Edexcel M5 (Mechanics 5) 2005 June

1. Two constant forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) are the only forces acting on a particle. \(\mathbf { F } _ { 1 }\) has magnitude 9 N and acts in the direction of \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } . \mathbf { F } _ { 2 }\) has magnitude 18 N and acts in the direction of \(\mathbf { i } + 8 \mathbf { j } - 4 \mathbf { k }\).

Find the total work done by the two forces in moving the particle from the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) to the point with position vector \(( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\).
(Total 6 marks)

2. At time \(t\) seconds the position vector of a particle \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres, where \(\mathbf { r }\) satisfies the differential equation $$\frac { \mathrm { d } \mathbf { r } } { \mathrm {~d} t } + 2 \mathbf { r } = 3 \mathrm { e } ^ { - t } \mathbf { j }$$ Given that \(\mathbf { r } = 2 \mathbf { i } - \mathbf { j }\) when \(t = 0\), find \(\mathbf { r }\) in terms of \(t\).
(Total 7 marks)

3. A system of forces acting on a rigid body consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting at a point \(A\) of the body, together with a couple of moment \(\mathbf { G } . \mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) N\). The position vector of the point \(A\) is \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and \(\mathbf { G } = ( 7 \mathbf { i } - 3 \mathbf { j } + 8 \mathbf { k } ) \mathrm { Nm }\). Given that the system is equivalent to a single force \(\mathbf { R }\),

1. find \(\mathbf { R }\),
2. find a vector equation for the line of action of \(\mathbf { R }\).
   (Total 9 marks) \section*{4.} \section*{Figure 1}
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   A thin uniform rod \(P Q\) has mass \(m\) and length \(3 a\). A thin uniform circular disc, of mass \(m\) and radius \(a\), is attached to the rod at \(Q\) in such a way that the rod and the diameter \(Q R\) of the disc are in a straight line with \(P R = 5 a\). The rod together with the disc form a composite body, as shown in Figure 1. The body is free to rotate about a fixed smooth horizontal axis \(L\) through \(P\), perpendicular to \(P Q\) and in the plane of the disc.

1. Show that the moment of inertia of the body about \(L\) is \(\frac { 77 m a ^ { 2 } } { 4 }\). When \(P R\) is vertical, the body has angular speed \(\omega\) and the centre of the disc strikes a stationary particle of mass \(\frac { 1 } { 2 } \mathrm {~m}\). Given that the particle adheres to the centre of the disc,
2. find, in terms of \(\omega\), the angular speed of the body immediately after the impact.

5. A uniform square lamina \(A B C D\), of mass \(m\) and side \(2 a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) which passes through \(A\) and is perpendicular to the plane of the lamina. The moment of inertia of the lamina about \(L\) is \(\frac { 8 m a ^ { 2 } } { 3 }\). Given that the lamina is released from rest when the line \(A C\) makes an angle of \(\frac { \pi } { 3 }\) with the downward vertical,

1. find the magnitude of the vertical component of the force acting on the lamina at \(A\) when the line \(A C\) is vertical. Given instead that the lamina now makes small oscillations about its position of stable equilibrium,
2. find the period of these oscillations.
   (5)
   (Total 12 marks)

6. A rocket-driven car moves along a straight horizontal road. The car has total initial mass \(M\). It propels itself forwards by ejecting mass backwards at a constant rate \(\lambda\) per unit time at a constant speed \(U\) relative to the car. The car starts from rest at time \(t = 0\). At time \(t\) the speed of the car is \(v\). The total resistance to motion is modelled as having magnitude \(k v\), where \(k\) is a constant. Given that \(t < \frac { M } { \lambda }\), show that

1. \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { \lambda U - k v } { M - \lambda t }\),
2. \(v = \frac { \lambda U } { k } \left\{ 1 - \left( 1 - \frac { \lambda t } { M } \right) ^ { \frac { k } { \lambda } } \right\}\).
   (6)
   (Total 13 marks)

7. A uniform lamina of mass \(m\) is in the shape of an equilateral triangle \(A B C\) of perpendicular height \(h\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\) through \(A\) and perpendicular to the lamina.

1. Show, by integration, that the moment of inertia of the lamina about \(L\) is \(\frac { 5 m h ^ { 2 } } { 9 }\). The centre of mass of the lamina is \(G\). The lamina is in equilibrium, with \(G\) below \(A\), when it is given an angular speed \(\sqrt { \left( \frac { 6 g } { 5 h } \right) }\).
2. Find the angle between \(A G\) and the downward vertical, when the lamina first comes to rest.
3. Find the greatest magnitude of the angular acceleration during the motion.
   (Total 17 marks)