Edexcel FM2 AS (Further Mechanics 2 AS) 2024 June

1. \begin{figure}[h] \end{figure} A uniform rod of length \(24 a\) is cut into seven pieces which are used to form the framework \(A B C D E F\) shown in Figure 1. It is given that

• \(A F = B E = C D = A B = F E = 4 a\)
• \(B C = E D = 2 a\)
• the rods \(A F , B E\) and \(C D\) are parallel
• the rods \(A B , B C , F E\) and \(E D\) are parallel
• \(A F\) is perpendicular to \(A B\)
• the rods all lie in the same plane

The distance of the centre of mass of the framework from \(A F\) is \(d\).

1. Show that \(d = \frac { 19 } { 6 } a\)
2. Find the distance of the centre of mass of the framework from \(A\).

2. \begin{figure}[h] \end{figure} A thin hollow hemisphere, with centre \(O\) and radius \(a\), is fixed with its axis vertical, as shown in Figure 2. A small ball \(B\) of mass \(m\) moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre \(C\) and radius \(r\). The point \(C\) is vertically below \(O\) such that \(O C = h\). The ball moves with constant angular speed \(\omega\)
The inner surface of the hemisphere is modelled as being smooth and \(B\) is modelled as a particle. Air resistance is modelled as being negligible.

1. Show that \(\omega ^ { 2 } = \frac { g } { h }\) Given that the magnitude of the normal reaction between \(B\) and the surface of the hemisphere is \(3 m g\)
2. find \(\omega\) in terms of \(g\) and \(a\).
3. State how, apart from ignoring air resistance, you have used the fact that \(B\) is modelled as a particle.

1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction and acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.

In a model of the motion of \(P\) $$a = 4 - 3 v$$ When \(t = 0 , v = 0\)

1. Use integration to show that \(v = k \left( 1 - \mathrm { e } ^ { - 3 t } \right)\), where \(k\) is a constant to be found. When \(t = 0 , P\) is at the origin \(O\)
2. Find, in terms of \(t\) only, the distance of \(P\) from \(O\) at time \(t\) seconds.

4. \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C\) has \(A B\) perpendicular to \(A C\), \(A B = 9 a\) and \(A C = 6 a\). The point \(D\) on \(A B\) is such that \(A D = a\). The rectangle \(D E F G\), with \(D E = 2 a\) and \(E F = 3 a\), is removed from the lamina to form the template shown shaded in Figure 3. The distance of the centre of mass of the template from \(A C\) is \(d\).

1. Show that \(d = \frac { 23 } { 7 } a\) The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical through \(A\).
2. Find the value of \(\theta\) A new piece, of exactly the same size and shape as the template, is cut from a lamina of a different uniform material. The template and the new piece are joined together to form the model shown in Figure 4. Both parts of the model lie in the same plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_369_1185_1667_440} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The weight of \(C P Q R S T A\) is \(W\)
   The weight of \(A D G F E B C\) is \(4 W\)
   The model is freely suspended from \(A\).
   A horizontal force of magnitude \(X\), acting in the same vertical plane as the model, is now applied to the model at \(T\) so that \(A C\) is vertical, as shown in Figure 4.
3. Find \(X\) in terms of \(W\).