6.02i Conservation of energy: mechanical energy principle

9 Two particles, \(A\) and \(B\), are connected by a light elastic string that passes through a hole at a point \(O\) in a rough horizontal table. The edges of the hole are smooth. Particle \(A\) has a mass of 8 kg and particle \(B\) has a mass of 3 kg . The elastic string has natural length 3 metres and modulus of elasticity 60 newtons.
Initially, particle \(A\) is held 3.5 metres from the point \(O\) on the surface of the table and particle \(B\) is held at a point 2 metres vertically below \(O\). The coefficient of friction between the table and particle \(A\) is 0.4 .
The two particles are released from rest.

1. 1. Show that initially particle \(A\) moves towards the hole in the table.
   2. Show that initially particle \(B\) also moves towards the hole in the table.
2. Calculate the initial elastic potential energy in the string.
3. Particle \(A\) comes permanently to rest when it has moved 0.46 metres, at which time particle \(B\) is still moving upwards. Calculate the distance that particle \(B\) has moved when it is at rest for the first time.

3 A diagram shows a children's slide, \(P Q R\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.

1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
   1. Find the kinetic energy of Simon when he reaches the point \(R\).
   2. Hence find the speed of Simon when he reaches the point \(R\).
2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
   Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}

8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
3. Find the maximum value of \(x\).
4. 1. Find the distance fallen by Carol when her speed is a maximum.
   2. Hence find Carol's maximum speed.

4. A small block of wood, of mass 0.5 kg , slides down a line of \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-1_219_501_2042_338}
greatest slope of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\). The block is given an initial impulse of magnitude 2 Ns , and reaches the bottom of the plane with kinetic energy 19 J.

1. Find, in J , the change in the potential energy of the block as it moves down the plane.
2. Hence find the distance travelled by the block down the plane.
3. State two modelling assumptions that you have made. \section*{MECHANICS 2 (A) TEST PAPER 6 Page 2}

1 A boy on a sledge slides down a straight track of length 180 m which descends a vertical distance of 40 m . The combined mass of the boy and the sledge is 75 kg . The initial speed is \(3 \mathrm {~ms} ^ { - 1 }\) and the final speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude, \(R \mathrm {~N}\), of the resistance to motion is constant. By considering the change in energy, calculate \(R\).

1 \includegraphics[max width=\textwidth, alt={}, center]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-2_314_931_242_607} A sledge with its load has mass 70 kg . It moves down a slope and the resistance to the motion of the sledge is 90 N . The speed of the sledge is controlled by the constant tension in a light rope, which is attached to the sledge and parallel to the slope (see diagram). While travelling 20 m down the slope, the speed of the sledge decreases from \(2.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it descends a vertical distance of 3 m .

1. Calculate the change in energy of the sledge and its load.
2. Calculate the tension in the rope.

5

1. A car of mass 800 kg is moving at a constant speed of \(20 \mathrm {~ms} ^ { - 1 }\) on a straight road down a hill inclined at an angle \(\alpha\) to the horizontal. The engine of the car works at a constant rate of 10 kW and there is a resistance to motion of 1300 N . Show that \(\sin \alpha = \frac { 5 } { 49 }\).
2. The car now travels up the same hill and its engine now works at a constant rate of 20 kW . The resistance to motion remains 1300 N . The car starts from rest and its speed is \(8 \mathrm {~ms} ^ { - 1 }\) after it has travelled a distance of 22.1 m . Calculate the time taken by the car to travel this distance.

1 A cyclist travels along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg and the resistance to motion is a constant 60 N .

1. The cyclist travels at a constant speed working at a constant rate of 480 W . Find the speed at which she travels.
2. The cyclist now instantaneously increases her power to 600 W . After travelling at this power for 14.2 s her speed reaches \(9.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance travelled at this power.

5 A small sphere of mass 0.2 kg is projected vertically downwards with a speed of \(5 \mathrm {~ms} ^ { - 1 }\) from a height of 1.6 m above horizontal ground. It hits the ground and rebounds vertically upwards coming to instantaneous rest at a height of \(h \mathrm {~m}\) above the ground. The coefficient of restitution between the sphere and the ground is 0.7 .

1. Find \(h\).
2. Find the magnitude and direction of the impulse exerted on the sphere by the ground.
3. Find the loss of energy of the sphere between the instant of projection and the instant it comes to instantaneous rest at height \(h \mathrm {~m}\).

4 A block of mass 20 kg is pulled by a light, horizontal string over a rough, horizontal plane. During 6 seconds, the work done against resistances is 510 J and the speed of the block increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the power of the pulling force. The block is now put on a rough plane that is at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The frictional resistance to sliding is \(11 g \mathrm {~N}\). A light string parallel to the plane is connected to the block. The string passes over a smooth pulley and is connected to a freely hanging sphere of mass \(m \mathrm {~kg}\), as shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-6_348_855_847_605} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} In parts (ii) and (iii), the sphere is pulled downwards and then released when travelling at a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards. The block never reaches the pulley.
2. Suppose that \(m = 5\) and that after the sphere is released the block moves \(x \mathrm {~m}\) up the plane before coming to rest.
   (A) Find an expression in terms of \(x\) for the change in gravitational potential energy of the system, stating whether this is a gain or a loss.
   (B) Find an expression in terms of \(x\) for the work done against friction.
   (C) Making use of your answers to parts (A) and (B), find the value of \(x\).
3. Suppose instead that \(m = 15\). Calculate the speed of the sphere when it has fallen a distance 0.5 m from its point of release.

2 A car of mass 1200 kg travels along a road for two minutes during which time it rises a vertical distance of 60 m and does \(1.8 \times 10 ^ { 6 } \mathrm {~J}\) of work against the resistance to its motion. The speeds of the car at the start and at the end of the two minutes are the same.

1. Calculate the average power developed over the two minutes. The car now travels along a straight level road at a steady speed of \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while developing constant power of 13.5 kW .
2. Calculate the resistance to the motion of the car. How much work is done against the resistance when the car travels 200 m ? While travelling at \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car starts to go down a slope inclined at \(5 ^ { \circ }\) to the horizontal with the power removed and its brakes applied. The total resistance to its motion is now 1500 N .
3. Use an energy method to determine how far down the slope the car travels before its speed is halved. Suppose the car is travelling along a straight level road and developing power \(P \mathrm {~W}\) while travelling at \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) against a resistance of \(R \mathrm {~N}\).
4. Show that \(P = ( R + 1200 a ) v\) and deduce that if \(P\) and \(R\) are constant then if \(a\) is not zero it cannot be constant.

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.

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?

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 .

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.

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

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

4

1. A small heavy object of mass 10 kg travels the path ABCD which is shown in Fig. 4. ABCD is in a vertical plane; CD and AEF are horizontal. The sections of the path AB and CD are smooth but section BC is rough. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-5_368_1323_402_338} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} You should assume that
   • the object does not leave the path when travelling along ABCD and does not lose energy when changing direction
   • there is no air resistance.
   Initially, the object is projected from A at a speed of \(16.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up the slope.
   1. Show that the object gets beyond B . The section of the path BC produces a constant resistance of 14 N to the motion of the object.
   2. Using an energy method, find the velocity of the object at D . At D , the object leaves the path and bounces on the smooth horizontal ground between E and F , shown in Fig. 4. The coefficient of restitution in the collision of the object with the ground is \(\frac { 1 } { 2 }\).
   3. Calculate the greatest height above the ground reached by the object after its first bounce.
2. A car of mass 1500 kg travelling along a straight, horizontal road has a steady speed of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its driving force has power \(P \mathrm {~W}\). When at this speed, the power is suddenly reduced by \(20 \%\). The resistance to the car's motion, \(F \mathrm {~N}\), does not change and the car begins to decelerate at \(0.08 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the values of \(P\) and \(F\).

2 Fig. 2 shows a wedge of angle \(30 ^ { \circ }\) fixed to a horizontal floor. Small objects P , of mass 8 kg , and Q , of mass 10 kg , are connected by a light inextensible string that passes over a smooth pulley at the top of the wedge. The part of the string between P and the pulley is parallel to a line of greatest slope of the wedge. Q hangs freely and at no time does either P or Q reach the pulley or P reach the floor. \begin{figure}[h] \end{figure}

1. Assuming the string remains taut, find the change in the gravitational potential energy of the system when Q descends \(h \mathrm {~m}\), stating whether it is a loss or a gain. Object P makes smooth contact with the wedge. The system is set in motion with the string taut.
2. Find the speed at which Q hits the floor if
   (A) the system is released from rest with Q a distance of 1.2 m above the floor,
   (B) instead, the system is set in motion with Q a distance of 0.3 m above the floor and P travelling down the slope at \(1.05 \mathrm {~ms} ^ { - 1 }\). The sloping face is roughened so that the coefficient of friction between object P and the wedge is 0.9 . The system is set in motion with the string taut and P travelling down the slope at \(2 \mathrm {~ms} ^ { - 1 }\).
3. How far does P move before it reaches its lowest point?
4. Determine what happens to the system after P reaches its lowest point.
5. Calculate the power of the frictional force acting on P in part (iii) at the moment the system is set in motion. \section*{Question 3 begins on page 4.}

2

1. A bullet of mass 0.04 kg is fired into a fixed uniform rectangular block along a line through the centres of opposite parallel faces, as shown in Fig. 2.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_209_1287_342_388} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
   \end{figure} The bullet enters the block at \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to rest after travelling 0.2 m into the block.
   1. Calculate the resistive force on the bullet, assuming that this force is constant. Another bullet of the same mass is fired, as before, with the same speed into a similar block of mass 3.96 kg . The block is initially at rest and is free to slide on a smooth horizontal plane.
   2. By considering linear momentum, find the speed of the block with the bullet embedded in it and at rest relative to the block.
   3. By considering mechanical energy, find the distance the bullet penetrates the block, given the resistance of the block to the motion of the bullet is the same as in part (i).
2. Fig. 2.2 shows a block of mass 6 kg on a uniformly rough plane that is inclined at \(30 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-3_348_636_1382_712} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A string with a constant tension of 91.5 N parallel to the plane pulls the block up a line of greatest slope. The speed of the block increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a distance of 8 m .

7. \begin{figure}[h] \end{figure} Figure 4 shows a particle \(P\) projected from the point \(A\) up the line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 4 } { 5 } . P\) is projected with speed \(5.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction between \(P\) and the plane is \(\frac { 4 } { 7 }\). Given that \(P\) first comes to rest at point \(B\),

1. use the Work-Energy principle to show that the distance \(A B\) is 1.4 m . The particle then slides back down the plane.
2. Find, correct to 2 significant figures, the speed of \(P\) when it returns to \(A\).

6 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of \(60 ^ { \circ }\) to the upward vertical, \(P\) is projected with speed \(2 \mathrm {~ms} ^ { - 1 }\) (see diagram). \(P\) begins to move without air resistance in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the upward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 8.9 - 9.8 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. \(P\) does not move in a complete circle. Calculate the angle through which \(O P\) turns before \(P\) leaves the circular path.

3 A light elastic string has natural length 3 m . One end is attached to a fixed point \(O\) and the other end is attached to a particle of mass 1.6 kg . The particle is released from rest in a position 5 m vertically below \(O\). Air resistance may be neglected.

1. Given that in the subsequent motion the particle just reaches \(O\), show that the modulus of elasticity of the string is 117.6 N .
2. Calculate the speed of the particle when it is 4.5 m below \(O\).

6 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_598_839_1480_706} One end of a light inextensible string of length 0.5 m is attached to a fixed point \(O\). A particle \(P\) of mass 0.3 kg is attached to the other end of the string. With the string taut and at an angle of \(60 ^ { \circ }\) to the upward vertical, \(P\) is projected with speed \(2 \mathrm {~ms} ^ { - 1 }\) (see diagram). \(P\) begins to move without air resistance in a vertical circle with centre \(O\). When the string makes an angle \(\theta\) with the upward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } = 8.9 - 9.8 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. \(P\) does not move in a complete circle. Calculate the angle through which \(O P\) turns before \(P\) leaves the circular path.

7 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-4_122_1009_265_571} As shown in the diagram, \(A\) and \(B\) are fixed points on a smooth horizontal table, where \(A B = 3 \mathrm {~m}\). A particle \(Q\) of mass 1.2 kg is attached to \(A\) by a light elastic string of natural length 1 m and modulus of elasticity \(180 \mathrm {~N} . Q\) is attached to \(B\) by a light elastic string of natural length 1.2 m and modulus of elasticity 360 N .

1. Verify that when \(Q\) is in equilibrium \(B Q = 1.5 \mathrm {~m}\). \(Q\) is projected towards \(B\) from the equilibrium position with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Subsequently \(Q\) oscillates with simple harmonic motion.
2. Show that the period of the motion is 0.314 s approximately.
3. Show that \(u \leqslant 6\).
4. Given that \(u = 6\), find the time taken for \(Q\) to move from the equilibrium position to a position 1.3 m from \(A\) for the first time.

1 A particle \(P\) of mass 0.6 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . While hanging at a distance 0.4 m vertically below \(O , P\) is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a complete vertical circle. Calculate the tension in the string when \(P\) is vertically above \(O\).