Questions — Edexcel M5 (185 questions)

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Edexcel M5 2003 June Q5
16 marks Challenging +1.2
5. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The rod is hanging in equilibrium with \(B\) below \(A\) when it is hit by a particle of mass \(m\) moving horizontally with speed \(v\) in a vertical plane perpendicular to the axis. The particle strikes the rod at \(B\) and immediately adheres to it.
  1. Show that the angular speed of the rod immediately after the impact is \(\frac { 3 v } { 8 a }\). Given that the rod rotates through \(120 ^ { \circ }\) before first coming to instantaneous rest,
  2. find \(v\) in terms of \(a\) and \(g\).
  3. find, in terms of \(m\) and \(g\), the magnitude of the vertical component of the force acting on the \(\operatorname { rod }\) at \(A\) immediately after the impact.
    (5)
Edexcel M5 2003 June Q6
18 marks Challenging +1.8
6. (a) Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through its centre \(O\) perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\). The line \(A B\) is a diameter of the disc and \(P\) is the mid-point of \(O A\). The disc is free to rotate about a fixed smooth horizontal axis \(L\). The axis lies in the plane of the disc, passes through \(P\) and is perpendicular to \(O A\). A particle of mass \(m\) is attached to the disc at \(A\) and a particle of mass \(2 m\) is attached to the disc at \(B\).
(b) Show that the moment of inertia of the loaded disc about \(L\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). At time \(t = 0 , P B\) makes a small angle with the downward vertical through \(P\) and the loaded disc is released from rest. By obtaining an equation of motion for the disc and using a suitable approximation,
(c) find the time when the loaded disc first comes to instantaneous rest. END
Edexcel M5 2004 June Q1
7 marks Challenging +1.2
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } , \mathbf { F } _ { 2 } = ( - 3 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\). The force \(\mathbf { F } _ { 3 }\) acts at the point with position vector \(( 2 \mathbf { i } - \mathbf { k } ) \mathrm { m }\).
Given that this set of forces is equivalent to a couple, find
  1. \(\mathbf { F } _ { 3 }\),
  2. the magnitude of the couple.
Edexcel M5 2004 June Q2
8 marks Standard +0.3
2. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle \(P\) of mass 2 kg . The particle is initially at rest at the point \(A\) with position vector \(( - 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } ) \mathrm { m }\). Four seconds later, \(P\) is at the point \(B\) with position vector \(( 6 \mathbf { i } - 5 \mathbf { j } + 8 \mathbf { k } ) \mathrm { m }\). Given that \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\), find
  1. \(\mathbf { F } _ { 2 }\),
  2. the work done on \(P\) as it moves from \(A\) to \(B\).
Edexcel M5 2004 June Q3
9 marks Standard +0.8
3. A uniform lamina of mass \(m\) is in the shape of a rectangle \(P Q R S\), where \(P Q = 8 a\) and \(Q R = 6 a\).
  1. Find the moment of inertia of the lamina about the edge \(P Q\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-3_336_772_528_642}
    \end{figure} The flap on a letterbox is modelled as such a lamina. The flap is free to rotate about an axis along its horizontal edge \(P Q\), as shown in Fig. 1. The flap is released from rest in a horizontal position. It then swings down into a vertical position.
  2. Show that the angular speed of the flap as it reaches the vertical position is \(\sqrt { \left( \frac { g } { 2 a } \right) }\).
  3. Find the magnitude of the vertical component of the resultant force of the axis \(P Q\) on the flap, as it reaches the vertical position.
Edexcel M5 2004 June Q4
10 marks Challenging +1.2
4. A uniform circular disc, of mass \(m\) and radius \(r\), has a diameter \(A B\). The point \(C\) on \(A B\) is such that \(A C = \frac { 1 } { 2 } r\). The disc can rotate freely in a vertical plane about a horizontal axis through \(C\), perpendicular to the plane of the disc. The disc makes small oscillations in a vertical plane about the position of equilibrium in which \(B\) is below \(A\).
  1. Show that the motion is approximately simple harmonic.
  2. Show that the period of this approximate simple harmonic motion is \(\pi \sqrt { \left( \frac { 6 r } { g } \right) }\). The speed of \(B\) when it is vertically below \(A\) is \(\sqrt { \left( \frac { g r } { 54 } \right) }\). The disc comes to rest when \(C B\) makes an angle \(\alpha\) with the downward vertical.
  3. Find an approximate value of \(\alpha\).
    (3)
Edexcel M5 2004 June Q5
10 marks Challenging +1.8
5. A rocket is launched vertically upwards under gravity from rest at time \(t = 0\). The rocket propels itself upward by ejecting burnt fuel vertically downwards at a constant speed \(u\) relative to the rocket. The initial mass of the rocket, including fuel, is \(M\). At time \(t\), before all the fuel has been used up, the mass of the rocket, including fuel, is \(M ( 1 - k t )\) and the speed of the rocket is \(v\).
  1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { k u } { 1 - k t } - g\).
  2. Hence find the speed of the rocket when \(t = \frac { 1 } { 3 k }\).
Edexcel M5 2004 June Q6
15 marks Challenging +1.3
6. A particle \(P\) of mass 2 kg moves in the \(x - y\) plane. At time \(t\) seconds its position vector is \(\mathbf { r }\) metres. When \(t = 0\), the position vector of \(P\) is \(\mathbf { i }\) metres and the velocity of \(P\) is ( \(- \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The vector \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = \mathbf { 0 }$$
  1. Find \(\mathbf { r }\) in terms of \(t\).
  2. Show that the speed of \(P\) at time \(t\) is \(\mathrm { e } ^ { - t } \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find, in terms of e, the loss of kinetic energy of \(P\) in the interval \(t = 0\) to \(t = 1\).
Edexcel M5 2004 June Q7
16 marks Challenging +1.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-5_313_1443_317_356}
\end{figure} A body consists of two uniform circular discs, each of mass \(m\) and radius \(a\), with a uniform rod. The centres of the discs are fixed to the ends \(A\) and \(B\) of the rod, which has mass \(3 m\) and length 8a. The discs and the rod are coplanar, as shown in Fig. 2. The body is free to rotate in a vertical plane about a smooth fixed horizontal axis. The axis is perpendicular to the plane of the discs and passes through the point \(O\) of the rod, where \(A O = 3 a\).
  1. Show that the moment of inertia of the body about the axis is \(54 m a ^ { 2 }\). The body is held at rest with \(A B\) horizontal and is then released. When the body has turned through an angle of \(30 ^ { \circ }\), the rod \(A B\) strikes a small fixed smooth peg \(P\) where \(O P = 3 a\). Given that the body rebounds from the peg with its angular speed halved by the impact,
  2. show that the magnitude of the impulse exerted on the body by the peg at the impact is $$9 m \sqrt { \left( \frac { 5 g a } { 6 } \right) } .$$ END
Edexcel M5 2005 June Q1
6 marks Standard +0.3
  1. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle. \(\mathbf { F } _ { 1 }\) has magnitude 9 N and acts in the direction of \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } . \mathbf { F } _ { 2 }\) has magnitude 18 N and acts in the direction of \(\mathbf { i } + 8 \mathbf { j } - 4 \mathbf { k }\).
Find the total work done by the two forces in moving the particle from the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) to the point with position vector \(( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\).
(Total 6 marks)
Edexcel M5 2005 June Q2
7 marks Standard +0.3
2. At time \(t\) seconds the position vector of a particle \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres, where \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 3 \mathrm { e } ^ { - t } \mathbf { j }$$ Given that \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j }\) when \(t = 0\), find \(\mathbf { r }\) in terms of \(t\).
(Total 7 marks)
Edexcel M5 2005 June Q3
9 marks Standard +0.3
3. A system of forces acting on a rigid body consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting at a point \(A\) of the body, together with a couple of moment \(\mathbf { G } . \mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) N\). The position vector of the point \(A\) is \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and \(\mathbf { G } = ( 7 \mathbf { i } - 3 \mathbf { j } + 8 \mathbf { k } ) \mathrm { Nm }\). Given that the system is equivalent to a single force \(\mathbf { R }\),
  1. find \(\mathbf { R }\),
  2. find a vector equation for the line of action of \(\mathbf { R }\).
    (Total 9 marks) \section*{4.} \section*{Figure 1}
    \includegraphics[max width=\textwidth, alt={}]{43ce237f-c8e4-428a-b8cd-04673e62abb9-3_896_515_276_772}
    A thin uniform rod \(P Q\) has mass \(m\) and length \(3 a\). A thin uniform circular disc, of mass \(m\) and radius \(a\), is attached to the rod at \(Q\) in such a way that the rod and the diameter \(Q R\) of the disc are in a straight line with \(P R = 5 a\). The rod together with the disc form a composite body, as shown in Figure 1. The body is free to rotate about a fixed smooth horizontal axis \(L\) through \(P\), perpendicular to \(P Q\) and in the plane of the disc.
Edexcel M5 2005 June Q5
12 marks Challenging +1.8
5. A uniform square lamina \(A B C D\), of mass \(m\) and side \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about \(L\) is \(\frac { 8 m a ^ { 2 } } { 3 }\). Given that the lamina is released from rest when the line \(A C\) makes an angle of \(\frac { \pi } { 3 }\) with the downward vertical,
  1. find the magnitude of the vertical component of the force acting on the lamina at \(A\) when the line \(A C\) is vertical. Given instead that the lamina now makes small oscillations about its position of stable equilibrium,
  2. find the period of these oscillations.
    (5)
    (Total 12 marks)
Edexcel M5 2005 June Q6
13 marks Challenging +1.8
6. A rocket-driven car moves along a straight horizontal road. The car has total initial mass \(M\). It propels itself forwards by ejecting mass backwards at a constant rate \(\lambda\) per unit time at a constant speed \(U\) relative to the car. The car starts from rest at time \(t = 0\). At time \(t\) the speed of the car is \(v\). The total resistance to motion is modelled as having magnitude \(k v\), where \(k\) is a constant. Given that \(t < \frac { M } { \lambda }\), show that
  1. \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }\),
  2. \(v = \frac { \lambda U } { k } \left\{ 1 - \left( 1 - \frac { \lambda t } { M } \right) ^ { \frac { k } { \lambda } } \right\}\).
    (6)
    (Total 13 marks)
Edexcel M5 2005 June Q7
17 marks Challenging +1.8
7. A uniform lamina of mass \(m\) is in the shape of an equilateral triangle \(A B C\) of perpendicular height \(h\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) through \(A\) and perpendicular to the lamina.
  1. Show, by integration, that the moment of inertia of the lamina about \(L\) is \(\frac { 5 m h ^ { 2 } } { 9 }\). The centre of mass of the lamina is \(G\). The lamina is in equilibrium, with \(G\) below \(A\), when it is given an angular speed \(\sqrt { \left( \frac { 6 g } { 5 h } \right) }\).
  2. Find the angle between \(A G\) and the downward vertical, when the lamina first comes to rest.
  3. Find the greatest magnitude of the angular acceleration during the motion.
    (Total 17 marks)
Edexcel M5 2007 June Q1
4 marks Standard +0.3
  1. A bead of mass 0.5 kg is threaded on a smooth straight wire. The only forces acting on the bead are a constant force ( \(4 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k }\) ) N and the normal reaction of the wire. The bead starts from rest at the point \(A\) with position vector \(( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and moves to the point \(B\) with position vector \(( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).
Find the speed of the bead when it reaches \(B\).
(4)
Edexcel M5 2007 June Q2
7 marks Moderate -0.3
2. At time \(t\) seconds, the position vector of a particle \(P\) is \(\mathbf { r }\) metres, where \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 3 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ When \(t = 0\), the velocity of \(P\) is \(( 8 \mathbf { i } - 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find the velocity of \(P\) when \(t = \frac { 2 } { 3 } \ln 2\).
(7)
Edexcel M5 2007 June Q3
5 marks Standard +0.8
3. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate about a fixed smooth axis which passes through \(A\) and is perpendicular to the rod. The rod has angular speed \(\omega\) when it strikes a particle \(P\) of mass \(m\) and adheres to it. Immediately before the rod strikes \(P , P\) is at rest and at a distance \(x\) from \(A\). Immediately after the rod strikes \(P\), the angular speed of the rod is \(\frac { 3 } { 4 } \omega\). Find \(x\) in terms of \(a\).
(5)
Edexcel M5 2007 June Q4
7 marks Challenging +1.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5180a4e0-dafe-4595-a517-e3501f7aed40-3_780_1175_242_420} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A region \(R\) is bounded by the curve \(y ^ { 2 } = 4 a x ( y > 0 )\), the \(x\)-axis and the line \(x = a ( a > 0 )\), as shown in Figure 1. A uniform solid \(S\) of mass \(M\) is formed by rotating \(R\) about the \(x\)-axis through \(360 ^ { \circ }\). Using integration, prove that the moment of inertia of \(S\) about the \(x\)-axis is \(\frac { 4 } { 3 } M a ^ { 2 }\).
(You may assume without proof that the moment of inertia of a uniform disc, of mass \(m\) and radius \(r\), about an axis through its centre perpendicular to its plane is \(\frac { 1 } { 2 } m r ^ { 2 }\).)
Edexcel M5 2007 June Q5
9 marks Standard +0.8
5. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\).
The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\), and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector ( \(3 \mathbf { i } - \mathbf { k }\) ) m. The two forces are equivalent to a single force \(\mathbf { F }\) acting at the point with position vector \(( \mathbf { i } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G }\).
  1. Find \(\mathbf { F }\).
  2. Find the magnitude of \(\mathbf { G }\).
    (8)
Edexcel M5 2007 June Q6
13 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5180a4e0-dafe-4595-a517-e3501f7aed40-4_419_773_196_664} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A lamina \(S\) is formed from a uniform disc, centre \(O\) and radius \(2 a\), by removing the disc of centre \(O\) and radius \(a\), as shown in Figure 2. The mass of \(S\) is \(M\).
  1. Show that the moment of inertia of \(S\) about an axis through \(O\) and perpendicular to its plane is \(\frac { 5 } { 2 } M a ^ { 2 }\).
    (3) The lamina is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) lies in the plane of \(S\) and is a tangent to its outer circumference, as shown in Figure 2.
  2. Show that the moment of inertia of \(S\) about \(L\) is \(\frac { 21 } { 4 } M a ^ { 2 }\).
    (4) \(S\) is displaced through a small angle from its position of stable equilibrium and, at time \(t = 0\), it is released from rest. Using the equation of motion of \(S\), with a suitable approximation,
  3. find the time when \(S\) first passes through its position of stable equilibrium.
    (6)
Edexcel M5 2007 June Q7
14 marks Challenging +1.8
7. A motor boat of mass \(M\) is moving in a straight line, with its engine switched off, across a stretch of still water. The boat is moving with speed \(U\) when, at time \(t = 0\), it develops a leak. The water comes in at a constant rate so that at time \(t\), the mass of water in the boat is \(\lambda t\). At time \(t\) the speed of the boat is \(v\) and it experiences a total resistance to motion of magnitude \(2 \lambda v\).
  1. Show that \(( M + \lambda t ) \frac { \mathrm { d } v } { \mathrm {~d} t } + 3 \lambda v = 0\).
    (6)
  2. Show that the time taken for the speed of the boat to reduce to \(\frac { 1 } { 2 } U\) is \(\frac { M } { \lambda } \left( 2 ^ { \frac { 1 } { 3 } } - 1 \right)\).
    (6) The boat sinks when the mass of water inside the boat is \(M\).
  3. Show that the boat does not sink before the speed of the boat is \(\frac { 1 } { 2 } U\).
Edexcel M5 2007 June Q8
16 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5180a4e0-dafe-4595-a517-e3501f7aed40-5_533_584_292_703} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform rod \(A B\) has mass \(3 m\) and length \(2 a\). It is free to rotate in a vertical plane about a smooth fixed horizontal axis through the point \(X\) on the rod, where \(A X = \frac { 1 } { 2 } a\). A particle of mass \(m\) is attached to the rod at \(B\). At time \(t = 0\), the rod is vertical, with \(B\) above \(A\), and is given an initial angular speed \(\sqrt { \frac { g } { a } }\). When the rod makes an angle \(\theta\) with the upward vertical, the angular speed of the rod is \(\omega\), as shown in Figure 3.
  1. By using the principle of the conservation of energy, show that $$\omega ^ { 2 } = \frac { g } { 2 a } ( 5 - 3 \cos \theta )$$
  2. Find the angular acceleration of the rod when it makes an angle \(\theta\) with the upward vertical. When \(\theta = \phi\), the resultant force of the axis on the rod is in a direction perpendicular to the rod.
  3. Find \(\cos \phi\).
Edexcel M5 2008 June Q1
6 marks Standard +0.8
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A small bead of mass 0.5 kg is threaded on a smooth horizontal wire. The bead is initially at rest at the point with position vector \(( \mathbf { i } - 6 \mathbf { j } ) \mathrm { m }\). A constant horizontal force \(\mathbf { P } \mathrm { N }\) then acts on the bead causing it to move along the wire. The bead passes through the point with position vector ( \(7 \mathbf { i } - 14 \mathbf { j }\) ) m with speed \(2 \sqrt { 7 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(\mathbf { P }\) is parallel to ( \(6 \mathbf { i } + \mathbf { j }\) ), find \(\mathbf { P }\).
(6)
Edexcel M5 2008 June Q2
7 marks Standard +0.3
2. The velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of a particle \(P\) at time \(t\) seconds satisfies the vector differential equation $$\frac { \mathrm { d } \mathbf { v } } { \mathrm {~d} t } + 4 \mathbf { v } = \mathbf { 0 }$$ The position vector of \(P\) at time \(t\) seconds is \(\mathbf { r }\) metres.
Given that at \(t = 0 , \mathbf { r } = ( \mathbf { i } - \mathbf { j } )\) and \(\mathbf { v } = ( - 8 \mathbf { i } + 4 \mathbf { j } )\), find \(\mathbf { r }\) at time \(t\) seconds.