Edexcel FM2 AS (Further Mechanics 2 AS) 2021 June

1. \begin{figure}[h] \end{figure} A uniform rod of length \(72 a\) is cut into pieces. The pieces are used to make two rigid squares, \(A B C D\) and \(P Q R S\), with sides of length \(10 a\) and \(8 a\) respectively. The two squares are joined to form the rigid framework shown in Figure 1. The squares both lie in the same plane with the rod \(A B\) parallel to the rod \(P Q\).
Given that

• \(A D\) cuts \(P Q\) in the ratio \(3 : 5\)
• \(D C\) cuts \(Q R\) in the ratio 5:3
  1. explain why the centre of mass of square \(A B C D\) is at \(Q\).
  2. Find the distance of the centre of mass of the framework from \(B\).

2. \begin{figure}[h] \end{figure} A small smooth ring \(P\), of mass \(m\), is threaded onto a light inextensible string of length 4a. One end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The other end of the string is attached to a fixed point \(B\) which is vertically above \(A\). The ring moves in a horizontal circle with centre \(A\) and radius \(a\), as shown in Figure 2. The ring moves with constant angular speed \(\sqrt { \frac { 2 g } { 3 a } }\) about \(A B\).
The string remains taut throughout the motion.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between \(P\) and the table. The angular speed of \(P\) is now gradually increased.
2. Find, in terms of \(a\) and \(g\), the angular speed of \(P\) at the instant when it loses contact with the table.
3. Explain how you have used the fact that \(P\) is smooth.

3. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E F G H I J\) is shown in Figure 3.
The lamina has \(A J = 8 a , A B = 5 a\) and \(B C = D E = E F = F G = G H = H I = I J = 2 a\).
All the corners are right angles.

1. Show that the distance of the centre of mass of the lamina from \(A J\) is \(\frac { 49 } { 26 } a\) A light inextensible rope is attached to the lamina at \(A\) and another light inextensible rope is attached to the lamina at \(B\). The lamina hangs in equilibrium with both ropes vertical and \(A B\) horizontal. The weight of the lamina is \(W\).
2. Find, in terms of \(W\), the tension in the rope attached to the lamina at \(B\). The rope attached to \(B\) breaks and subsequently the lamina hangs freely in equilibrium, suspended from \(A\).
3. Find the size of the angle between \(A J\) and the downward vertical.

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = 5 \sin 2 t$$ When \(t = 0 , x = 1\) and \(P\) is at rest.

1. Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
2. Show that \(1 \leqslant x \leqslant 6\)
3. Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
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