Questions M2 (1391 questions)

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CAIE M2 2010 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-2_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 \(m\).
CAIE M2 2010 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-2_279_905_1560_621} 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 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 \(30 ^ { \circ }\) to the horizontal (see diagram).
  1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\).
  2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\).
CAIE M2 2010 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-3_560_894_258_628} A uniform beam \(A B\) has length 2 m and weight 70 N . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point 1.7 m vertically above the hinge. The other end of the rope is attached to the beam at a point 0.8 m from \(A\). The rope is at right angles to \(A B\). The beam carries a load of weight 220 N at \(B\) (see diagram).
  1. Find the tension in the rope.
  2. Find the direction of the force exerted on the beam at \(A\).
CAIE M2 2010 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{8425223a-1924-43ef-bd7c-e9b424fdc311-4_433_841_255_653} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30 ^ { \circ }\) from \(O\) (see diagram). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward 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 \(t\), and hence obtain the equation of the trajectory of \(P\).
  2. Calculate the value of \(x\) when \(P\) is at \(A\).
  3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). \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.
    University of 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 2010 November Q1
1
\(A B C D\) is a uniform lamina with \(A B = 1.8 \mathrm {~m} , A D = D C = 0.9 \mathrm {~m}\), and \(A D\) perpendicular to \(A B\) and \(D C\) (see diagram).
  1. Find the distance of the centre of mass of the lamina from \(A B\) and the distance from \(A D\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
  2. Calculate the angle between \(A B\) and the vertical.
CAIE M2 2010 November Q2
2 A particle \(P\) is projected with speed \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane.
  1. For the instant when the vertical component of the velocity of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards, find the direction of motion of \(P\) and the height of \(P\) above the plane.
  2. \(P\) strikes the plane at the point \(A\). Calculate the time taken by \(P\) to travel from \(O\) to \(A\) and the distance \(O A\).
    \includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-2_679_455_1544_845} Particles \(P\) and \(Q\) have masses 0.8 kg and 0.4 kg respectively. \(P\) is attached to a fixed point \(A\) by a light inextensible string which is inclined at an angle \(\alpha ^ { \circ }\) to the vertical. \(Q\) is attached to a fixed point \(B\), which is vertically below \(A\), by a light inextensible string of length 0.3 m . The string \(B Q\) is horizontal. \(P\) and \(Q\) are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius 0.3 m about the axis through \(A\) and \(B\) with constant angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram).
  3. By considering the motion of \(Q\), find the tensions in the strings \(P Q\) and \(B Q\).
  4. Find the tension in the string \(A P\) and the value of \(\alpha\).
CAIE M2 2010 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-3_371_570_258_790} A uniform \(\operatorname { rod } A B\) has weight 15 N and length 1.2 m . The end \(A\) of the rod is in contact with a rough plane inclined at \(30 ^ { \circ }\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).
  1. Show that the magnitude of the force applied at \(B\) is 4.33 N , correct to 3 significant figures.
  2. Find the magnitude of the frictional force exerted by the plane on the rod.
  3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane.
CAIE M2 2010 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-3_287_1068_1306_536} A light elastic string has natural length 2 m and modulus of elasticity \(\lambda \mathrm { N }\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and 2.4 m apart. A particle \(P\) of mass 0.6 kg is attached to the mid-point of the string and hangs in equilibrium at a point 0.5 m below \(A B\) (see diagram).
  1. Show that \(\lambda = 26\).
    \(P\) is projected vertically downwards from the equilibrium position, and comes to instantaneous rest at a point 0.9 m below \(A B\).
  2. Calculate the speed of projection of \(P\).
CAIE M2 2010 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-4_341_572_258_790} A particle \(P\) of mass 0.2 kg is projected with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards along a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal (see diagram). Air resistance of magnitude \(0.5 v \mathrm {~N}\) opposes the motion of \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after projection. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 \sqrt { } 3 }\). The particle \(P\) reaches a position of instantaneous rest when \(t = T\).
  1. Show that, while \(P\) is moving up the plane, \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 2.5 ( 3 + v )\).
  2. Calculate \(T\).
  3. Calculate the speed of \(P\) when \(t = 2 T\). \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. University of 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 2011 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_334_679_258_731} A non-uniform \(\operatorname { rod } A B\), of length 0.6 m and weight 9 N , has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\) (see diagram).
  1. Calculate \(T\).
  2. Find the least possible value of the coefficient of friction at \(A\).
CAIE M2 2011 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_525_913_1123_616} A particle \(P\) is projected from a point \(O\) at an angle of \(60 ^ { \circ }\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45 ^ { \circ }\) (see diagram).
  1. Show that the speed of projection of \(P\) is \(8.20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Calculate the time after projection when the direction of motion of \(P\) is \(45 ^ { \circ }\) above the horizontal.
CAIE M2 2011 November Q3
3 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(0.25 \mathrm {~kg} . P\) hangs in equilibrium below \(O\).
  1. Calculate the distance \(O P\). The particle \(P\) is raised, and is released from rest at \(O\).
  2. Calculate the speed of \(P\) when it passes through the equilibrium position.
  3. Calculate the greatest value of the distance \(O P\) in the subsequent motion.
CAIE M2 2011 November Q4
4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.
  1. Find the value of \(\theta\) at which the cone is about to topple.
  2. Given that the cylinder does not topple, find the greatest possible value of \(h\). The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-3_476_1211_836_466}
  3. Given that the solid immediately topples, find the least possible value of \(h\).
CAIE M2 2011 November Q5
5 A ball of mass 0.05 kg is released from rest at a height \(h \mathrm {~m}\) above the ground. At time \(t \mathrm {~s}\) after its release, the downward velocity of the ball is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Air resistance opposes the motion of the ball with a force of magnitude \(0.01 \nu \mathrm {~N}\).
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 - 0.2 v\). Hence find \(v\) in terms of \(t\).
  2. Given that the ball reaches the ground when \(t = 2\), calculate \(h\).
CAIE M2 2011 November Q6
6 A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m . One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is \(T \mathrm {~N}\), and the bead rotates with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle about the vertical axis through \(A\) and \(C\).
  1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m , calculate \(\omega\), and hence find the kinetic energy of \(B\).
  2. Given instead that angle \(A B C = 90 ^ { \circ }\), and that \(A B\) makes an angle \(\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)\) with the vertical, calculate \(T\) and \(\omega\).
CAIE M2 2011 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-2_334_679_258_731} A non-uniform \(\operatorname { rod } A B\), of length 0.6 m and weight 9 N , has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\) (see diagram).
  1. Calculate \(T\).
  2. Find the least possible value of the coefficient of friction at \(A\).
CAIE M2 2011 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-2_525_913_1123_616} A particle \(P\) is projected from a point \(O\) at an angle of \(60 ^ { \circ }\) above horizontal ground. At an instant 0.6 s after projection, the angle of elevation of \(P\) from \(O\) is \(45 ^ { \circ }\) (see diagram).
  1. Show that the speed of projection of \(P\) is \(8.20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
  2. Calculate the time after projection when the direction of motion of \(P\) is \(45 ^ { \circ }\) above the horizontal.
CAIE M2 2011 November Q4
4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.
  1. Find the value of \(\theta\) at which the cone is about to topple.
  2. Given that the cylinder does not topple, find the greatest possible value of \(h\). The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram).
    \includegraphics[max width=\textwidth, alt={}, center]{ef21fd99-b1d5-44bc-bc71-80a34d8e3b05-3_476_1211_836_466}
  3. Given that the solid immediately topples, find the least possible value of \(h\).
CAIE M2 2011 November Q1
1 A particle is projected with speed \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) above the horizontal from a point on horizontal ground. Calculate the speed of the particle 2 s after the instant of projection.
CAIE M2 2011 November Q2
2 An object is made from two identical uniform rods \(A B\) and \(B C\) each of length 0.6 m and weight 7 N . The rods are rigidly joined to each other at \(B\) and angle \(A B C = 90 ^ { \circ }\).
  1. Calculate the distance of the centre of mass of the object from \(B\). The object is freely suspended at \(A\) and a force of magnitude \(F \mathrm {~N}\) is applied to the rod \(B C\) at \(C\). The object is in equilibrium with \(A B\) inclined at \(45 ^ { \circ }\) to the horizontal.
  2. (a) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_401_314_799_995} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Calculate \(F\) given that the force acts horizontally as shown in Fig. 1.
    (b) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_503_273_1446_1014} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Calculate \(F\) given instead that the force acts perpendicular to the rod as shown in Fig. 2.
CAIE M2 2011 November Q3
3 A particle \(P\) is projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 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 upward 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 path of \(P\) is \(y = x - 0.016 x ^ { 2 }\).
  2. Calculate the horizontal distance between the two positions at which \(P\) is 2.4 m above the ground.
CAIE M2 2011 November Q4
4 A particle \(P\) of mass 0.4 kg is projected horizontally with velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on a smooth horizontal surface. The motion of \(P\) is opposed by a resisting force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after projection.
  1. Show that \(v = \frac { 8 } { 1 + 4 t }\).
  2. Calculate the distance \(O P\) when \(t = 1.5\).
CAIE M2 2011 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-3_927_1022_689_559} One end of a light inextensible string of length 1.2 m is attached to a fixed point \(O\) on a smooth horizontal surface. Particles \(P , Q\) and \(R\) are attached to the string so that \(O P = P Q = Q R = 0.4 \mathrm {~m}\). The particles rotate in horizontal circles about \(O\) with constant angular speed \(\omega \operatorname { rads } ^ { - 1 }\) and with \(O , P\), \(Q\) and \(R\) in a straight line (see diagram). \(R\) has mass 0.2 kg , and the tensions in the parts of the string attached to \(Q\) are 6 N and 10 N .
  1. Show that \(\omega = 5\).
  2. Calculate the mass of \(Q\).
  3. Given that the kinetic energy of \(P\) is equal to the kinetic energy of \(R\), calculate the tension in the part of the string attached to \(O\).
CAIE M2 2011 November Q6
6 A uniform solid consists of a hemisphere with centre \(O\) and radius 0.6 m joined to a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere coincides with one of the plane faces of the cylinder.
  1. Calculate the distance of the centre of mass of the solid from \(O\).
    [0pt] [The volume of a hemisphere of radius \(r\) is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]

  2. \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-4_547_631_593_797} A cylindrical hole, of length 0.48 m , starting at the plane face of the solid, is made along the axis of symmetry (see diagram). The resulting solid has its centre of mass at \(O\). Show that the area of the cross-section of the hole is \(\frac { 3 } { 16 } \pi \mathrm {~m} ^ { 2 }\).
  3. It is possible to increase the length of the cylindrical hole so that the solid still has its centre of mass at \(O\). State the increase in the length of the hole.
CAIE M2 2011 November Q7
7 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a particle \(P\) of mass 0.8 kg . The other end of the string is attached to a fixed point \(O\) at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The particle rests in equilibrium on the plane.
  1. Calculate the extension of the string.
    \(P\) is projected from its equilibrium position up the plane along a line of greatest slope. In the subsequent motion \(P\) just reaches \(O\), and later just reaches the foot of the plane. Calculate
  2. the speed of projection of \(P\),
  3. the length of the line of greatest slope of the plane.