Questions — OCR MEI M2 (72 questions)

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OCR MEI M2 2012 January Q4
4
  1. A large nail of mass 0.02 kg has been driven a short distance horizontally into a fixed block of wood, as shown in Fig. 4.1, and is to be driven horizontally further into the block. The wood produces a constant resistance of 2.43 N to the motion of the nail. The situation is modelled by assuming that linear momentum is conserved when the nail is struck, that all the impacts with the nail are direct and that the head of the nail never reaches the wood. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_279_711_482_676} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure} The nail is first struck by an object of mass 0.1 kg that is moving parallel to the nail with linear momentum of magnitude 0.108 Ns . The object becomes firmly attached to the nail.
    1. Calculate the speed of the nail and object immediately after the impact.
    2. Calculate the time for which the nail and object move, and the distance they travel in that time. On a second attempt to drive in the nail, it is struck by the same object of mass 0.1 kg moving parallel to the nail with the same linear momentum of magnitude 0.108 Ns . This time the object does not become attached to the nail and after the contact is still moving parallel to the nail. The coefficient of restitution in the impact is \(\frac { 1 } { 3 }\).
    3. Calculate the speed of the nail immediately after this impact.
  2. A small ball slides on a smooth horizontal plane and bounces off a smooth straight vertical wall. The speed of the ball is \(u\) before the impact and, as shown in Fig. 4.2, the impact turns the path of the ball through \(90 ^ { \circ }\). The coefficient of restitution in the collision between the ball and the wall is \(e\). Before the collision, the path is inclined at \(\alpha\) to the wall. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-5_294_590_1804_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure}
    1. Write down, in terms of \(u , e\) and \(\alpha\), the components of the velocity of the ball parallel and perpendicular to the wall before and after the impact.
    2. Show that \(\tan \alpha = \frac { 1 } { \sqrt { e } }\).
    3. Hence show that \(\alpha \geqslant 45 ^ { \circ }\).
OCR MEI M2 2013 January Q1
1
  1. Fig. 1.1 shows the velocities of a tanker of mass 120000 tonnes before and after it changed speed and direction. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-2_237_917_360_577} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} Calculate the magnitude of the impulse that acted on the tanker.
  2. An object of negligible size is at rest on a horizontal surface. It explodes into two parts, P and Q , which then slide along the surface. Part P has mass 0.4 kg and speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Part Q has mass 0.5 kg .
    1. Calculate the speed of Q immediately after the explosion. State how the directions of motion of P and Q are related. The explosion takes place at a distance of 0.75 m from a raised vertical edge, as shown in Fig. 1.2. P travels along a line perpendicular to this edge. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-2_238_1205_1366_429} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
      \end{figure} After the explosion, P has a perfectly elastic direct collision with the raised edge and then collides again directly with Q . The collision between P and Q occurs \(\frac { 2 } { 3 } \mathrm {~s}\) after the explosion. Both collisions are instantaneous. The contact between P and the surface is smooth but there is a constant frictional force between Q and the surface.
    2. Show that Q has speed \(2.7 \mathrm {~ms} ^ { - 1 }\) just before P collides with it.
    3. Calculate the coefficient of friction between Q and the surface.
    4. Given that the coefficient of restitution between P and Q is \(\frac { 1 } { 8 }\), calculate the speed of Q immediately after its collision with P .
OCR MEI M2 2013 January Q2
2 This question is about 'kart gravity racing' in which, after an initial push, unpowered home-made karts race down a sloping track. The moving karts have only the following resistive forces and these both act in the direction opposite to the motion.
  • A force \(R\), called rolling friction, with magnitude \(0.01 M g \cos \theta \mathrm {~N}\) where \(M \mathrm {~kg}\) is the mass of the kart and driver and \(\theta\) is the angle of the track with the horizontal
  • A force \(F\) of varying magnitude, due to air resistance
A kart with its driver has a mass of 80 kg .
One stretch of track slopes uniformly downwards at \(4 ^ { \circ }\) to the horizontal. The kart travels 12 m down this stretch of track. The total work done by the kart against both rolling friction and air resistance is 455 J .
  1. Calculate the work done against air resistance.
  2. During this motion, the kart's speed increases from \(2 \mathrm {~ms} ^ { - 1 }\) to \(v \mathrm {~ms} ^ { - 1 }\). Use an energy method to calculate \(v\). To reach the starting line, the kart (with the driver seated) is pushed up a slope against rolling friction and air resistance. At one point the slope is at \(5 ^ { \circ }\) to the horizontal, the air resistance is 15 N , the acceleration of the kart is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up the slope and the power of the pushing force is 405 W .
  3. Calculate the speed of the kart at this point.
OCR MEI M2 2013 January Q3
5 marks
3 The object shown shaded in Fig. 3.1 is cut from a flat sheet of thin rigid uniform material; LMJK, OAIJ, AEFH and CDEB are rectangles. The grid-lines in Fig. 3.1 are 1 cm apart. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-4_825_1077_210_822} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Calculate the coordinates of the centre of mass of the object referred to the axes shown in Fig. 3.1. [5] The object is freely suspended from the point K and hangs in equilibrium.
  2. Calculate the angle that KI makes with the vertical. The mass of the object is 0.3 kg .
    A particle of mass \(m \mathrm {~kg}\) is attached to the object at a point on the line OJ so that the new centre of mass is at the centre of the square OAIJ.
  3. Calculate the value of \(m\) and the position of the particle referred to the axes shown in Fig. 3.1. The extra particle is now removed and the object shown in Fig. 3.1 is folded: LMJK is folded along JM so that it is perpendicular to OAIJ; ABCDEFH is folded along AH so that it is perpendicular to OAIJ and on the same side of OAIJ as LMJK. The folded object is placed on a horizontal table with the edges KL and FED in contact with the table. A plan view and a 3D representation are shown in Fig. 3.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-4_609_648_1836_246} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-4_332_695_2001_1144} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  4. On the plan, indicate the region corresponding to positions of the centre of mass for which the folded object is stable. You are given that the \(x\)-coordinate of the centre of mass of the folded object is 1.7 . Determine whether the object is stable.
OCR MEI M2 2013 January Q4
4 A rigid thin uniform rod AB with length 2.4 m and weight 30 N is used in different situations.
  1. In the first situation, the rod rests on a small support 0.6 m from B and is held horizontally in equilibrium by a vertical string attached to A, as shown in Fig. 4.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_196_707_456_680} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure} Calculate the tension in the string and the force of the support on the rod.
  2. In the second situation, the rod rests in equilibrium on the point of slipping with end A on a horizontal floor and the rod resting at P on a fixed block of height 0.9 m , as shown in Fig. 4.2. The rod is perpendicular to the edge of the block on which it rests and is inclined at \(\theta\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_208_746_1101_657} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
    \end{figure} (A) Suppose that the contact between the block and the rod is rough with coefficient of friction 0.6 and contact between the end A and the floor is smooth. Show that \(\tan \theta = 0.6\).
    (B) Suppose instead that the contact between the block and the rod is smooth and the contact between the end A and the floor is rough. The rod is now in limiting equilibrium at a different angle \(\theta\) such that the distance AP is 1.5 m . Calculate the normal reaction of the block on the rod. Calculate the coefficient of friction between the rod and the floor.
OCR MEI M2 2005 June Q1
1
  1. Roger of mass 70 kg and Sheuli of mass 50 kg are skating on a horizontal plane containing the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\). The resistances to the motion of the skaters are negligible. The two skaters are locked in a close embrace and accelerate from rest until they reach a velocity of \(2 \mathrm { ims } ^ { - 1 }\), as shown in Fig. 1.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-2_191_181_543_740} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-2_177_359_589_1051} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure}
    1. What impulse has acted on them? During a dance routine, the skaters separate on three occasions from their close embrace when travelling at a constant velocity of \(2 \mathrm { i } \mathrm { ms } ^ { - 1 }\).
    2. Calculate the velocity of Sheuli after the separation in the following cases.
      (A) Roger has velocity \(\mathrm { ims } ^ { - 1 }\) after the separation.
      (B) Roger and Sheuli have equal speeds in opposite senses after the separation, with Roger moving in the \(\mathbf { i }\) direction.
      (C) Roger has velocity \(4 ( \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) after the separation.
  2. Two discs with masses 2 kg and 3 kg collide directly in a horizontal plane. Their velocities just before the collision are shown in Fig. 1.2. The coefficient of restitution in the collision is 0.5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-2_278_970_1759_594} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure}
    1. Calculate the velocity of each disc after the collision. The disc of mass 3 kg moves freely after the collision and makes a perfectly elastic collision with a smooth wall inclined at \(60 ^ { \circ }\) to its direction of motion, as shown in Fig. 1.2.
    2. State with reasons the speed of the disc and the angle between its direction of motion and the wall after the collision.
OCR MEI M2 2005 June Q2
2 A car of mass 850 kg is travelling along a road that is straight but not level.
On one section of the road the car travels at constant speed and gains a vertical height of 60 m in 20 seconds. Non-gravitational resistances to its motion (e.g. air resistance) are negligible.
  1. Show that the average power produced by the car is about 25 kW . On a horizontal section of the road, the car develops a constant power of exactly 25 kW and there is a constant resistance of 800 N to its motion.
  2. Calculate the maximum possible steady speed of the car.
  3. Find the driving force and acceleration of the car when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). When travelling along the horizontal section of road, the car accelerates from \(15 \mathrm {~ms} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6.90 seconds with the same constant power and constant resistance.
  4. By considering work and energy, find how far the car travels while it is accelerating. When the car is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a constant slope inclined at \(\arcsin ( 0.05 )\) to the horizontal, the driving force is removed. Subsequently, the resistance to the motion of the car remains constant at 800 N .
  5. What is the speed of the car when it has travelled a further 105 m up the slope?
OCR MEI M2 2005 June Q3
3 Fig. 3.1 shows an object made up as follows. ABCD is a uniform lamina of mass \(16 \mathrm {~kg} . \mathrm { BE } , \mathrm { EF }\), FG, HI, IJ and JD are each uniform rods of mass 2 kg . ABCD, BEFG and HIJD are squares lying in the same plane. The dimensions in metres are shown in the figure. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-4_627_648_429_735} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Find the coordinates of the centre of mass of the object, referred to the axes shown in Fig.3.1. The rods are now re-positioned so that BEFG and HIJD are perpendicular to the lamina, as shown in Fig. 3.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-4_442_666_1510_722} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{figure}
  2. Find the \(x\)-, \(y\)-and \(z\)-coordinates of the centre of mass of the object, referred to the axes shown in Fig. 3.2. Calculate the distance of the centre of mass from A . The object is now freely suspended from A and hangs in equilibrium with AC at \(\alpha ^ { \circ }\) to the vertical.
  3. Calculate \(\alpha\).
OCR MEI M2 2006 June Q1
1
  1. Two small spheres, \(P\) of mass 2 kg and \(Q\) of mass 6 kg , are moving in the same straight line along a smooth, horizontal plane with the velocities shown in Fig. 1.1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-2_252_647_404_708} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
    \end{figure} Consider the direct collision of P and Q in the following two cases.
    1. The spheres coalesce on collision.
      (A) Calculate the common velocity of the spheres after the collision.
      (B) Calculate the energy lost in the collision.
    2. The spheres rebound with a coefficient of restitution of \(\frac { 2 } { 3 }\) in the collision.
      (A) Calculate the velocities of P and Q after the collision.
      (B) Calculate the impulse on P in the collision.
  2. A small ball bounces off a smooth, horizontal plane. The ball hits the plane with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\arcsin \frac { 12 } { 13 }\) to it. The ball rebounds at an angle of \(\arcsin \frac { 3 } { 5 }\) to the plane, as shown in Fig. 1.2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-2_238_545_1695_767} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
    \end{figure} Calculate the speed with which the ball rebounds from the plane.
    Calculate also the coefficient of restitution in the impact.
OCR MEI M2 2006 June Q2
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
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
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
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
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
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
3
  1. 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.]
  2. 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 \(\frac { \sqrt { 3 } } { 6 } W \mathrm {~N}\).
  3. \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
      • there is a force opposing the motion that results in an energy loss of 0.6 J for every metre travelled by the sphere,
  4. the sphere is given an initial speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (and it is descending) with AB at \(40 ^ { \circ }\) to the vertical.
    5. 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
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
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
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
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
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
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
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
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
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.