OCR MEI M2 2011 January — Question 2

Exam BoardOCR MEI
ModuleM2 (Mechanics 2)
Year2011
SessionJanuary
TopicMoments
2
  1. A firework is instantaneously at rest in the air when it explodes into two parts. One part is the body B of mass 0.06 kg and the other a cap C of mass 0.004 kg . The total kinetic energy given to B and C is 0.8 J . B moves off horizontally in the \(\mathbf { i }\) direction. By considering both kinetic energy and linear momentum, calculate the velocities of B and C immediately after the explosion.
  2. A car of mass 800 kg is travelling up some hills. In one situation the car climbs a vertical height of 20 m while its speed decreases from \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car is subject to a resistance to its motion but there is no driving force and the brakes are not being applied.
    1. Using an energy method, calculate the work done by the car against the resistance to its motion. In another situation the car is travelling at a constant speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and climbs a vertical height of 20 m in 25 s up a uniform slope. The resistance to its motion is now 750 N .
    2. Calculate the power of the driving force required. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{680f1be3-13a2-4f75-a324-fb6aadf07607-4_579_851_258_646} \captionsetup{labelformat=empty} \caption{Fig. 3}
      \end{figure} Fig. 3 shows a framework in equilibrium in a vertical plane. The framework is made from the equal, light, rigid rods \(\mathrm { AB } , \mathrm { AD } , \mathrm { BC } , \mathrm { BD }\) and CD so that ABD and BCD are equilateral triangles of side 2 m . AD and BC are horizontal. The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . The pin-joint at A is fixed to a wall and the pin-joint at B rests on a smooth horizontal support. Fig. 3 also shows the external forces acting on the framework: there is a vertical load of 45 N at C and a horizontal force of 50 N applied at D ; the normal reaction of the support on the framework at B is \(R \mathrm {~N}\); horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) act at A .
    3. Write down equations for the horizontal and vertical equilibrium of the framework.
    4. Show that \(R = 135\) and \(Y = 90\).
    5. On the diagram in your printed answer book, show the forces internal to the rods acting on the pin-joints.
      [0pt]
    6. Calculate the forces internal to the five rods, stating whether each rod is in tension or compression (thrust). [You may leave your answers in surd form. Your working in this part should correspond to your diagram in part (iii).]
    7. Suppose that the force of magnitude 50 N applied at D is no longer horizontal, and the system remains in equilibrium in the same position. By considering the equilibrium at C , show that the forces in rods CD and BC are not changed.
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