Edexcel M5 (Mechanics 5) 2013 June

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 \(t\).

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.

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 \(M _ { 0 }\).

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 \(\frac { 1 } { 4 } m r ^ { 2 }\).]

5. \begin{figure}[h] \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 $$2 \pi \sqrt { } \left( \frac { 151 a } { 48 g } \right)$$

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 \(\frac { \pi } { 2 }\).