Questions M5 (185 questions)

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Edexcel M5 2013 June Q3
7 marks Challenging +1.8
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
15 marks Challenging +1.2
  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
10 marks Challenging +1.2
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
10 marks Challenging +1.2
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
17 marks Challenging +1.2
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
5 marks Standard +0.3
  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
8 marks Challenging +1.2
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
8 marks Challenging +1.8
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)
Edexcel M5 2014 June Q4
8 marks Challenging +1.8
4. A uniform solid sphere has mass \(M\) and radius \(a\). Prove, using integration, that the moment of inertia of the sphere about a diameter is \(\frac { 2 M a ^ { 2 } } { 5 }\) [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 \(\frac { 1 } { 2 } m r ^ { 2 }\).]
Edexcel M5 2014 June Q5
8 marks Standard +0.8
  1. A particle moves in a plane so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds satisfies the differential equation
$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + ( \tan t ) \mathbf { r } = \left( \cos ^ { 2 } t \right) \mathbf { i } - ( 3 \cos t ) \mathbf { j } , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ When \(t = 0\), the particle is at the point with position vector \(4 \mathbf { j } \mathrm {~m}\). Find \(\mathbf { r }\) in terms of \(t\).
Edexcel M5 2014 June Q6
11 marks Standard +0.3
6. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body at the points with position vectors, \(\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }\) and \(\mathbf { r } _ { 3 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N }\) \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\) \(\mathbf { F } _ { 3 } = ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { k } ) \mathrm { m }\) \(\mathbf { r } _ { 2 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\) \(\mathbf { r } _ { 3 } = ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }\) The system of the three forces is equivalent to a single force \(\mathbf { R }\) acting at the point with position vector ( \(\mathbf { 3 i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\), together with a couple of moment \(\mathbf { G }\).
  1. Find \(\mathbf { R }\).
  2. Find \(\mathbf { G }\).
Edexcel M5 2014 June Q7
9 marks Challenging +1.8
7. A raindrop absorbs water as it falls vertically under gravity through a cloud. In a model of the motion the cloud is assumed to consist of stationary water particles. At time \(t\), the mass of the raindrop is \(m\) and the speed of the raindrop is \(v\). At time \(t = 0\), the raindrop is at rest. The rate of increase of the mass of the raindrop with respect to time is modelled as being \(m k v\), where \(k\) is a positive constant.
  1. Ignoring air resistance, show from first principles, that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - k v ^ { 2 }$$
  2. Find the time taken for the raindrop to reach a speed of \(\frac { 1 } { 2 } \sqrt { } \left( \frac { g } { k } \right)\)
Edexcel M5 2014 June Q8
18 marks Challenging +1.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57b98cdd-4121-4495-b500-185cbf3ff1a8-13_739_739_276_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform circular disc of radius \(2 a\) has centre \(O\). The points \(P , Q , R\) and \(S\) on the disc are the vertices of a square with centre \(O\) and \(O P = a\). Four circular holes, each of radius \(\frac { a } { 2 }\), and with centres \(P , Q , R\) and \(S\), are drilled in the disc to produce the lamina \(L\), shown shaded in Figure 1. The mass of \(L\) is \(M\).
  1. Show that the moment of inertia of \(L\) about an axis through \(O\), and perpendicular to the plane of \(L\), is \(\frac { 55 M a ^ { 2 } } { 24 }\) The lamina \(L\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(L\) and which passes through a point \(A\) on the circumference of \(L\). At time \(t , A O\) makes an angle \(\theta\) with the downward vertical through \(A\).
  2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { 48 g } { 151 a } \sin \theta\)
  3. Hence find the period of small oscillations of \(L\) about its position of stable equilibrium. The magnitude of the component, in a direction perpendicular to \(A O\), of the force exerted on \(L\) by the axis is \(X\).
  4. Find \(X\) in terms of \(M , g\) and \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{57b98cdd-4121-4495-b500-185cbf3ff1a8-14_159_1662_2416_173}
Edexcel M5 2015 June Q1
6 marks Moderate -0.3
  1. A particle \(P\) moves from the point \(A\), with position vector ( \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) ) m , where \(a\) is a positive constant, to the point \(B\), with position vector ( \(- \mathbf { i } + a \mathbf { j } - \mathbf { k }\) ) m , under the action of a constant force \(\mathbf { F } = ( 2 \mathbf { i } + a \mathbf { j } - 3 \mathbf { k } )\) N. The work done by \(\mathbf { F }\), as it moves the particle \(P\) from \(A\) to \(B\), is 3 J . Find the value of \(a\).
    (6)
Edexcel M5 2015 June Q2
8 marks Standard +0.8
2. A particle \(P\) moves so that its position vector, \(\mathbf { r }\) metres, at time \(t\) seconds, where \(0 \leqslant t < \frac { \pi } { 2 }\), satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - ( \tan t ) \mathbf { r } = ( \sin t ) \mathbf { i }$$ When \(t = 0 , \mathbf { r } = - \frac { 1 } { 2 } \mathbf { i }\).
Find \(\mathbf { r }\) in terms of \(t\).
Edexcel M5 2015 June Q3
12 marks Challenging +1.2
  1. A rigid body is in equilibrium under the action of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 } \mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act at the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N } \quad \mathbf { r } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { m } \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { N } \quad \mathbf { r } _ { 2 } = ( - \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\)
    1. Find the magnitude of \(\mathbf { F } _ { 3 }\)
    2. Find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a scalar parameter.
    \includegraphics[max width=\textwidth, alt={}, center]{cac4dd38-796c-414b-9b80-fe39ab12d41b-11_62_49_2643_1886}
Edexcel M5 2015 June Q4
12 marks Challenging +1.8
4. A particle \(P\), whose initial mass is \(m _ { 0 }\), is projected vertically upwards from the ground at time \(t = 0\) with speed \(\frac { g } { k }\), where \(k\) is a constant. As the particle moves upwards it gains mass by picking up small droplets of moisture from the atmosphere. The droplets are at rest before they are picked up. At time \(t\) the speed of \(P\) is \(v\) and its mass has increased to \(m _ { 0 } \mathrm { e } ^ { k t }\). Assuming that, during the motion, the acceleration due to gravity is constant,
  1. show that, while \(P\) is moving upwards, $$k v + \frac { \mathrm { d } v } { \mathrm {~d} t } = - g$$
  2. find, in terms of \(m _ { 0 }\), the mass of \(P\) when it reaches its greatest height above the ground.
    (6)
Edexcel M5 2015 June Q5
9 marks Challenging +1.8
  1. A uniform circular disc, of mass \(m\) and radius \(a\), is free to rotate about a fixed smooth horizontal axis \(L\). The axis \(L\) is a tangent to the disc at the point \(A\). The centre \(O\) of the disc moves in a vertical plane that is perpendicular to \(L\).
The disc is held at rest with its plane horizontal and released.
  1. Find the angular acceleration of the disc when it has turned through an angle of \(\frac { \pi } { 3 }\)
  2. Find the magnitude of the component, in a direction perpendicular to the disc, of the force of the axis \(L\) acting on the disc at \(A\), when the disc has turned through an angle of \(\frac { \pi } { 3 }\)
Edexcel M5 2015 June Q6
16 marks Challenging +1.3
  1. A pendulum is modelled as a uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), which has a particle of mass \(2 m\) attached at \(B\). The pendulum is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\). The vertical plane is perpendicular to the axis \(L\).
    1. Find the period of small oscillations of the pendulum about its position of stable equilibrium.
    The pendulum is hanging at rest in a vertical position, with \(B\) below \(A\), when it is given a horizontal impulse of magnitude \(J\). The impulse acts at \(B\) in a vertical plane which is perpendicular to the axis \(L\). Given that the pendulum turns through an angle of \(60 ^ { \circ }\) before first coming to instantaneous rest,
  2. find \(J\).
Edexcel M5 2015 June Q7
12 marks Hard +2.3
7. (a) Find, using integration, the moment of inertia of a uniform solid hemisphere, of mass \(m\) and radius \(a\), about a diameter of its plane face.
[0pt] [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about a diameter is \(\frac { 1 } { 4 } m r ^ { 2 }\).]
(b) Hence find the moment of inertia of a uniform solid sphere, of mass \(M\) and radius \(a\), about a diameter.
Edexcel M5 2016 June Q1
7 marks Standard +0.8
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A bead \(P\) of mass 0.4 kg is threaded on a smooth straight horizontal wire. The wire lies along the line with vector equation \(\mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } ) + \lambda ( - 2 \mathbf { i } + 3 \mathbf { j } )\). The bead is initially at rest at the point \(A\) with position vector \(( - \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). A constant horizontal force \(( 0.5 \mathbf { i } + \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it along the wire to the point \(B\). At \(B\) the speed of \(P\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the position vector of \(B\).
Edexcel M5 2016 June Q2
13 marks Challenging +1.3
  1. A particle \(P\) is moving in a plane. At time \(t\) seconds the position vector of \(P\) is \(\mathbf { r }\) metres and the velocity of \(P\) is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = \frac { \pi } { 2 } , P\) is instantaneously at rest at the point with position vector \(( \mathbf { i } - \mathbf { j } ) \mathrm { m }\).
Given that \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 4 \mathbf { r } = ( 3 \sin t ) \mathbf { i }$$ find \(\mathbf { v }\) in terms of \(t\).
(13)
Edexcel M5 2016 June Q3
11 marks Standard +0.8
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body at the points with position vectors \(\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }\) and \(\mathbf { r } _ { 3 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) \(\mathbf { F } _ { 3 } = ( \mathbf { i } + \mathbf { j } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) \(\mathbf { r } _ { 3 } = ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }\)
$$\begin{aligned} & \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { k } ) \mathrm { N } \\ & \mathbf { r } _ { 2 } = ( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \end{aligned}$$ j The system of the three forces is equivalent to a single force \(\mathbf { R }\) acting through the point with position vector \(( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\), together with a couple of moment \(\mathbf { G }\).
  1. Find \(\mathbf { R }\).
  2. Find \(\mathbf { G }\). respectively, where The
Edexcel M5 2016 June Q4
10 marks Challenging +1.8
4. Find, using integration, the moment of inertia of a uniform cylindrical shell of radius \(r\), height \(h\) and mass \(M\), about a diameter of one end.
(10)
Edexcel M5 2016 June Q5
11 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f932d7cb-1299-41d1-8248-cfbf639795ed-08_613_649_221_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform piece of wire \(A B C\), of mass \(2 m\) and length \(4 a\), is bent into two straight equal portions, \(A B\) and \(B C\), which are at right angles to each other, as shown in Figure 1. The wire rotates freely in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the wire.
  1. Show that the moment of inertia of the wire about \(L\) is \(\frac { 20 m a ^ { 2 } } { 3 }\)
  2. By writing down an equation of rotational motion for the wire as it rotates about \(L\), find the period of small oscillations of the wire about its position of stable equilibrium.