Edexcel M5 (Mechanics 5) 2014 June

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A bead \(P\) of mass 0.2 kg is threaded on a smooth straight horizontal wire. The bead is at rest at the point \(A\) with position vector \(( 4 \mathbf { i } - \mathbf { j } ) \mathrm { m }\). A force \(( 0.2 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it to the point \(B\) with position vector \(( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the speed of \(P\) at \(B\).

2. A uniform equilateral triangular lamina \(A B C\) has mass \(m\) and sides of length \(\sqrt { } 3 a\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\), which passes through \(A\) and is perpendicular to the lamina. The midpoint of \(B C\) is \(D\). The lamina is held with \(A D\) making an angle of \(60 ^ { \circ }\) with the upward vertical through \(A\) and released from rest. The moment of inertia of the lamina about the axis \(L\) is \(\frac { 5 m a ^ { 2 } } { 4 }\) Find the speed of \(D\) when \(A D\) is vertical.
(8)

3. A uniform rectangular lamina \(A B C D\), where \(A B = a\) and \(B C = 2 a\), has mass \(2 m\). The lamina is free to rotate about its edge \(A B\), which is fixed and vertical. The lamina is at rest when it is struck at \(C\) by a particle \(P\) of mass \(m\). The particle \(P\) is moving horizontally with speed \(U\) in a direction which is perpendicular to the lamina. The coefficient of restitution between \(P\) and the lamina is 0.5 Find the angular speed of the lamina immediately after the impact.
(8)

4. A uniform solid sphere has mass \(M\) and radius \(a\). Prove, using integration, that the moment of inertia of the sphere about a diameter is \(\frac { 2 M a ^ { 2 } } { 5 }\)
[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 \(\frac { 1 } { 2 } m r ^ { 2 }\).]

1. A particle moves in a plane so that its position vector \(\mathbf { r }\) metres at time \(t\) seconds satisfies the differential equation

$$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + ( \tan t ) \mathbf { r } = \left( \cos ^ { 2 } t \right) \mathbf { i } - ( 3 \cos t ) \mathbf { j } , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ When \(t = 0\), the particle is at the point with position vector \(4 \mathbf { j } \mathrm {~m}\). Find \(\mathbf { r }\) in terms of \(t\).

6. 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, where
\(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N }\)
\(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\)
\(\mathbf { F } _ { 3 } = ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\)
\(\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { k } ) \mathrm { m }\)
\(\mathbf { r } _ { 2 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\)
\(\mathbf { r } _ { 3 } = ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }\) The system of the three forces is equivalent to a single force \(\mathbf { R }\) acting at the point with position vector ( \(\mathbf { 3 i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\), together with a couple of moment \(\mathbf { G }\).

1. Find \(\mathbf { R }\).
2. Find \(\mathbf { G }\).

7. A raindrop absorbs water as it falls vertically under gravity through a cloud. In a model of the motion the cloud is assumed to consist of stationary water particles. At time \(t\), the mass of the raindrop is \(m\) and the speed of the raindrop is \(v\). At time \(t = 0\), the raindrop is at rest. The rate of increase of the mass of the raindrop with respect to time is modelled as being \(m k v\), where \(k\) is a positive constant.

1. Ignoring air resistance, show from first principles, that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - k v ^ { 2 }$$
2. Find the time taken for the raindrop to reach a speed of \(\frac { 1 } { 2 } \sqrt { } \left( \frac { g } { k } \right)\)

8. \begin{figure}[h] \end{figure} A uniform circular disc of radius \(2 a\) has centre \(O\). The points \(P , Q , R\) and \(S\) on the disc are the vertices of a square with centre \(O\) and \(O P = a\). Four circular holes, each of radius \(\frac { a } { 2 }\), and with centres \(P , Q , R\) and \(S\), are drilled in the disc to produce the lamina \(L\), shown shaded in Figure 1. The mass of \(L\) is \(M\).

1. Show that the moment of inertia of \(L\) about an axis through \(O\), and perpendicular to the plane of \(L\), is \(\frac { 55 M a ^ { 2 } } { 24 }\) The lamina \(L\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(L\) and which passes through a point \(A\) on the circumference of \(L\). At time \(t , A O\) makes an angle \(\theta\) with the downward vertical through \(A\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - \frac { 48 g } { 151 a } \sin \theta\)
3. Hence find the period of small oscillations of \(L\) about its position of stable equilibrium. The magnitude of the component, in a direction perpendicular to \(A O\), of the force exerted on \(L\) by the axis is \(X\).
4. Find \(X\) in terms of \(M , g\) and \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{57b98cdd-4121-4495-b500-185cbf3ff1a8-14_159_1662_2416_173}