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CAIE M2 2015 June Q4
6 marks Moderate -0.3
4 A small ball \(B\) is projected from a point 1.5 m above horizontal ground with initial speed \(29 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal.
  1. Show that \(B\) strikes the ground 3 s after projection.
  2. Find the speed and direction of motion of \(B\) immediately before it strikes the ground.
CAIE M2 2015 June Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{8f8492a7-8a83-4eb2-81ee-99b4a385b704-3_876_483_260_840} A uniform triangular prism of weight 20 N rests on a horizontal table. \(A B C\) is the cross-section through the centre of mass of the prism, where \(B C = 0.5 \mathrm {~m} , A B = 0.4 \mathrm {~m} , A C = 0.3 \mathrm {~m}\) and angle \(B A C = 90 ^ { \circ }\). The vertical plane \(A B C\) is perpendicular to the edge of the table. The point \(D\) on \(A C\) is at the edge of the table, and \(A D = 0.25 \mathrm {~m}\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 48 N is attached to \(C\) and a particle of mass 2.5 kg is attached to the other end of the string. The particle is released from rest at \(C\) and falls vertically (see diagram).
  1. Show that the tension in the string is 60 N at the instant when the prism topples.
  2. Calculate the speed of the particle at the instant when the prism topples.
CAIE M2 2015 June Q6
9 marks Standard +0.3
6 A cyclist and her bicycle have a total mass of 60 kg . The cyclist rides in a horizontal straight line, and exerts a constant force in the direction of motion of 150 N . The motion is opposed by a resistance of magnitude \(12 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after passing through a fixed point \(A\).
  1. Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 12.5 - v\).
  2. Given that the cyclist passes through \(A\) with speed \(11.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), solve this differential equation to show that \(v = 12.5 - \mathrm { e } ^ { - 0.2 t }\).
  3. Express the displacement of the cyclist from \(A\) in terms of \(t\).
CAIE M2 2015 June Q7
11 marks Standard +0.8
7 A particle \(P\) of mass 0.7 kg is attached to one end of a light inextensible string of length 0.5 m . The other end of the string is attached to a fixed point \(A\) which is \(h \mathrm {~m}\) above a smooth horizontal surface. \(P\) moves in contact with the surface with uniform circular motion about the point on the surface which is vertically below \(A\).
  1. Given that \(h = 0.14\), find an inequality for the angular speed of \(P\).
  2. Given instead that the magnitude of the force exerted by the surface on \(P\) is 1.4 N and that the speed of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate the tension in the string and the value of \(h\).
CAIE M2 2016 June Q1
4 marks Standard +0.3
1 A small ball is projected with speed \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) above the horizontal from a point on horizontal ground. Calculate the period of time, before the ball lands, for which the speed of the ball is less than \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
CAIE M2 2016 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{e5d70ccb-cec0-4390-a500-b550957a4ac6-2_515_463_484_842} A uniform wire has the shape of a semicircular arc, with diameter \(A B\) of length 0.8 m . The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(A B\) inclined at \(70 ^ { \circ }\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall 0.8 m vertically above \(A\). The tension in the string is 15 N (see diagram).
  1. Show that the horizontal distance of the centre of mass of the wire from the wall is 0.463 m , correct to 3 significant figures.
  2. Calculate the weight of the wire.
CAIE M2 2016 June Q3
6 marks Standard +0.8
3 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) down the plane, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(0.8 \mathrm { e } ^ { - x } \mathrm {~N}\) acts on \(P\) up the plane along the line of greatest slope through \(O\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 2 \mathrm { e } ^ { - x }\).
  2. Find \(v\) when \(x = 0.6\). \includegraphics[max width=\textwidth, alt={}, center]{e5d70ccb-cec0-4390-a500-b550957a4ac6-3_905_604_251_769} A uniform solid cone has base radius 0.4 m and height 4.4 m . A uniform solid cylinder has radius 0.4 m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal (see diagram).
  3. Calculate the least possible value of the coefficient of friction between the plane and the object.
  4. Calculate the greatest possible height of the cylinder.
CAIE M2 2016 June Q5
9 marks Standard +0.3
5 A particle is projected at an angle of \(\theta ^ { \circ }\) below the horizontal from a point at the top of a vertical cliff 26 m high. The particle strikes horizontal ground at a distance 8 m from the foot of the cliff 2 s after the instant of projection. Find
  1. the speed of projection of the particle and the value of \(\theta\),
  2. the direction of motion of the particle immediately before it strikes the ground.
CAIE M2 2016 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{e5d70ccb-cec0-4390-a500-b550957a4ac6-4_503_805_260_671} A light inextensible string passes through a small smooth bead \(B\) of mass 0.4 kg . One end of the string is attached to a fixed point \(A 0.4 \mathrm {~m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and 0.3 m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius 0.3 m (see diagram).
  1. Given that the tension in the string is 2 N , calculate
    (a) the angular speed of the bead,
    (b) the magnitude of the contact force exerted on the bead by the surface.
  2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead.
CAIE M2 2016 June Q7
11 marks Standard +0.8
7 A particle \(P\) is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(O\) on a smooth plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(P\) rests in equilibrium on the plane, 1.6 m from \(O\).
  1. Calculate the mass of \(P\). A particle \(Q\), with mass equal to the mass of \(P\), is projected up the plane along a line of greatest slope. When \(Q\) strikes \(P\) the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving 0.2 m .
  2. Show that the initial kinetic energy of the combined particle is 1 J . The combined particle subsequently moves down the plane.
  3. Calculate the greatest speed of the combined particle in the subsequent motion. \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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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 2016 June Q1
5 marks Standard +0.3
1 A small ball \(B\) is projected with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. At the instant 0.8 s after projection, \(B\) is 0.5 m vertically above the top of a vertical post.
  1. Calculate the height of the top of the post above the ground.
  2. Show that \(B\) is at its greatest height 0.2 s before passing over the post.
CAIE M2 2016 June Q2
8 marks Standard +0.3
2 One end of a light elastic string of natural length 0.4 m is attached to a fixed point \(O\). The other end of the string is attached to a particle of weight 5 N which hangs in equilibrium 0.6 m vertically below \(O\).
  1. Find the modulus of elasticity of the string. The particle is projected vertically upwards from the equilibrium position and comes to instantaneous rest after travelling 0.3 m upwards.
  2. Calculate the speed of projection of the particle.
  3. Calculate the greatest extension of the string in the subsequent motion.
CAIE M2 2016 June Q3
7 marks Standard +0.3
3 The point \(O\) is 8 m above a horizontal plane. A particle \(P\) is projected from \(O\). After projection, the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is $$y = 2 x - x ^ { 2 }$$
  1. Find the value of \(x\) for the point where \(P\) strikes the plane.
  2. Find the angle and speed of projection of \(P\).
  3. Calculate the speed of \(P\) immediately before it strikes the plane.
CAIE M2 2016 June Q4
8 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-3_784_556_260_790} A uniform object is made by drilling a cylindrical hole through a rectangular block. The axis of the cylindrical hole is perpendicular to the cross-section \(A B C D\) through the centre of mass of the object. \(A B = C D = 0.7 \mathrm {~m} , B C = A D = 0.4 \mathrm {~m}\), and the centre of the hole is 0.1 m from \(A B\) and 0.2 m from \(A D\) (see diagram). The hole has a cross-section of area \(0.03 \mathrm {~m} ^ { 2 }\).
  1. Show that the distance of the centre of mass of the object from \(A B\) is 0.212 m , and calculate the distance of the centre of mass from \(A D\). The object has weight 70 N and is placed on a rough horizontal surface, with \(A D\) in contact with the surface. A vertically upwards force of magnitude \(F \mathrm {~N}\) acts on the object at \(C\). The object is on the point of toppling.
  2. Find the value of \(F\). The force acting at \(C\) is removed, and the object is placed on a rough plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal. \(A D\) lies along a line of greatest slope, with \(A\) higher than \(D\). The plane is sufficiently rough to prevent sliding, and the object does not topple.
  3. Find the greatest possible value of \(\theta\).
CAIE M2 2016 June Q5
10 marks Standard +0.3
5 A particle \(P\) of mass 0.4 kg is placed at rest at a point \(A\) on a rough horizontal surface. A horizontal force, directed away from \(A\) and with magnitude \(0.6 t \mathrm {~N}\), acts on \(P\), where \(t \mathrm {~s}\) is the time after \(P\) is placed at \(A\). The coefficient of friction between \(P\) and the surface is 0.3 , and \(P\) has displacement from \(A\) of \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).
  1. Show that \(P\) starts to move when \(t = 2\). Show also that when \(P\) is in motion it has acceleration \(( 1.5 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Express the velocity of \(P\) in terms of \(t\), for \(t \geqslant 2\).
  3. Express \(x\) in terms of \(t\), for \(t \geqslant 2\).
CAIE M2 2016 June Q6
12 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{f8633b64-b20c-4471-9641-ccc3e6854f2c-4_479_499_255_824} \(O A\) is a rod which rotates in a horizontal circle about a vertical axis through \(O\). A particle \(P\) of mass 0.2 kg is attached to the mid-point of a light inextensible string. One end of the string is attached to the \(\operatorname { rod }\) at \(A\) and the other end of the string is attached to a point \(B\) on the axis. It is given that \(O A = O B\), angle \(O A P =\) angle \(O B P = 30 ^ { \circ }\), and \(P\) is 0.4 m from the axis. The rod and the particle rotate together about the axis with \(P\) in the plane \(O A B\) (see diagram).
  1. Calculate the tensions in the two parts of the string when the speed of \(P\) is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The angular speed of the rod is increased to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), and it is given that the system now rotates with angle \(O A P =\) angle \(O B P = 60 ^ { \circ }\).
  2. Show that the tension in the part \(A P\) of the string is zero. \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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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 2016 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-2_515_463_484_842} A uniform wire has the shape of a semicircular arc, with diameter \(A B\) of length 0.8 m . The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(A B\) inclined at \(70 ^ { \circ }\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall 0.8 m vertically above \(A\). The tension in the string is 15 N (see diagram).
  1. Show that the horizontal distance of the centre of mass of the wire from the wall is 0.463 m , correct to 3 significant figures.
  2. Calculate the weight of the wire.
CAIE M2 2016 June Q3
6 marks Standard +0.8
3 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) down the plane, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(0.8 \mathrm { e } ^ { - x } \mathrm {~N}\) acts on \(P\) up the plane along the line of greatest slope through \(O\).
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 - 2 \mathrm { e } ^ { - x }\).
  2. Find \(v\) when \(x = 0.6\). \includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-3_905_604_251_769} A uniform solid cone has base radius 0.4 m and height 4.4 m . A uniform solid cylinder has radius 0.4 m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal (see diagram).
  3. Calculate the least possible value of the coefficient of friction between the plane and the object.
  4. Calculate the greatest possible height of the cylinder.
CAIE M2 2016 June Q6
9 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{76a47bf6-1982-4cdb-bcaa-2cdf84cc4f37-4_503_805_260_671} A light inextensible string passes through a small smooth bead \(B\) of mass 0.4 kg . One end of the string is attached to a fixed point \(A 0.4 \mathrm {~m}\) above a fixed point \(O\) on a smooth horizontal surface. The other end of the string is attached to a fixed point \(C\) which is vertically below \(A\) and 0.3 m above the surface. The bead moves with constant speed on the surface in a circle with centre \(O\) and radius 0.3 m (see diagram).
  1. Given that the tension in the string is 2 N , calculate
    (a) the angular speed of the bead,
    (b) the magnitude of the contact force exerted on the bead by the surface.
  2. Given instead that the bead is about to lose contact with the surface, calculate the speed of the bead.
CAIE M2 2017 June Q1
4 marks Standard +0.3
1 A particle is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) above the horizontal. Calculate the time after projection when the particle is descending at an angle of \(40 ^ { \circ }\) below the horizontal.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7800deca-98e8-4eb4-9176-288bb1f44fec-04_376_713_260_715} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m \mathrm {~kg}\) which hangs vertically below \(A\). The particle is also attached to one end of a light elastic string of natural length 0.25 m . The other end of this string is attached to a point \(B\) which is 0.6 m from \(P\) and on the same horizontal level as \(P\). Equilibrium is maintained by a horizontal force of magnitude 7 N applied to \(P\) (see Fig. 1).
  1. Calculate the modulus of elasticity of the elastic string.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{7800deca-98e8-4eb4-9176-288bb1f44fec-05_348_488_262_826} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \(P\) is released from rest by removing the 7 N force. In its subsequent motion \(P\) first comes to instantaneous rest at a point where \(B P = 0.3 \mathrm {~m}\) and the elastic string makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2).
  2. Find the value of \(m\).
CAIE M2 2017 June Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-06_351_607_269_769} An object is made from a uniform solid hemisphere of radius 0.56 m and centre \(O\) by removing a hemisphere of radius 0.28 m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.
  1. Calculate the distance of the centre of mass of the object from \(O\).
    [0pt] [The volume of a hemisphere is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
    The object has weight 24 N . A uniform hemisphere \(H\) of radius 0.28 m is placed in the hollow part of the object to create a non-uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is 0.15 m from \(O\).
  2. Calculate the weight of \(H\).
CAIE M2 2017 June Q4
8 marks Moderate -0.3
4 A particle is projected from a point \(O\) on horizontal ground. The initial components of the velocity of the particle are \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) horizontally and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically. At time \(t \mathrm {~s}\) after projection, the horizontal and vertically upwards displacements of the particle from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(t\), and hence find the equation of the trajectory of the particle. \includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-08_63_1569_488_328}
    The horizontal ground is at the top of a vertical cliff. The point \(O\) is at a distance \(d \mathrm {~m}\) from the edge of the cliff. The particle is projected towards the edge of the cliff and does not strike the ground before it passes over the edge of the cliff.
  2. Show that \(d\) is less than 30 .
  3. Find the value of \(x\) when the particle is 14 m below the level of \(O\). \includegraphics[max width=\textwidth, alt={}, center]{7800deca-98e8-4eb4-9176-288bb1f44fec-10_501_614_258_762} A uniform semicircular lamina of radius 0.7 m and weight 14 N has diameter \(A B\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(A B\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(A B\) and the horizontal is \(30 ^ { \circ }\) and \(A P = 0.9 \mathrm {~m}\) (see diagram).
  4. Show that the magnitude of the force exerted by the peg on the lamina is 7.12 N , correct to 3 significant figures.
  5. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\).
CAIE M2 2017 June Q6
9 marks Standard +0.8
6 A particle \(P\) of mass 0.15 kg is attached to one end of a light elastic string of natural length 0.4 m and modulus of elasticity 12 N . The other end of the string is attached to a fixed point \(A\). The particle \(P\) moves in a horizontal circle which has its centre vertically below \(A\), with the string inclined at \(\theta ^ { \circ }\) to the vertical and \(A P = 0.5 \mathrm {~m}\).
  1. Find the angular speed of \(P\) and the value of \(\theta\).
  2. Calculate the difference between the elastic potential energy stored in the string and the kinetic energy of \(P\). \(7 \quad\) A particle \(P\) of mass 0.5 kg is at rest at a point \(O\) on a rough horizontal surface. At time \(t = 0\), where \(t\) is in seconds, a horizontal force acting in a fixed direction is applied to \(P\). At time \(t \mathrm {~s}\) the magnitude of the force is \(0.6 t ^ { 2 } \mathrm {~N}\) and the velocity of \(P\) away from \(O\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(P\) remains at rest at \(O\) until \(t = 0.5\).
  3. Calculate the coefficient of friction between \(P\) and the surface, and show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 1.2 t ^ { 2 } - 0.3 \quad \text { for } t > 0.5$$
  4. Express \(v\) in terms of \(t\) for \(t > 0.5\).
  5. Find the displacement of \(P\) from \(O\) when \(t = 1.2\).
CAIE M2 2017 June Q1
5 marks Moderate -0.5
1 A particle \(P\) of mass 0.2 kg moves with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and angular speed \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle on a smooth surface. \(P\) is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to the point on the surface which is the centre of the circular motion of \(P\).
  1. Find the radius of this circle.
  2. Find the modulus of elasticity of the string.
CAIE M2 2017 June Q2
5 marks Standard +0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-04_444_455_260_845} The ends of two light inextensible strings of length 0.7 m are attached to a particle \(P\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) which lie in the same vertical line with \(A\) above \(B\). The particle \(P\) moves in a horizontal circle which has its centre at the mid-point of \(A B\). Both strings are inclined at \(60 ^ { \circ }\) to the vertical. The tension in the string attached to \(A\) is 6 N and the tension in the string attached to \(B\) is 4 N (see diagram).
  1. Find the mass of \(P\).
  2. Calculate the speed of \(P\).