Edexcel M5 (Mechanics 5) 2006 June

1. (a) Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2 a\), about an axis perpendicular to the rod through one end is \(\frac { 4 } { 3 } m a ^ { 2 }\).
   (b) Hence, or otherwise, find the moment of inertia of a uniform square lamina, of mass \(M\) and side \(2 a\), about an axis through one corner and perpendicular to the plane of the lamina.
2. A particle of mass 0.5 kg is at rest at the point with position vector ( \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) ) m. The particle is then acted upon by two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \(( 4 \mathbf { i } + 5 \mathbf { j } - 5 \mathbf { k } ) \mathrm { m }\) with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that \(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\), find \(\mathbf { F } _ { 2 }\).
3. A particle \(P\) moves in the \(x - y\) plane and has position vector \(\mathbf { r }\) metres at time \(t\) seconds. It is given that \(\mathbf { r }\) satisfies the differential equation

$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } = 2 \frac { \mathrm {~d} \mathbf { r } } { \mathrm {~d} t }$$ When \(t = 0 , P\) is at the point with position vector \(3 \mathbf { i }\) metres and is moving with velocity \(\mathbf { j } \mathrm { m } \mathrm { s } ^ { - 1 }\).
(a) Find \(\mathbf { r }\) in terms of \(t\).
(b) Describe the path of \(P\), giving its cartesian equation.

4. A force system consists of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) acting on a rigid body.
\(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and acts at the point with position vector \(( - \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\).
\(\mathbf { F } _ { 2 } = ( - \mathbf { j } + \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector ( \(\left. 2 \mathbf { i } + \mathbf { j } + \mathbf { k } \right) \mathrm { m }\).
\(\mathbf { F } _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\).
It is given that this system can be reduced to a single force \(\mathbf { R }\).

1. Find \(\mathbf { R }\).
   (2)
2. Find a vector equation of the line of action of \(\mathbf { R }\), 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.
   (10)

5. A space-ship is moving in a straight line in deep space and needs to reduce its speed from \(U\) to \(V\). This is done by ejecting fuel from the front of the space-ship at a constant speed \(k\) relative to the space-ship. When the speed of the space-ship is \(v\), its mass is \(m\).

1. Show that, while the space-ship is ejecting fuel, \(\frac { \mathrm { d } m } { \mathrm {~d} v } = \frac { m } { k }\). The initial mass of the space-ship is \(M\).
2. Find, in terms of \(U , V , k\) and \(M\), the amount of fuel which needs to be used to reduce the speed of the space-ship from \(U\) to \(V\).
   (6)

6. A uniform circular disc, of mass \(m\), radius \(a\) and centre \(O\), is free to rotate in a vertical plane about a fixed smooth horizontal axis. The axis passes through the mid-point \(A\) of a radius of the disc.

1. Find an equation of motion for the disc when the line \(A O\) makes an angle \(\theta\) with the downward vertical through \(A\).
   (5)
2. Hence find the period of small oscillations of the disc about its position of stable equilibrium. When the line \(A O\) makes an angle \(\theta\) with the downward vertical through \(A\), the force acting on the disc at \(A\) is \(\mathbf { F }\).
3. Find the magnitude of the component of \(\mathbf { F }\) perpendicular to \(A O\).
   (5)

7. Particles \(P\) and \(Q\) have mass \(3 m\) and \(m\) respectively. Particle \(P\) is attached to one end of a light inextensible string and \(Q\) is attached to the other end. The string passes over a circular pulley which can freely rotate in a vertical plane about a fixed horizontal axis through its centre \(O\). The pulley is modelled as a uniform circular disc of mass \(2 m\) and radius \(a\). The pulley is sufficiently rough to prevent the string slipping. The system is at rest with the string taut. A third particle \(R\) of mass \(m\) falls freely under gravity from rest for a distance \(a\) before striking and adhering to \(Q\). Immediately before \(R\) strikes \(Q\), particles \(P\) and \(Q\) are at rest with the string taut.

1. Show that, immediately after \(R\) strikes \(Q\), the angular speed of the pulley is \(\frac { 1 } { 3 } \sqrt { \left( \frac { g } { 2 a } \right) }\). When \(R\) strikes \(Q\), there is an impulse in the string attached to \(Q\).
2. Find the magnitude of this impulse. Given that \(P\) does not hit the pulley,
3. find the distance that \(P\) moves upwards before first coming to instantaneous rest.