Edexcel M5 (Mechanics 5) 2013 June

1. Solve the differential equation

$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } - 2 \mathbf { r } = \mathbf { 0 }$$ given that when \(t = 0 , \mathbf { r } . \mathbf { j } = 0\) and \(\mathbf { r } \times \mathbf { j } = \mathbf { i } + \mathbf { k }\).

2. A uniform square lamina \(S\) has side \(2 a\). The radius of gyration of \(S\) about an axis through a vertex, perpendicular to \(S\), is \(k\).

1. Show that \(k ^ { 2 } = \frac { 8 a ^ { 2 } } { 3 }\). The lamina \(S\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(S\) and passes through a vertex.
2. By writing down an equation of rotational motion for \(S\), find the period of small oscillations of \(S\) about its position of stable equilibrium.

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 $$\frac { \mathrm { d } v } { \mathrm {~d} t } + v \left( k + \frac { c } { m _ { 0 } + c t } \right) = g$$

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.

5. \begin{figure}[h] \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 \(\frac { 1 } { 3 } m l ^ { 2 }\).]

6. \begin{figure}[h] \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.

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.