Edexcel M2 2003 June — Question 1

Exam BoardEdexcel
ModuleM2 (Mechanics 2)
Year2003
SessionJune
TopicMoments
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v = 6 t - 2 t ^ { 2 }\). When \(t = 0 , P\) is at the origin \(O\). Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest after leaving \(O\).
  2. A tennis ball of mass 0.2 kg is moving with velocity \(( - 10 \mathbf { i } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) when it is struck by a tennis racket. Immediately after being struck, the ball has velocity \(( 15 \mathbf { i } + 15 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find
    1. the magnitude of the impulse exerted by the racket on the ball,
    2. the angle, to the nearest degree, between the vector \(\mathbf { i }\) and the impulse exerted by the racket,
    3. the kinetic energy gained by the ball as a result of being struck.
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{b6c32af8-4b73-4ac0-94a5-e046bda06939-2_473_721_1302_625}
    \end{figure} A uniform lamina \(A B C D\) is made by taking a uniform sheet of metal in the form of a rectangle \(A B E D\), with \(A B = 3 a\) and \(A D = 2 a\), and removing the triangle \(B C E\), where \(C\) lies on \(D E\) and \(C E = a\), as shown in Fig. 1.
  3. Find the distance of the centre of mass of the lamina from \(A D\). The lamina has mass \(M\). A particle of mass \(m\) is attached to the lamina at \(B\). When the loaded lamina is freely suspended from the mid-point of \(A B\), it hangs in equilibrium with \(A B\) horizontal.
  4. Find \(m\) in terms of \(M\).
    (4) \section*{4.} \section*{Figure 2}
    \includegraphics[max width=\textwidth, alt={}]{b6c32af8-4b73-4ac0-94a5-e046bda06939-3_471_618_402_678}
    A uniform steel girder \(A B\), of mass 40 kg and length 3 m , is freely hinged at \(A\) to a vertical wall. The girder is supported in a horizontal position by a steel cable attached to the girder at \(B\). The other end of the cable is attached to the point \(C\) vertically above \(A\) on the wall, with \(\angle A B C = \alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). A load of mass 60 kg is suspended by another cable from the girder at the point \(D\), where \(A D = 2 \mathrm {~m}\), as shown in Fig. 2. The girder remains horizontal and in equilibrium. The girder is modelled as a rod, and the cables as light inextensible strings.
  5. Show that the tension in the cable \(B C\) is 980 N .
  6. Find the magnitude of the reaction on the girder at \(A\).
  7. Explain how you have used the modelling assumption that the cable at \(D\) is light.
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