OCR MEI M2 (Mechanics 2) 2013 June

1

1. In this part-question, all the objects move along the same straight line on a smooth horizontal plane. All their collisions are direct. The masses of the objects \(\mathrm { P } , \mathrm { Q }\) and R and the initial velocities of P and Q (but not R ) are shown in Fig. 1.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_177_1011_488_529} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} A force of 21 N acts on P for 2 seconds in the direction \(\mathrm { PQ } . \mathrm { P }\) does not reach Q in this time.
   1. Calculate the speed of P after the 2 seconds. The force of 21 N is removed after the 2 seconds. When P collides with Q they stick together (coalesce) to form an object S of mass 6 kg .
   2. Show that immediately after the collision S has a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards R . The collision between S and R is elastic with coefficient of restitution \(\frac { 1 } { 4 }\). After the collision, S has a velocity of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of its motion before the collision.
   3. Find the velocities of R before and after the collision. \section*{(b) In this part-question take \(\boldsymbol { g } = \mathbf { 1 0 }\).} A particle of mass 0.2 kg is projected vertically downwards with initial speed \(5 \mathrm {~ms} ^ { - 1 }\) and it travels 10 m before colliding with a fixed smooth plane. The plane is inclined at \(\alpha\) to the vertical where \(\tan \alpha = \frac { 3 } { 4 }\). Immediately after its collision with the plane, the particle has a speed of \(13 \mathrm {~ms} ^ { - 1 }\). This information is shown in Fig. 1.2. Air resistance is negligible. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-2_383_341_1795_854} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure}
   4. Calculate the angle between the direction of motion of the particle and the plane immediately after the collision. Calculate also the coefficient of restitution in the collision.
   5. Calculate the magnitude of the impulse of the plane on the particle.

2 A fairground ride consists of raising vertically a bench with people sitting on it, allowing the bench to drop and then bringing it to rest using brakes. Fig. 2 shows the bench and its supporting tower. The tower provides lifting and braking mechanisms. The resistances to motion are modelled as having a constant value of 400 N whenever the bench is moving up or down; the only other resistance to motion comes from the action of the brakes. \begin{figure}[h] \end{figure} On one occasion, the mass of the bench (with its riders) is 800 kg .
With the brakes not applied, the bench is lifted a distance of 6 m in 12 seconds. It starts from rest and ends at rest.

1. Show that the work done in lifting the bench in this way is 49440 J and calculate the average power required. For a short period while the bench is being lifted it has a constant speed of \(0.55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Calculate the power required during this period. With neither the lifting mechanism nor the brakes applied, the bench is now released from rest and drops 3 m .
3. Using an energy method, calculate the speed of the bench when it has dropped 3 m . The brakes are now applied and they halve the speed of the bench while it falls a further 0.8 m .
4. Using an energy method, calculate the work done by the brakes.

3 Fig. 3.1 shows a rigid, thin, non-uniform 20 cm by 80 cm rectangular panel ABCD of weight 60 N that is in a vertical plane. Its dimensions and the position of its centre of mass, \(G\), are shown in centimetres. The panel is free to rotate about a fixed horizontal axis through A perpendicular to its plane; the panel rests on a small smooth fixed peg at B positioned so that AB is at \(40 ^ { \circ }\) to the horizontal. A horizontal force in the plane of ABCD of magnitude \(P \mathrm {~N}\) acts at D away from the panel. \begin{figure}[h] \end{figure}

1. Show that the clockwise moment of the weight about A is 9.93 Nm , correct to 3 significant figures.
2. Calculate the value of \(P\) for which the panel is on the point of turning about the axis through A .
3. In the situation where \(P = 0\), calculate the vertical component of the force exerted on the panel by the axis through A . The panel is now placed on a line of greatest slope of a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. The panel is at all times in a vertical plane. A horizontal force in the plane ABCD of magnitude 200 N acts at D towards the panel. This situation is shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-4_497_842_1653_616} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
4. Given that the panel is moving up the plane with acceleration up the plane of \(1.75 \mathrm {~ms} ^ { - 2 }\), calculate the coefficient of friction between the panel and the plane.

4

1. Fig. 4.1 shows a framework constructed from 4 uniform heavy rigid rods \(\mathrm { OP } , \mathrm { OQ } , \mathrm { PR }\) and RS , rigidly joined at \(\mathrm { O } , \mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S and with OQ perpendicular to PR . Fig. 4.1 also shows the dimensions of the rods and axes \(\mathrm { O } x\) and \(\mathrm { O } y\) : the units are metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-5_454_994_408_548} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Each rod has a mass of 0.8 kg per metre.
   1. Show that, referred to the axes in Fig. 4.1, the \(x\)-coordinate of the centre of mass of the framework is 1.5 and calculate the \(y\)-coordinate. The framework is freely suspended from S and a small object of mass \(m \mathrm {~kg}\) is attached to it at O . The framework is in equilibrium with OQ horizontal.
   2. Calculate \(m\).
2. Fig. 4.2 shows a framework in equilibrium in a vertical plane. The framework is made from 5 light, rigid rods \(\mathrm { OP } , \mathrm { OQ } , \mathrm { OR } , \mathrm { PQ }\) and QR . Its dimensions are indicated. PQ is horizontal and OR vertical. The rods are freely pin-jointed to each other at \(\mathrm { O } , \mathrm { P } , \mathrm { Q }\) and R . The pin-joint at O is fixed to a wall.
   Fig. 4.2 also shows the external forces acting on the framework: there are vertical loads of 120 N and 60 N at Q and P respectively; a horizontal string attached to Q has tension \(T \mathrm {~N}\); horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) act on the framework from the pin-joint at O . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-6_566_453_625_788} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. By considering only the pin-joint at R , explain why the rods OR and RQ must have zero internal force.
   2. Find the values of \(T , X\) and \(Y\).
   3. Using the diagram in your printed answer book, show all the forces acting on the pin-joints, including those internal to the rods.
      [0pt]
   4. Calculate the forces internal to the rods OP and PQ , stating whether each rod is in tension or compression (thrust). [You may leave answers in surd form. Your working in this part should correspond to your diagram in part (iii).]