } { v + 10 } \frac { \mathrm {~d} v } { \mathrm {~d} t } = - 1\).
  2. Hence find an expression for \(v\) in terms of \(t\), and explain why it is valid only for \(0 \leqslant t \leqslant \ln 3\).
  3. Find the initial acceleration of \(P\).
CAIE M2 2009 November Q3
6 marks Standard +0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_408_1164_248_493} A particle \(P\) is released from rest at a point \(A\) which is 7 m above horizontal ground. At the same instant that \(P\) is released a particle \(Q\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(A\) is 24 m . Particle \(Q\) moves in the vertical plane containing \(O\) and \(A\), with initial speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle \(\theta\) above the horizontal, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram). Show that the particles collide.
CAIE M2 2009 November Q4
7 marks Standard +0.3
4 One end of a light elastic string of natural length 3 m and modulus of elasticity 15 mN is attached to a fixed point \(O\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. \(P\) is released from rest at \(O\) and moves vertically downwards. When the extension of the string is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } = 5 \left( 12 + 4 x - x ^ { 2 } \right)\).
  2. Find the magnitude of the acceleration of \(P\) when it is at its lowest point, and state the direction of this acceleration.
CAIE M2 2009 November Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_593_828_1530_660} A horizontal disc of radius 0.5 m is rotating with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis through its centre \(O\). One end of a light inextensible string of length 0.8 m is attached to a point \(A\) of the circumference of the disc. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. The string is taut and the system rotates so that the string is always in the same vertical plane as the radius \(O A\) of the disc. The string makes a constant angle \(\theta\) with the vertical (see diagram). The speed of \(P\) is 1.6 times the speed of \(A\).
  1. Show that \(\sin \theta = \frac { 3 } { 8 }\).
  2. Find the tension in the string.
  3. Find the value of \(\omega\).
CAIE M2 2009 November Q6
10 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-4_447_736_269_701} \(P\) is the vertex of a uniform solid cone of mass 5 kg , and \(O\) is the centre of its base. Strings are attached to the cone at \(P\) and at \(O\). The cone hangs in equilibrium with \(P O\) horizontal and the strings taut. The strings attached at \(P\) and \(O\) make angles of \(\theta ^ { \circ }\) and \(20 ^ { \circ }\), respectively, with the vertical (see diagram, which shows a cross-section).
  1. By taking moments about \(P\) for the cone, find the tension in the string attached at \(O\).
  2. Find the value of \(\theta\) and the tension in the string attached at \(P\).
CAIE M2 2009 November Q7
10 marks Standard +0.8
7 A particle \(P\) of mass 0.3 kg is projected vertically upwards from the ground with an initial speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(P\) is at height \(x \mathrm {~m}\) above the ground, its upward speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that $$3 v - 90 \ln ( v + 30 ) + x = A ,$$ where \(A\) is a constant.
  1. Differentiate this equation with respect to \(x\) and hence show that the acceleration of the particle is \(- \frac { 1 } { 3 } ( v + 30 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find, in terms of \(v\), the resisting force acting on the particle.
  3. Find the time taken for \(P\) to reach its maximum height.
CAIE M2 2010 November Q1
3 marks Moderate -0.8
1 A horizontal circular disc rotates with constant angular speed \(9 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about its centre \(O\). A particle of mass 0.05 kg is placed on the disc at a distance 0.4 m from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc.
CAIE M2 2010 November Q2
6 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-2_673_401_525_872} A bow consists of a uniform curved portion \(A B\) of mass 1.4 kg , and a uniform taut string of mass \(m \mathrm {~kg}\) which joins \(A\) and \(B\). The curved portion \(A B\) is an arc of a circle centre \(O\) and radius 0.8 m . Angle \(A O B\) is \(\frac { 2 } { 3 } \pi\) radians (see diagram). The centre of mass of the bow (including the string) is 0.65 m from \(O\). Calculate \(m\).