Edexcel M1 2013 January — Question 1

Exam BoardEdexcel
ModuleM1 (Mechanics 1)
Year2013
SessionJanuary
TopicMoments
  1. Two particles \(P\) and \(Q\) have masses \(4 m\) and \(m\) respectively. The particles are moving towards each other on a smooth horizontal plane and collide directly. The speeds of \(P\) and \(Q\) immediately before the collision are \(2 u\) and \(5 u\) respectively. Immediately after the collision, the speed of \(P\) is \(\frac { 1 } { 2 } u\) and its direction of motion is reversed.
    1. Find the speed and direction of motion of \(Q\) after the collision.
    2. Find the magnitude of the impulse exerted on \(P\) by \(Q\) in the collision.
    3. A steel girder \(A B\), of mass 200 kg and length 12 m , rests horizontally in equilibrium on two smooth supports at \(C\) and at \(D\), where \(A C = 2 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\). A man of mass 80 kg stands on the girder at the point \(P\), where \(A P = 4 \mathrm {~m}\), as shown in Figure 1.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_318_1061_420_440} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The man is modelled as a particle and the girder is modelled as a uniform rod.
  2. Find the magnitude of the reaction on the girder at the support at \(C\). The support at \(D\) is now moved to the point \(X\) on the girder, where \(X B = x\) metres. The man remains on the girder at \(P\), as shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{6e887c0f-911a-4f18-81a1-2f90025f5410-03_324_1058_1292_443} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Given that the magnitudes of the reactions at the two supports are now equal and that the girder again rests horizontally in equilibrium, find
  3. the magnitude of the reaction at the support at \(X\),
  4. the value of \(x\).
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