AQA M2 (Mechanics 2) 2016 June

1 A stone, of mass 0.3 kg , is thrown with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 5 metres above a horizontal surface.

1. Calculate the initial kinetic energy of the stone.
2. 1. Find the kinetic energy of the stone when it hits the surface.
   2. Hence find the speed of the stone when it hits the surface.
   3. State one modelling assumption that you have made.

2 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons.
The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = \left( 8 t - t ^ { 4 } \right) \mathbf { i } + 6 \mathrm { e } ^ { - 3 t } \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 2 kg .
   1. Find an expression for the force \(\mathbf { F }\) acting on the particle at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 1\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due south.
4. When \(t = 0\), the particle is at the point with position vector \(( 3 \mathbf { i } - 5 \mathbf { j } )\) metres. Find an expression for the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
   [0pt] [4 marks]

3 The diagram shows a uniform lamina \(A B C D E F G H I J K L\).
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-08_474_1378_351_370}

1. Explain why the centre of mass of the lamina is 35 cm from \(A L\).
2. Find the distance of the centre of mass from \(A F\).
3. The lamina is freely suspended from \(A\). Find the angle between \(A B\) and the vertical when the lamina is in equilibrium.
4. Explain, briefly, how you have used the fact that the lamina is uniform.

4 A particle \(P\), of mass 6 kg , is attached to one end of a light inextensible string. The string passes through a small smooth ring, fixed at a point \(O\). A second particle \(Q\), of mass 8 kg , is attached to the other end of the string. The particle \(Q\) hangs at rest vertically below the ring, and the particle \(P\) moves with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle, as shown in the diagram. The angle between \(O P\) and the vertical is \(\theta\).
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-10_517_433_683_790}

1. Find the tension in the string.
2. \(\quad\) Find \(\theta\).
3. Find the radius of the horizontal circle.
   [0pt] [4 marks]

5 A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(R\) vertically below \(O\). The particle is set into motion with a horizontal velocity \(u\) so that it moves in a complete vertical circle with centre \(O\). The point \(T\) on the circle is such that angle \(R O T\) is \(30 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-12_766_736_644_651}

1. Find, in terms of \(g , l\) and \(u\), the speed of the particle at the point \(T\).
2. Find, in terms of \(g , l , m\) and \(u\), the tension in the string when the particle is at the point \(T\).
3. Find, in terms of \(g , l , m\) and \(u\), the tension in the string when the particle returns to the point \(R\).
4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\).
   [0pt] [4 marks]

6 A stone, of mass \(m\), falls vertically downwards under gravity through still water. At time \(t\), the stone has speed \(v\) and it experiences a resistance force of magnitude \(\lambda m v\), where \(\lambda\) is a constant.

1. Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - \lambda v$$
2. The initial speed of the stone is \(u\). Find an expression for \(v\) at time \(t\).
   [0pt] [6 marks] \(7 \quad\) A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2 \mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.
3. Draw a diagram to show the forces acting on the ladder.
4. Find \(\tan \theta\) in terms of \(\mu\).

8 A particle, \(P\), of mass 5 kg is placed at the point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The points \(Q\) and \(R\) are also on the surface of the inclined plane, with \(Q R = 15\) metres. The point \(A\) is between \(Q\) and \(R\) so that \(A Q = 4\) metres and \(A R = 11\) metres. The three points \(Q , A\) and \(R\) are on a line of greatest slope of the plane.
\includegraphics[max width=\textwidth, alt={}, center]{7c2c50e0-4976-4301-9898-61b2760a2aee-20_391_882_676_587} The particle is attached to two light elastic strings, \(P Q\) and \(P R\).
One of the strings, \(P Q\), has natural length 4 metres and modulus of elasticity 160 N , the other string, \(P R\), has natural length 6 metres and modulus of elasticity 120 N . The particle is released from rest at the point \(A\).
The coefficient of friction between the particle and the plane is 0.4 .
Find the distance of the particle from \(Q\) when it is next at rest.
[0pt] [8 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}