OCR MEI M2 2012 June — Question 3

Exam BoardOCR MEI
ModuleM2 (Mechanics 2)
Year2012
SessionJune
TopicCentre of Mass 1
3
  1. You are given that the position of the centre of mass, G , of a right-angled triangle cut from thin uniform material in the position shown in Fig. 3.1 is at the point \(\left( \frac { 1 } { 3 } a , \frac { 1 } { 3 } b \right)\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-4_328_382_360_845} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} A plane thin uniform sheet of metal is in the shape OABCDEFHIJO shown in Fig. 3.2. BDEA and CDIJ are rectangles and FEH is a right angle. The lengths of the sides are shown with each unit representing 1 cm . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-4_862_906_1032_584} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
    1. Calculate the coordinates of the centre of mass of the metal sheet, referred to the axes shown in Fig. 3.2. The metal sheet is freely suspended from corner B and hangs in equilibrium.
    2. Calculate the angle between BD and the vertical.
  2. Part of a framework of light rigid rods freely pin-jointed at their ends is shown in Fig. 3.3. The framework is in equilibrium. All the rods meeting at the pin-joints at \(\mathrm { A } , \mathrm { B }\) and C are shown. The rods connected to \(\mathrm { A } , \mathrm { B }\) and C are connected to the rest of the framework at \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\) and T . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-5_499_734_493_662} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
    \end{figure} There is a tension of 18 N in rod AP and a thrust (compression) of 5 N in rod AQ.
    1. Show the forces internal to the rods acting on the pin-joints at \(\mathrm { A } , \mathrm { B }\) and C .
    2. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CA , stating whether each rod is in tension or compression. [You may leave your answers in surd form. Your working in this part should be consistent with your diagram in part (i).]
      \(4 P\) and \(Q\) are circular discs of mass 3 kg and 10 kg respectively which slide on a smooth horizontal surface. The discs have the same diameter and move in the line joining their centres with no resistive forces acting on them. The surface has vertical walls which are perpendicular to the line of centres of the discs. This information is shown in Fig. 4 together with the direction you should take as being positive. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-6_430_1404_443_328} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
    3. For what time must a force of 26 N act on P to accelerate it from rest to \(13 \mathrm {~ms} ^ { - 1 }\) ? P is travelling at \(13 \mathrm {~ms} ^ { - 1 }\) when it collides with Q , which is at rest. The coefficient of restitution in this collision is \(e\).
    4. Show that, after the collision, the velocity of P is \(( 3 - 10 e ) \mathrm { ms } ^ { - 1 }\) and find an expression in terms of \(e\) for the velocity of Q.
    5. For what set of values of \(e\) does the collision cause P to reverse its direction of motion?
    6. Determine the set of values of \(e\) for which P has a greater speed than Q immediately after the collision. You are now given that \(e = \frac { 1 } { 2 }\). After P and Q collide with one another, each has a perfectly elastic collision with a wall. P and Q then collide with one another again and in this second collision they stick together (coalesce).
    7. Determine the common velocity of P and Q .
    8. Determine the impulse of Q on P in this collision.
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