Questions — OCR MEI M2 (75 questions)

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OCR MEI M2 2006 June Q2
18 marks Standard +0.3
2 Two heavy rods AB and BC are freely jointed together at B and to a wall at A . AB has weight 90 N and centre of mass at \(\mathrm { P } ; \mathrm { BC }\) has weight 75 N and centre of mass at Q . The lengths of the rods and the positions of P and Q are shown in Fig. 2.1, with the lengths in metres. Initially, AB and BC are horizontal. There is a support at R , as shown in Fig. 2.1. The system is held in equilibrium by a vertical force acting at C . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_381_703_584_680} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Draw diagrams showing all the forces acting on \(\operatorname { rod } \mathrm { AB }\) and on \(\operatorname { rod } \mathrm { BC }\). Calculate the force exerted on AB by the hinge at B and hence the force required at C . The rods are now set up as shown in Fig. 2.2. AB and BC are each inclined at \(60 ^ { \circ }\) to the vertical and C rests on a rough horizontal table. Fig. 2.3 shows all the forces acting on AB , including the forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) due to the hinge at A and the forces \(U \mathrm {~N}\) and \(V \mathrm {~N}\) in the hinge at B . The rods are in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_393_661_1615_429} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_355_438_1530_1178} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
    \end{figure}
  2. By considering the equilibrium of \(\operatorname { rod } \mathrm { AB }\), show that \(60 \sqrt { 3 } = U + V \sqrt { 3 }\).
  3. Draw a diagram showing all the forces acting on rod BC .
  4. Find a further equation connecting \(U\) and \(V\) and hence find their values. Find also the frictional force at C .
OCR MEI M2 2006 June Q3
18 marks Standard +0.3
3
  1. A car of mass 900 kg is travelling at a steady speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at arcsin 0.1 to the horizontal. The power required to do this is 20 kW . Calculate the resistance to the motion of the car.
  2. A small box of mass 11 kg is placed on a uniform rough slope inclined at arc \(\cos \frac { 12 } { 13 }\) to the horizontal. The coefficient of friction between the box and the slope is \(\mu\).
    1. Show that if the box stays at rest then \(\mu \geqslant \frac { 5 } { 12 }\). For the remainder of this question, the box moves on a part of the slope where \(\mu = 0.2\).
      The box is projected up the slope from a point P with an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It travels a distance of 1.5 m along the slope before coming instantaneously to rest. During this motion, the work done against air resistance is 6 joules per metre.
    2. Calculate the value of \(v\). As the box slides back down the slope, it passes through its point of projection P and later reaches its initial speed at a point Q . During this motion, once again the work done against air resistance is 6 joules per metre.
    3. Calculate the distance PQ.
OCR MEI M2 2007 June Q2
17 marks Standard +0.3
2 The position of the centre of mass, \(G\), of a uniform wire bent into the shape of an arc of a circle of radius \(r\) and centre C is shown in Fig. 2.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_325_1132_365_669} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Use this information to show that the centre of mass, G , of the uniform wire bent into the shape of a semi-circular arc of radius 8 shown in Fig. 2.2 has coordinates \(\left( - \frac { 16 } { \pi } , 8 \right)\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_586_871_1016_806} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} A walking-stick is modelled as a uniform rigid wire. The walking-stick and coordinate axes are shown in Fig. 2.3. The section from O to A is a semi-circular arc and the section OB lies along the \(x\)-axis. The lengths are in centimetres. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_394_958_1937_552} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
    \end{figure}
  2. Show that the coordinates of the centre of mass of the walking-stick are ( \(25.37,2.07\) ), correct to two decimal places. The walking-stick is now hung from a shelf as shown in Fig. 2.4. The only contact between the walking-stick and the shelf is at A . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-4_339_374_388_842} \captionsetup{labelformat=empty} \caption{Fig. 2.4}
    \end{figure}
  3. When the walking-stick is in equilibrium, OB is at an angle \(\alpha\) to the vertical. Draw a diagram showing the position of the centre of mass of the walking-stick in relation to A .
    Calculate \(\alpha\).
  4. The walking-stick is now held in equilibrium, with OB vertical and A still resting on the shelf, by means of a vertical force, \(F \mathrm {~N}\), at B . The weight of the walking-stick is 12 N . Calculate \(F\).
OCR MEI M2 2007 June Q3
19 marks Standard +0.3
3 A uniform plank is 2.8 m long and has weight 200 N . The centre of mass is G.
  1. Fig. 3.1 shows the plank horizontal and in equilibrium, resting on supports at A and B . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_229_1125_434_459} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
    \end{figure} Calculate the reactions of the supports on the plank at A and at B .
  2. Fig. 3.2 shows the plank horizontal and in equilibrium between a support at C and a peg at D . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_236_1141_993_461} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} Calculate the reactions of the support and the peg on the plank at C and at D , showing the directions of these forces on a diagram. Fig. 3.3 shows the plank in equilibrium between a support at P and a peg at Q . The plank is inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_424_1099_1692_475} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
    \end{figure}
  3. Calculate the normal reactions at P and at Q .
  4. Just one of the contacts is rough. Determine which one it is if the value of the coefficient of friction is as small as possible. Find this value of the coefficient of friction.
OCR MEI M2 2007 June Q4
17 marks Moderate -0.3
4 Jack and Jill are raising a pail of water vertically using a light inextensible rope. The pail and water have total mass 20 kg . In parts (i) and (ii), all non-gravitational resistances to motion may be neglected.
  1. How much work is done to raise the pail from rest so that it is travelling upwards at \(0.5 \mathrm {~ms} ^ { - 1 }\) when at a distance of 4 m above its starting position?
  2. What power is required to raise the pail at a steady speed of \(0.5 \mathrm {~ms} ^ { - 1 }\) ? Jack falls over and hurts himself. He then slides down a hill.
    His mass is 35 kg and his speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(3 \mathrm {~ms} ^ { - 1 }\) while descending through a vertical height of 3 m .
  3. How much work is done against friction? In Jack's further motion, he slides down a slope at an angle \(\alpha\) to the horizontal where \(\sin \alpha = 0.1\). The frictional force on him is now constant at 150 N . For this part of the motion, Jack's initial speed is \(3 \mathrm {~ms} ^ { - 1 }\).
  4. How much further does he slide before coming to rest?
OCR MEI M2 2008 June Q1
17 marks Moderate -0.8
1
  1. Disc A of mass 6 kg and disc B of mass 0.5 kg are moving in the same straight line. The relative positions of the discs and the \(\mathbf { i }\) direction are shown in Fig. 1.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_282_1325_402_450} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} The discs collide directly. The impulse on A in the collision is \(- 12 \mathbf { i }\) Ns and after the collision A has velocity \(3 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and B has velocity \(11 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    1. Show that the velocity of A just before the collision is \(5 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and find the velocity of B at this time.
    2. Calculate the coefficient of restitution in the collision.
    3. After the collision, a force of \(- 2 \mathbf { i } \mathrm {~N}\) acts on B for 7 seconds. Find the velocity of B after this time.
  2. A ball bounces off a smooth plane. The angles its path makes with the plane before and after the impact are \(\alpha\) and \(\beta\), as shown in Fig. 1.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-2_317_1082_1468_575} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure} The velocity of the ball before the impact is \(u \mathbf { i } - v \mathbf { j }\) and the coefficient of restitution in the impact is \(e\). Write down an expression in terms of \(u , v , e , \mathbf { i }\) and \(\mathbf { j }\) for the velocity of the ball immediately after the impact. Hence show that \(\tan \beta = e \tan \alpha\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_581_486_274_383} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-3_593_392_264_1370} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure} A uniform wire is bent to form a bracket OABCD . The sections \(\mathrm { OA } , \mathrm { AB }\) and BC lie on three sides of a square and CD is parallel to AB . This is shown in Fig. 2.1 where the dimensions, in centimetres, are also given.
    1. Show that, referred to the axes shown in Fig. 2.1, the \(x\)-coordinate of the centre of mass of the bracket is 3.6 . Find also the \(y\)-coordinate of its centre of mass.
    2. The bracket is now freely suspended from D and hangs in equilibrium. Draw a diagram showing the position of the centre of mass and calculate the angle of CD to the vertical.
    3. The bracket is now hung by means of vertical, light strings BP and DQ attached to B and to D , as shown in Fig. 2.2. The bracket has weight 5 N and is in equilibrium with OA horizontal. Calculate the tensions in the strings BP and DQ . The original bracket shown in Fig. 2.1 is now changed by adding the section OE, where AOE is a straight line. This section is made of the same type of wire and has length \(L \mathrm {~cm}\), as shown in Fig. 2.3. \(\begin{array} { l l l l } \begin{array} { l } \text { not to } \\ \text { scale } \end{array} & 2 & 6 & \\ \mathrm {~L} \longrightarrow & \mathrm {~L} & & \\ \mathrm {~L} & \mathrm { O } & 6 & \mathrm {~A} \end{array}\) Fig. 2.3 The value of \(L\) is chosen so that the centre of mass is now on the \(y\)-axis.
    4. Calculate \(L\).
OCR MEI M2 2008 June Q3
18 marks Standard +0.8
3 \begin{enumerate}[label=(\alph*)] \item Fig. 3.1 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\) and BD . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and D and to a vertical wall at C and D. There are vertical loads of \(L \mathrm {~N}\) at A and \(3 L \mathrm {~N}\) at B . Angle DAB is \(30 ^ { \circ }\), angle DBC is \(60 ^ { \circ }\) and ABC is a straight, horizontal line. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_538_617_497_804} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Draw a diagram showing the loads and the internal forces in the four rods.
  2. Find the internal forces in the rods in terms of \(L\), stating whether each rod is in tension or in thrust (compression). [You may leave answers in surd form. Note that you are not required to find the external forces acting at C and at D.]
\item Fig. 3.2 shows uniform beams PQ and QR , each of length 2 lm and of weight \(W \mathrm {~N}\). The beams are freely hinged at Q and are in equilibrium on a rough horizontal surface when inclined at \(60 ^ { \circ }\) to the horizontal. You are given that the total force acting at Q on QR due to the hinge is horizontal. This force, \(U \mathrm {~N}\), is shown in Fig. 3.3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_428_566_1699_536} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_296_282_1699_1407} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
\end{figure} Show that the frictional force between the floor and each beam is \(\frac { \sqrt { 3 } } { 6 } W \mathrm {~N}\).
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-5_641_885_269_671} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure} A small sphere of mass 0.15 kg is attached to one end, B, of a light, inextensible piece of fishing line of length 2 m . The other end of the line, A , is fixed and the line can swing freely. The sphere swings with the line taut from a point where the line is at an angle of \(40 ^ { \circ }\) with the vertical, as shown in Fig. 4.
    1. Explain why no work is done on the sphere by the tension in the line.
    2. Show that the sphere has dropped a vertical distance of about 0.4679 m when it is at the lowest point of its swing and calculate the amount of gravitational potential energy lost when it is at this point.
    3. Assuming that there is no air resistance and that the sphere swings from rest from the position shown in Fig. 4, calculate the speed of the sphere at the lowest point of its swing.
    4. Now consider the case where
      Calculate the speed of the sphere at the lowest point of its swing.
    5. A block of mass 3 kg slides down a uniform, rough slope that is at an angle of \(30 ^ { \circ }\) to the horizontal. The acceleration of the block is \(\frac { 1 } { 8 } g\). Show that the coefficient of friction between the block and the slope is \(\frac { 1 } { 4 } \sqrt { 3 }\).
OCR MEI M2 2009 June Q1
18 marks Moderate -0.3
1
  1. Two small objects, P of mass \(m \mathrm {~kg}\) and Q of mass \(k m \mathrm {~kg}\), slide on a smooth horizontal plane. Initially, P and Q are moving in the same straight line towards one another, each with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After a direct collision with P , the direction of motion of Q is reversed and it now has a speed of \(\frac { 1 } { 3 } u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of P is now \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where the positive direction is the original direction of motion of P .
    1. Draw a diagram showing the velocities of P and Q before and after the impact.
    2. By considering the linear momentum of the objects before and after the collision, show that \(v = \left( 1 - \frac { 4 } { 3 } k \right) u\).
    3. Hence find the condition on \(k\) for the direction of motion of P to be reversed. The coefficient of restitution in the collision is 0.5 .
    4. Show that \(v = - \frac { 2 } { 3 } u\) and calculate the value of \(k\).
  2. Particle \(A\) has a mass of 5 kg and velocity \(\binom { 3 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Particle \(B\) has mass 3 kg and is initially at rest. A force \(\binom { 1 } { - 2 } \mathrm {~N}\) acts for 9 seconds on B and subsequently (in the absence of the force), \(A\) and \(B\) collide and stick together to form an object \(C\) that moves off with a velocity \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\).
    1. Show that \(\mathbf { V } = \binom { 3 } { - 1 }\). The object C now collides with a smooth barrier which lies in the direction \(\binom { 0 } { 1 }\). The coefficient of restitution in the collision is 0.5 .
    2. Calculate the velocity of C after the impact.
OCR MEI M2 2009 June Q2
17 marks Standard +0.3
2
  1. A small block of mass 25 kg is on a long, horizontal table. Each side of the block is connected to a small sphere by means of a light inextensible string passing over a smooth pulley. Fig. 2 shows this situation. Sphere A has mass 5 kg and sphere B has mass 20 kg . Each of the spheres hangs freely. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-3_487_1123_466_552} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} Initially the block moves on a smooth part of the table. With the block at a point O , the system is released from rest with both strings taut.
    1. (A) Is mechanical energy conserved in the subsequent motion? Give a brief reason for your answer.
      (B) Why is no work done by the block against the reaction of the table on it? The block reaches a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at point P .
    2. Use an energy method to calculate the distance OP. The block continues moving beyond P , at which point the table becomes rough. After travelling two metres beyond P , the block passes through point Q . The block does 180 J of work against resistances to its motion from P to Q .
    3. Use an energy method to calculate the speed of the block at Q .
  2. A tree trunk of mass 450 kg is being pulled up a slope inclined at \(20 ^ { \circ }\) to the horizontal. Calculate the power required to pull the trunk at a steady speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) against a frictional force of 2000 N .
OCR MEI M2 2009 June Q3
18 marks Standard +0.3
3 A non-uniform beam AB has weight 85 N . The length of the beam is 5 m and its centre of mass is 3 m from A . In this question all the forces act in the same vertical plane. Fig. 3.1 shows the beam in horizontal equilibrium, supported at its ends. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_215_828_466_660} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Calculate the reactions of the supports on the beam. Using a smooth hinge, the beam is now attached at A to a vertical wall. The beam is held in equilibrium at an angle \(\alpha\) to the horizontal by means of a horizontal force of magnitude 27.2 N acting at B . This situation is shown in Fig. 3.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_725_675_1153_347} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_732_565_1153_1231} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
    \end{figure}
  2. Show that \(\tan \alpha = \frac { 15 } { 8 }\). The hinge and 27.2 N force are removed. The beam now rests with B on a rough horizontal floor and A on a smooth vertical wall, as shown in Fig. 3.3. It is at the same angle \(\alpha\) to the horizontal. There is now a force of 34 N acting at right angles to the beam at its centre in the direction shown. The beam is in equilibrium and on the point of slipping.
  3. Draw a diagram showing the forces acting on the beam. Show that the frictional force acting on the beam is 7.4 N .
    Calculate the value of the coefficient of friction between the beam and the floor.
OCR MEI M2 2009 June Q4
19 marks Standard +0.3
4 In this question you may use the following facts: as illustrated in Fig. 4.1, the centre of mass, G, of a uniform thin open hemispherical shell is at the mid-point of OA on its axis of symmetry; the surface area of this shell is \(2 \pi r ^ { 2 }\), where \(r\) is the distance OA. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-5_344_542_445_804} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure} A perspective view and a cross-section of a dog bowl are shown in Fig. 4.2. The bowl is made throughout from thin uniform material. An open hemispherical shell of radius 8 cm is fitted inside an open circular cylinder of radius 8 cm so that they have a common axis of symmetry and the rim of the hemisphere is at one end of the cylinder. The height of the cylinder is \(k \mathrm {~cm}\). The point O is on the axis of symmetry and at the end of the cylinder. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-5_494_947_1238_267} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-5_478_492_1238_1370} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
\end{figure}
  1. Show that the centre of mass of the bowl is a distance \(\frac { 64 + k ^ { 2 } } { 16 + 2 k } \mathrm {~cm}\) from O . A version of the bowl for the 'senior dog' has \(k = 12\) and an end to the cylinder, as shown in Fig. 4.3. The end is made from the same material as the original bowl.
  2. Show that the centre of mass of this bowl is a distance \(6 \frac { 1 } { 3 } \mathrm {~cm}\) from O . This bowl is placed on a rough slope inclined at \(\theta\) to the horizontal.
  3. Assume that the bowl is prevented from sliding and is on the point of toppling. Draw a diagram indicating the position of the centre of mass of the bowl with relevant lengths marked. Calculate the value of \(\theta\).
  4. If the bowl is not prevented from sliding, determine whether it will slide when placed on the slope when there is a coefficient of friction between the bowl and the slope of 1.5.
OCR MEI M2 2010 June Q1
17 marks Moderate -0.3
1 Two sledges P and Q, with their loads, have masses of 200 kg and 250 kg respectively and are sliding on a horizontal surface against negligible resistance. There is an inextensible light rope connecting the sledges that is horizontal and parallel to the direction of motion. Fig. 1 shows the initial situation with both sledges travelling with a velocity of \(5 \mathbf { i m ~ } \mathbf { m } ^ { - 1 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a605f0b-f595-4bb9-9624-f816c789ad86-2_397_1379_520_383} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A mechanism on Q jerks the rope so that there is an impulse of \(250 \mathbf { i N s }\) on P .
  1. Show that the new velocity of \(P\) is \(6.25 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\) and find the new velocity of \(Q\). There is now a direct collision between the sledges and after the impact P has velocity \(4.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\).
  2. Show that the velocity of Q becomes \(5.4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the coefficient of restitution in the collision. Before the rope becomes taut again, the velocity of P is increased so that it catches up with Q . This is done by throwing part of the load from sledge P in the \(- \mathbf { i }\) direction so that P 's velocity increases to \(5.5 \mathrm { i } \mathrm { m } \mathrm { s } ^ { - 1 }\). The part of the load thrown out is an object of mass 20 kg .
  3. Calculate the speed of separation of the object from P . When the sledges directly collide again they are held together and move as a single object.
  4. Calculate the common velocity of the pair of sledges, giving your answer correct to 3 significant figures. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1a605f0b-f595-4bb9-9624-f816c789ad86-3_987_524_258_769} \captionsetup{labelformat=empty} \caption{not to scale he lengths are}
    \end{figure} Fig. 2 Fig. 2 shows a stand on a horizontal floor and horizontal and vertical coordinate axes \(\mathrm { O } x\) and \(\mathrm { O } y\). The stand is modelled as
    • a thin uniform rectangular base PQRS, 30 cm by 40 cm with mass 15 kg ; the sides QR and PS are parallel to \(\mathrm { O } x\),
    • a thin uniform vertical rod of length 200 cm and mass 3 kg that is fixed to the base at O , the mid-point of PQ and the origin of coordinates,
    • a thin uniform top rod AB of length 50 cm and mass \(2 \mathrm {~kg} ; \mathrm { AB }\) is parallel to \(\mathrm { O } x\).
    Coordinates are referred to the axes shown in the figure.
OCR MEI M2 2010 June Q3
19 marks Challenging +1.2
3 Fig. 3 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DA }\) and BD . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D and to a vertical wall at A . ABCD is a parallelogram with AD horizontal and BD vertical; the dimensions of the framework, in metres, are shown. There is a vertical load of 300 N acting at C and a vertical wire attached to D , with tension \(T \mathrm {~N}\), holds the framework in equilibrium. The horizontal and vertical forces, \(X \mathrm {~N}\) and \(Y \mathrm {~N}\), acting on the framework at A due to the wall are also shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a605f0b-f595-4bb9-9624-f816c789ad86-4_737_860_568_641} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that \(T = 600\) and calculate the values of \(X\) and \(Y\).
  2. Draw a diagram showing all the forces acting on the framework, and also the internal forces in the rods.
  3. Calculate the internal forces in the five rods, indicating 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 (ii).) Suppose that the vertical wire is attached at B instead of D and that the framework is still in equilibrium.
  4. Without doing any further calculations, state which four of the rods have the same internal forces as in part (iii) and say briefly why this is the case. Determine the new force in the fifth rod.
OCR MEI M2 2010 June Q4
18 marks Standard +0.3
4 A box of mass 16 kg is on a uniformly rough horizontal floor with an applied force of fixed direction but varying magnitude \(P\) N acting as shown in Fig. 4. You may assume that the box does not tip for any value of \(P\). The coefficient of friction between the box and the floor is \(\mu\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1a605f0b-f595-4bb9-9624-f816c789ad86-5_348_863_429_643} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure} Initially the box is at rest and on the point of slipping with \(P = 58\).
  1. Show that \(\mu\) is about 0.25 . In the rest of this question take \(\mu\) to be exactly 0.25 .
    The applied force on the box is suddenly increased so that \(P = 70\) and the box moves against friction with the floor and another horizontal retarding force, \(S\). The box reaches a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest after 5 seconds; during this time the box slides 3 m .
  2. Calculate the work done by the applied force of 70 N and also the average power developed by this force over the 5 seconds.
  3. By considering the values of time, distance and speed, show that an object starting from rest that travels 3 m while reaching a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 5 seconds cannot be moving with constant acceleration. The reason that the acceleration is not constant is that the retarding force \(S\) is not constant.
  4. Calculate the total work done by the retarding force \(S\).
OCR MEI M2 2011 June Q1
19 marks Moderate -0.8
1
  1. Sphere P , of mass 10 kg , and sphere Q , of mass 15 kg , move with their centres on a horizontal straight line and have no resistances to their motion. \(\mathrm { P } , \mathrm { Q }\) and the positive direction are shown in Fig. 1.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_332_803_434_712} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} Initially, P has a velocity of \(- 1.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is acted on by a force of magnitude 13 N acting in the direction PQ . After \(T\) seconds, P has a velocity of \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and has not reached Q .
    1. Calculate \(T\). The force of magnitude 13 N is removed. P is still travelling at \(4.75 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides directly with Q , which has a velocity of \(- 0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Suppose that P and Q coalesce in the collision to form a single object.
    2. Calculate their common velocity after the collision. Suppose instead that P and Q separate after the collision and that P has a velocity of \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) afterwards.
    3. Calculate the velocity of Q after the collision and also the coefficient of restitution in the collision.
  2. Fig. 1.2 shows a small ball projected at a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal over smooth horizontal ground. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-2_424_832_1918_699} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure} The ball is initially 3.125 m above the ground. The coefficient of restitution between the ball and the ground is 0.6 . Calculate the angle with the horizontal of the ball's trajectory immediately after the second bounce on the ground.
OCR MEI M2 2011 June Q2
17 marks Standard +0.3
2 Any non-exact answers to this question should be given correct to four significant figures.
A thin, straight beam AB is 2 m long. It has a weight of 600 N and its centre of mass G is 0.8 m from end A. The beam is freely pivoted about a horizontal axis through A. The beam is held horizontally in equilibrium.
Initially this is done by means of a support at B, as shown in Fig.2.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_222_805_644_669} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure}
  1. Calculate the force on the beam due to the support at B . The support at B is now removed and replaced by a wire attached to the beam 0.3 m from A and inclined at \(50 ^ { \circ }\) to the beam, as shown in Fig. 2.2. The beam is still horizontal and in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_275_803_1226_671} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure}
  2. Calculate the tension in the wire. The forces acting on the beam at A due to the hinge are a horizontal force \(X \mathrm {~N}\) in the direction AB , and a downward vertical force \(Y \mathrm {~N}\), which have a resultant of magnitude \(R\) at \(\alpha\) to the horizontal.
  3. Calculate \(X , Y , R\) and \(\alpha\). The dotted lines in Fig. 2.3 are the lines of action of the tension in the wire and the weight of the beam. These lines of action intersect at P . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_460_791_2074_678} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
    \end{figure}
  4. State with a reason the size of the angle GAP.
OCR MEI M2 2011 June Q3
18 marks Standard +0.3
3 A bracket is being made from a sheet of uniform thin metal. Firstly, a plate is cut from a square of the sheet metal in the shape OABCDEFHJK, shown shaded in Fig. 3.1. The dimensions shown in the figure are in centimetres; axes \(\mathrm { O } x\) and \(\mathrm { O } y\) are also shown. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-4_561_569_429_788} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Show that, referred to the axes given in Fig. 3.1, the centre of mass of the plate OABCDEFHJK has coordinates (0.8, 2.5). The plate is hung using light vertical strings attached to \(\mathbf { J }\) and \(\mathbf { H }\). The edge \(\mathbf { J H }\) is horizontal and the plate is in equilibrium. The weight of the plate is 3.2 N .
  2. Calculate the tensions in each of the strings. The plate is now bent to form the bracket. This is shown in Fig. 3.2: the rectangle IJKO is folded along the line IA so that it is perpendicular to the plane ABCGHI ; the rectangle DEFG is folded along the line DG so it is also perpendicular to the plane ABCGHI but on the other side of it. Fig. 3.2 also shows the axes \(\mathrm { O } x , \mathrm { O } y\) and \(\mathrm { O } z\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-4_611_782_1713_678} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  3. Show that, referred to the axes given in Fig. 3.2, the centre of mass of the bracket has coordinates ( \(1,2.7,0\) ). The bracket is now hung freely in equilibrium from a string attached to O .
  4. Calculate the angle between the edge OI and the vertical.
OCR MEI M2 2011 June Q4
18 marks Standard +0.3
4
  1. A parachutist and her equipment have a combined mass of 80 kg . During a descent where the parachutist loses 1600 m in height, her speed reduces from \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and she does \(1.3 \times 10 ^ { 6 } \mathrm {~J}\) of work against resistances. Use an energy method to calculate the value of \(V\).
  2. A vehicle of mass 800 kg is climbing a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = 0.1\). At one time the vehicle has a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating up the hill at \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of 1150 N .
    1. Show that the driving force on the vehicle is 2134 N and calculate its power at this time. The vehicle is pulling a sledge, of mass 300 kg , which is sliding up the hill. The sledge is attached to the vehicle by a light, rigid coupling parallel to the slope. The force in the coupling is 900 N .
    2. Assuming that the only resistance to the motion of the sledge is due to friction, calculate the coefficient of friction between the sledge and the ground.
OCR MEI M2 2012 June Q1
18 marks Moderate -0.3
1
  1. A stone of mass 0.6 kg falls vertically 1.5 m from A to B against resistance. Its downward speeds at A and \(B\) are \(5.5 \mathrm {~ms} ^ { - 1 }\) and \(7.5 \mathrm {~ms} ^ { - 1 }\) respectively.
    1. Calculate the change in kinetic energy and the change in gravitational potential energy of the stone as it falls from A to B .
    2. Calculate the work done against resistance to the motion of the stone as it falls from A to B .
    3. Assuming the resistive force is constant, calculate the power with which the resistive force is retarding the stone when it is at A .
  2. A uniform plank is inclined at \(40 ^ { \circ }\) to the horizontal. A box of mass 0.8 kg is on the point of sliding down it. The coefficient of friction between the box and the plank is \(\mu\).
    1. Show that \(\mu = \tan 40 ^ { \circ }\). The plank is now inclined at \(20 ^ { \circ }\) to the horizontal.
    2. Calculate the work done when the box is pushed 3 m up the plank, starting and finishing at rest.
OCR MEI M2 2012 June Q2
16 marks Challenging +1.2
2 The rigid object shown in Fig. 2.1 is made of thin non-uniform rods. ABC is a straight line; \(\mathrm { BC } , \mathrm { BE }\) and ED form three sides of a rectangle. The centre of mass of the object is at G. The lengths are in centimetres. The weight of the object is 15 N . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-3_273_444_397_813} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\end{figure} Initially, the object is suspended by light vertical strings attached to B and to C and hangs in equilibrium with AC horizontal.
  1. Calculate the tensions in each of the strings. In a new situation the strings are removed. The object can rotate freely in a vertical plane about a fixed horizontal axis through A and perpendicular to ABCDE. The object is held in equilibrium with AC horizontal by a force of magnitude \(T \mathrm {~N}\) in the plane ABCDE acting at C at an angle of \(30 ^ { \circ }\) to CA . This situation is shown in Fig. 2.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-3_356_451_1292_808} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
    \end{figure}
  2. Calculate \(T\). Calculate also the magnitude of the force exerted on the object by the axis at A . The object is now placed on a rough horizontal table and is in equilibrium with ABCDE in a vertical plane and DE in contact with the table. The coefficient of friction between the edge DE and the table is 0.65 . A force of slowly increasing magnitude (starting at 0 N ) is applied at A in the direction AB . Assume that the object remains in a vertical plane.
  3. Determine whether the object slips before it tips.
OCR MEI M2 2012 June Q3
18 marks Standard +0.3
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}
      1. 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 \(e\).
      2. 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.
      3. For what set of values of \(e\) does the collision cause P to reverse its direction of motion?
      4. 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).
      5. Determine the common velocity of P and Q .
      6. Determine the impulse of Q on P in this collision.
OCR MEI M2 2013 June Q1
20 marks Moderate -0.3
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.
OCR MEI M2 2013 June Q2
16 marks Standard +0.3
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]
\includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-3_552_741_479_628} \captionsetup{labelformat=empty} \caption{Fig. 2}
\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.
OCR MEI M2 2013 June Q3
17 marks Standard +0.3
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]
\includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-4_451_737_493_646} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\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.
OCR MEI M2 2013 June Q4
19 marks Standard +0.3
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).]