Questions — Edexcel M5 (158 questions)

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Edexcel M5 2011 June Q6
6. A uniform rod \(A B\) of mass \(4 m\) is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), through \(A\). The rod is hanging vertically at rest when it is struck at its end \(B\) by a particle of mass \(m\). The particle is moving with speed \(u\), in a direction which is horizontal and perpendicular to \(L\), and after striking the rod it rebounds in the opposite direction with speed \(v\). The coefficient of restitution between the particle and the rod is 1 . Show that \(u = 7 v\).
Edexcel M5 2011 June Q7
7. Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass \(M\) and base radius \(a\), about its axis is \(\frac { 3 } { 10 } M a ^ { 2 }\).
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about an axis through its centre and perpendicular to its plane is \(\left. \frac { 1 } { 2 } m r ^ { 2 } .\right]\)
Edexcel M5 2011 June Q8
  1. A pendulum consists of a uniform rod \(P Q\), of mass \(3 m\) and length \(2 a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.
    1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac { 33 } { 4 } m a ^ { 2 }\).
    The pendulum is released from rest in the position where \(P Q\) makes an angle \(\alpha\) with the downward vertical. At time \(t , P Q\) makes an angle \(\theta\) with the downward vertical.
  2. Show that the angular speed, \(\dot { \theta }\), of the pendulum satisfies $$\dot { \theta } ^ { 2 } = \frac { 40 g ( \cos \theta - \cos \alpha ) } { 33 a }$$
  3. Hence, or otherwise, find the angular acceleration of the pendulum. Given that \(\alpha = \frac { \pi } { 20 }\) and that \(P Q\) has length \(\frac { 8 } { 33 } \mathrm {~m}\),
  4. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest.
Edexcel M5 2012 June Q1
  1. A particle \(P\) moves in a plane such that its position vector \(\mathbf { r }\) metres at time \(t\) seconds \(( t > 0 )\) satisfies the differential equation
$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - \frac { 2 } { t } \mathbf { r } = 4 \mathbf { i }$$ When \(t = 1\), the particle is at the point with position vector \(( \mathbf { i } + \mathbf { j } ) \mathrm { m }\).
Find \(\mathbf { r }\) in terms of \(t\).
Edexcel M5 2012 June Q2
2. A rocket, with initial mass 1500 kg , including 600 kg of fuel, is launched vertically upwards from rest. The rocket burns fuel at a rate of \(15 \mathrm {~kg} \mathrm {~s} ^ { - 1 }\) and the burnt fuel is ejected vertically downwards with a speed of \(1000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the rocket. At time \(t\) seconds after launch \(( t \leqslant 40 )\) the rocket has mass \(m \mathrm {~kg}\) and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1000 } { m } \frac { \mathrm {~d} m } { \mathrm {~d} t } = - 9.8$$
  2. Find \(v\) at time \(t , 0 \leqslant t \leqslant 40\)
Edexcel M5 2012 June Q3
  1. A uniform rod \(P Q\), of mass \(m\) and length \(3 a\), is free to rotate about a fixed smooth horizontal axis \(L\), which passes through the end \(P\) of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with \(Q\) vertically below \(P\). One end of a light inextensible string of length \(2 a\) is attached to the rod at \(P\) and the other end is attached to a particle of mass \(3 m\). The particle is held with the string taut, and horizontal and perpendicular to \(L\), and is then released. After colliding, the particle sticks to the rod forming a body \(B\).
    1. Show that the moment of inertia of \(B\) about \(L\) is \(15 m a ^ { 2 }\).
    2. Show that \(B\) first comes to instantaneous rest after it has turned through an angle \(\arccos \left( \frac { 9 } { 25 } \right)\).
Edexcel M5 2012 June Q4
  1. A body consists of a uniform plane circular disc, of radius \(r\) and mass \(2 m\), with a particle of mass \(3 m\) attached to the circumference of the disc at the point \(P\).
    The line \(P Q\) is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), which is perpendicular to the plane of the disc and passes through \(Q\). The body is held with \(Q P\) making an angle \(\beta\) with the downward vertical through \(Q\), where \(\sin \beta = 0.25\), and released from rest. Find the magnitude of the component, perpendicular to \(P Q\), of the force acting on the body at \(Q\) at the instant when it is released.
    [0pt] [You may assume that the moment of inertia of the body about \(L\) is \(15 m r ^ { 2 }\).]
  2. The points \(P\) and \(Q\) have position vectors \(4 \mathbf { i } - 6 \mathbf { j } - 12 \mathbf { k }\) and \(2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }\) respectively, relative to a fixed origin \(O\).
Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\), act along \(\overrightarrow { O P } , \overrightarrow { Q O }\) and \(\overrightarrow { Q P }\) respectively, and have magnitudes \(7 \mathrm {~N} , 3 \mathrm {~N}\) and \(3 \sqrt { } 10 \mathrm {~N}\) respectively.
  1. Express \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) in vector form.
  2. Show that the resultant of \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) is \(( 2 \mathbf { i } - 10 \mathbf { j } - 16 \mathbf { k } ) \mathrm { N }\).
  3. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a parameter.
Edexcel M5 2012 June Q6
6. A uniform circular pulley, of mass \(4 m\) and radius \(r\), is free to rotate about a fixed smooth horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. A light inextensible string passes over the pulley and has a particle of mass \(2 m\) attached to one end and a particle of mass \(3 m\) attached to the other end. The particles hang with the string vertical and taut on each side of the pulley. The rim of the pulley is sufficiently rough to prevent the string slipping. The system is released from rest.
  1. Find the angular acceleration of the pulley. When the angular speed of the pulley is \(\Omega\), the string breaks and a constant braking couple of magnitude \(G\) is applied to the pulley which brings it to rest.
  2. Find an expression for the angle turned through by the pulley from the instant when the string breaks to the instant when the pulley first comes to rest.
Edexcel M5 2012 June Q7
  1. (a) A uniform lamina of mass \(m\) is in the shape of a triangle \(A B C\). The perpendicular distance of \(C\) from the line \(A B\) is \(h\). Prove, using integration, that the moment of inertia of the lamina about \(A B\) is \(\frac { 1 } { 6 } m h ^ { 2 }\).
    (b) Deduce the radius of gyration of a uniform square lamina of side \(2 a\), about a diagonal.
The points \(X\) and \(Y\) are the mid-points of the sides \(R Q\) and \(R S\) respectively of a square \(P Q R S\) of side \(2 a\). A uniform lamina of mass \(M\) is in the shape of \(P Q X Y S\).
(c) Show that the moment of inertia of this lamina about \(X Y\) is \(\frac { 79 } { 84 } M a ^ { 2 }\).
Edexcel M5 2013 June Q1
  1. A particle moves in a plane in such a way that its position vector \(\mathbf { r }\) metres at time \(t\) seconds satisfies the differential equation
$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t } = \mathbf { 0 }$$ When \(t = 0\), the particle is at the origin and is moving with velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Find \(\mathbf { r }\) in terms of \(t\).
Edexcel M5 2013 June Q2
2. Three forces \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 2 \mathbf { i } - \mathbf { k } ) \mathrm { N }\), and \(\mathbf { F } _ { 3 }\) act on a rigid body. The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\), the force \(\mathbf { F } _ { 2 }\) acts through the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\) and the force \(\mathbf { F } _ { 3 }\) acts through the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\). Given that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) reduces to a couple \(\mathbf { G }\),
  1. find \(\mathbf { G }\). The line of action of \(\mathbf { F } _ { 3 }\) is changed so that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) now reduces to a couple \(( 6 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) m.
  2. Find an equation of the new line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.
Edexcel M5 2013 June Q3
  1. A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the spacecraft. The burnt fuel is ejected at a constant rate of \(c \mathrm {~kg} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds the total mass of the spacecraft, including fuel, is \(m \mathrm {~kg}\) and the speed of the spacecraft is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that, while the spacecraft is ejecting burnt fuel,
    $$m \frac { \mathrm {~d} v } { \mathrm {~d} t } = 2000 c$$ At time \(t = 0\), the mass of the spacecraft is \(M _ { 0 } \mathrm {~kg}\) and the speed of the spacecraft is \(2000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 50\), the spacecraft is still ejecting burnt fuel and its speed is \(6000 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find \(c\) in terms of \(M _ { 0 }\).
Edexcel M5 2013 June Q4
4. Show, using integration, that the moment of inertia of a uniform solid right circular cone of mass \(M\), height \(h\) and base radius \(a\), about an axis through the vertex, parallel to the base, is $$\frac { 3 M } { 20 } \left( a ^ { 2 } + 4 h ^ { 2 } \right)$$ [You may assume without proof that the moment of inertia of a uniform circular disc, of radius \(r\) and mass \(m\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]
Edexcel M5 2013 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d3e55cec-05f7-4db3-8eb5-5d0adca38d4c-09_723_707_214_621} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform circular lamina has radius \(2 a\) and centre \(C\). The points \(P , Q , R\) and \(S\) on the lamina are the vertices of a square with centre \(C\) and \(C P = a\). Four circular discs, each of radius \(\frac { a } { 2 }\), with centres \(P , Q , R\) and \(S\), are removed from the lamina. The remaining lamina forms a template \(T\), as shown in Figure 1. The radius of gyration of \(T\) about an axis through \(C\), perpendicular to \(T\), is \(k\).
  1. Show that \(k ^ { 2 } = \frac { 55 a ^ { 2 } } { 24 }\) The template \(T\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(T\) and passes through a point on its outer rim.
  2. Write down an equation of rotational motion for \(T\) and deduce that the period of small oscillations of \(T\) about its stable equilibrium position is $$2 \pi \sqrt { } \left( \frac { 151 a } { 48 g } \right)$$
Edexcel M5 2013 June Q6
6. A uniform circular disc, of radius \(r\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which is perpendicular to the plane of the disc and passes through a point which is \(\frac { 1 } { 4 } r\) from the centre of the disc. The disc is held at rest with its centre vertically above the axis. The disc is then slightly disturbed from its rest position. You may assume without proof that the moment of inertia of the disc about \(L\) is \(\frac { 9 m r ^ { 2 } } { 16 }\).
  1. Show that the angular speed of the disc when it has turned through \(\frac { \pi } { 2 }\) is \(\sqrt { } \left( \frac { 8 g } { 9 r } \right)\).
  2. Find the magnitude of the force exerted on the disc by the axis when the disc has turned through \(\frac { \pi } { 2 }\).
Edexcel M5 2013 June Q1
  1. Solve the differential equation
$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - 2 \mathbf { r } = \mathbf { 0 }$$ given that when \(t = 0 , \mathbf { r } . \mathbf { j } = 0\) and \(\mathbf { r } \times \mathbf { j } = \mathbf { i } + \mathbf { k }\).
Edexcel M5 2013 June Q2
2. A uniform square lamina \(S\) has side \(2 a\). The radius of gyration of \(S\) about an axis through a vertex, perpendicular to \(S\), is \(k\).
  1. Show that \(k ^ { 2 } = \frac { 8 a ^ { 2 } } { 3 }\). The lamina \(S\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(S\) and passes through a vertex.
  2. By writing down an equation of rotational motion for \(S\), find the period of small oscillations of \(S\) about its position of stable equilibrium.
Edexcel M5 2013 June Q3
3. A raindrop falls vertically under gravity through a stationary cloud. At time \(t = 0\), the raindrop is at rest and has mass \(m _ { 0 }\). As the raindrop falls, water condenses onto it from the cloud so that the mass of the raindrop increases at a constant rate \(c\). At time \(t\), the mass of the raindrop is \(m\) and the speed of the raindrop is \(v\). The resistance to the motion of the raindrop has magnitude \(m k v\), where \(k\) is a constant. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } + v \left( k + \frac { c } { m _ { 0 } + c t } \right) = g$$
Edexcel M5 2013 June Q4
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. The forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act through the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively.
    \(\mathbf { r } _ { 1 } = ( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m }\),
    \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\)
    \(\mathbf { r } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { k } ) \mathrm { m }\),
    \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\)
    Given that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) is in equilibrium,
    1. find \(\mathbf { F } _ { 3 }\),
    2. find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
    The force \(\mathbf { F } _ { 3 }\) is replaced by a force \(\mathbf { F } _ { 4 }\) acting through the point with position vector \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\). The system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) is equivalent to a single force ( \(3 \mathbf { i } + \mathbf { j } + \mathbf { k }\) ) N acting through the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) together with a couple.
  2. Find the magnitude of this couple.
Edexcel M5 2013 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{90c52724-f7db-481f-acef-95a24f75b16a-07_561_545_205_705} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform triangular lamina \(A B C\), of mass \(M\), has \(A B = A C\) and \(B C = 2 a\). The mid-point of \(B C\) is \(D\) and \(A D = h\), as shown in Figure 1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), perpendicular to the plane of the lamina, is $$\frac { M } { 6 } \left( a ^ { 2 } + 3 h ^ { 2 } \right)$$ [You may assume without proof that the moment of inertia of a uniform rod, of length \(2 l\) and mass \(m\), about an axis through its midpoint and perpendicular to the rod, is \(\frac { 1 } { 3 } m l ^ { 2 }\).]
Edexcel M5 2013 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{90c52724-f7db-481f-acef-95a24f75b16a-09_951_305_212_808} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string has a particle of mass \(m\) attached to one end and a particle of mass \(4 m\) attached to the other end. The string passes over a rough pulley which is modelled as a uniform circular disc of radius \(a\) and mass \(2 m\), as shown in Figure 2. The pulley can rotate in a vertical plane about a fixed horizontal axis which passes through the centre of the pulley and is perpendicular to the plane of the pulley. As the pulley rotates, a frictional couple of constant magnitude \(2 m g a\) acts on it. The system is held with the string vertical and taut on each side of the pulley and released from rest. Given that the string does not slip on the pulley, find the initial angular acceleration of the pulley.
Edexcel M5 2013 June Q7
7. A uniform circular disc, of radius \(r\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis is perpendicular to the plane of the disc and passes through a point \(A\) on the circumference of the disc. The disc is held with \(A B\) horizontal, where \(A B\) is a diameter of the disc, and released from rest.
  1. Find the magnitude of
    1. the horizontal component,
    2. the vertical component
      of the force exerted on the disc by the axis immediately after the disc is released. When \(A B\) is vertical the disc is instantaneously brought to rest by a horizontal impulse which acts in the plane of the disc and is applied to the disc at \(B\).
  2. Find the magnitude of the impulse.
Edexcel M5 2014 June Q1
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A bead \(P\) of mass 0.2 kg is threaded on a smooth straight horizontal wire. The bead is at rest at the point \(A\) with position vector \(( 4 \mathbf { i } - \mathbf { j } ) \mathrm { m }\). A force \(( 0.2 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it to the point \(B\) with position vector \(( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the speed of \(P\) at \(B\).
Edexcel M5 2014 June Q2
2. A uniform equilateral triangular lamina \(A B C\) has mass \(m\) and sides of length \(\sqrt { } 3 a\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\), which passes through \(A\) and is perpendicular to the lamina. The midpoint of \(B C\) is \(D\). The lamina is held with \(A D\) making an angle of \(60 ^ { \circ }\) with the upward vertical through \(A\) and released from rest. The moment of inertia of the lamina about the axis \(L\) is \(\frac { 5 m a ^ { 2 } } { 4 }\) Find the speed of \(D\) when \(A D\) is vertical.
(8)
Edexcel M5 2014 June Q3
3. A uniform rectangular lamina \(A B C D\), where \(A B = a\) and \(B C = 2 a\), has mass \(2 m\). The lamina is free to rotate about its edge \(A B\), which is fixed and vertical. The lamina is at rest when it is struck at \(C\) by a particle \(P\) of mass \(m\). The particle \(P\) is moving horizontally with speed \(U\) in a direction which is perpendicular to the lamina. The coefficient of restitution between \(P\) and the lamina is 0.5 Find the angular speed of the lamina immediately after the impact.
(8)