Edexcel M5 (Mechanics 5) 2014 June

1. A small bead is threaded on a smooth, straight horizontal wire which passes through the point \(A ( - 3,1 )\) and the point \(B ( 2,5 )\) in the \(x - y\) plane. The bead moves under the action of a horizontal force \(\mathbf { F }\) of magnitude 8.5 N whose line of action is parallel to the line with equation \(15 x - 8 y + 4 = 0\). The unit on both the \(x\) and \(y\) axes has length one metre. Find the work done by \(\mathbf { F }\) as it moves the bead from \(A\) to \(B\).
   (8)
   
2. A particle \(P\) moves in a plane so that its position vector, \(\mathbf { r }\) metres at time \(t\) seconds, satisfies the differential equation

$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + \mathbf { r } = t \mathbf { i } + \mathrm { e } ^ { - t } \mathbf { j }$$ When \(t = 0\) the particle is at the point with position vector \(( \mathbf { i } + \mathbf { j } ) \mathrm { m }\). Find \(\mathbf { r }\) in terms of \(t\).

3. 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.
\(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { r } _ { 1 } = ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\),
\(\mathbf { F } _ { 2 } = ( \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { r } _ { 2 } = ( 3 \mathbf { i } - \mathbf { j } - \mathbf { k } ) \mathrm { m }\),
\(\mathbf { F } _ { 3 } = ( - 3 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { r } _ { 3 } = ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\).
Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple.

4. A spacecraft is travelling in a straight line in deep space where all external forces can be assumed to be negligible. The spacecraft decelerates by ejecting fuel at a constant speed \(k\) relative to the spacecraft, in the direction of motion of the spacecraft. At time \(t\), the spacecraft has speed \(v\) and mass \(m\).

1. Show, from first principles, that while the spacecraft is ejecting fuel, $$\frac { \mathrm { d } v } { \mathrm {~d} m } - \frac { k } { m } = 0$$ At time \(t = 0\), the spacecraft has speed \(U\) and mass \(M\).
2. Find the mass of the spacecraft when it comes to rest. Given that \(m = M \mathrm { e } ^ { - \alpha t ^ { 2 } }\), where \(\alpha\) is a positive constant, and that the spacecraft comes to rest at time \(t = T\),
3. find, in terms of \(U\) and \(T\) only, the distance travelled by the spacecraft in decelerating from speed \(U\) to rest.

1. A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\). The axis \(L\) is perpendicular to the rod and passes through the point \(P\) of the rod, where \(A P = \frac { 2 } { 3 } a\).
   1. Find the moment of inertia of the rod about \(L\).
   The rod is held at rest with \(B\) vertically above \(P\) and is slightly displaced.
2. Find the angular speed of the rod when \(P B\) makes an angle \(\theta\) with the upward vertical.
3. Find the magnitude of the angular acceleration of the rod when \(P B\) makes an angle \(\theta\) with the upward vertical.
4. Find, in terms of \(g\) and \(a\) only, the angular speed of the rod when the force acting on the rod at \(P\) is perpendicular to the rod.

6. (a) Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through the centre of the disc and perpendicular to the plane of the disc is \(\frac { 1 } { 2 } m a ^ { 2 }\).
[0pt] [You may assume without proof that the moment of inertia of a uniform hoop of mass \(m\) and radius \(r\) about an axis through its centre and perpendicular to its plane is \(m r ^ { 2 }\).] \begin{figure}[h] \end{figure} A uniform plane shape \(S\) of mass \(M\) is formed by removing a uniform circular disc with centre \(O\) and radius \(a\) from a uniform circular disc with centre \(O\) and radius \(2 a\), as shown in Figure 1. The shape \(S\) is free to rotate about a fixed smooth axis \(L\), which passes through \(O\) and lies in the plane of the shape.
(b) Show that the moment of inertia of \(S\) about \(L\) is \(\frac { 5 } { 4 } M a ^ { 2 }\). The shape \(S\) is at rest in a horizontal plane and is free to rotate about the axis \(L\). A particle of mass \(M\) falls vertically and strikes \(S\) at the point \(A\), where \(O A = \frac { 3 } { 2 } a\) and \(O A\) is perpendicular to \(L\). The particle adheres to \(S\) at \(A\). Immediately before the particle strikes \(S\) the speed of the particle is \(u\).
(c) Find, in terms of \(M\) and \(u\), the loss in kinetic energy due to the impact.