Edexcel M5 (Mechanics 5) 2011 June

1. A particle moves from the point \(A\) with position vector \(( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) to the point \(B\) with position vector \(( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } ) \mathrm { m }\) under the action of the force \(( 2 \mathbf { i } - 3 \mathbf { j } - \mathbf { k } ) \mathrm { N }\). Find the work done by the force.
   (4)
   
2. A particle \(P\) moves in the \(x - y\) plane so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds satisfies the differential equation

$$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } - 4 \mathbf { r } = - 3 \mathrm { e } ^ { t } \mathbf { j }$$ When \(t = 0\), the particle is at the origin and is moving with velocity ( \(2 \mathbf { i } + \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\).
Find \(\mathbf { r }\) in terms of \(t\).

1. A rocket propels itself by its engine ejecting burnt fuel. Initially the rocket has total mass \(M\), of which a mass \(k M , k < 1\), is fuel. The rocket is at rest when its engine is started. The burnt fuel is ejected with constant speed \(c\), relative to the rocket, in a direction opposite to that of the rocket's motion. Assuming that there are no external forces, find the speed of the rocket when all its fuel has been burnt.
2. Two forces \(\mathbf { F } _ { 1 } = ( 3 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 4 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) act on a rigid body.

The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector ( \(2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) ) m and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector ( \(- 3 \mathbf { i } + 2 \mathbf { k }\) ) m.
The two forces are equivalent to a single force \(\mathbf { R }\) acting at the point with position vector \(( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\) together with a couple of moment \(\mathbf { G }\). Find,

1. \(\mathbf { R }\),
2. \(\mathbf { G }\). A third force \(\mathbf { F } _ { 3 }\) is now added to the system. The force \(\mathbf { F } _ { 3 }\) acts at the point with position vector ( \(2 \mathbf { i } - \mathbf { k }\) ) m and the three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a couple.
3. Find the magnitude of the couple.

5. A uniform rod \(P Q\), of mass \(m\) and length \(2 a\), is made to rotate in a vertical plane with constant angular speed \(\sqrt { } \left( \frac { g } { a } \right)\) about a fixed smooth horizontal axis through the end \(P\) of the rod. Show that, when the rod is inclined at an angle \(\theta\) to the downward vertical, the magnitude of the force exerted on the axis by the rod is \(2 m g \left| \cos \left( \frac { 1 } { 2 } \theta \right) \right|\).

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 \(u = 7 v\).

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 \(\left. \frac { 1 } { 2 } m r ^ { 2 } .\right]\)

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.