6.02i Conservation of energy: mechanical energy principle

6 A lorry of mass 15000 kg climbs a hill of length 500 m at a constant speed. The hill is inclined at \(2.5 ^ { \circ }\) to the horizontal. The resistance to the lorry's motion is constant and equal to 800 N .

1. Find the work done by the lorry's driving force. On its return journey the lorry reaches the top of the hill with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and continues down the hill with a constant driving force of 2000 N . The resistance to the lorry's motion is again constant and equal to 800 N .
2. Find the speed of the lorry when it reaches the bottom of the hill.

5 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-3_485_874_255_638} The diagram shows the vertical cross-section \(O A B\) of a slide. The straight line \(A B\) is tangential to the curve \(O A\) at \(A\). The line \(A B\) is inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The point \(O\) is 10 m higher than \(B\), and \(A B\) has length 10 m (see diagram). The part of the slide containing the curve \(O A\) is smooth and the part containing \(A B\) is rough. A particle \(P\) of mass 2 kg is released from rest at \(O\) and moves down the slide.

1. Find the speed of \(P\) when it passes through \(A\). The coefficient of friction between \(P\) and the part of the slide containing \(A B\) is \(\frac { 1 } { 12 }\). Find
2. the acceleration of \(P\) when it is moving from \(A\) to \(B\),
3. the speed of \(P\) when it reaches \(B\).

6 A car of mass 1250 kg travels from the bottom to the top of a straight hill which has length 400 m and is inclined to the horizontal at an angle of \(\alpha\), where \(\sin \alpha = 0.125\). The resistance to the car's motion is 800 N . Find the work done by the car's engine in each of the following cases.

1. The car's speed is constant.
2. The car's initial speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car's driving force is 3 times greater at the top of the hill than it is at the bottom, and the car's power output is 5 times greater at the top of the hill than it is at the bottom.

5 A lorry of mass 16000 kg moves on a straight hill inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The length of the hill is 500 m .

1. While the lorry moves from the bottom to the top of the hill at constant speed, the resisting force acting on the lorry is 800 N and the work done by the driving force is 2800 kJ . Find the value of \(\alpha\).
2. On the return journey the speed of the lorry is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill. While the lorry travels down the hill, the work done by the driving force is 2400 kJ and the work done against the resistance to motion is 800 kJ . Find the speed of the lorry at the bottom of the hill.
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2 A car of mass 1250 kg travels from the bottom to the top of a straight hill of length 600 m , which is inclined at an angle of \(2.5 ^ { \circ }\) to the horizontal. The resistance to motion of the car is constant and equal to 400 N . The work done by the driving force is 450 kJ . The speed of the car at the bottom of the hill is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of the car at the top of the hill.

2 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-2_529_691_529_726} Particle \(A\) of mass 1.6 kg and particle \(B\) of mass 2 kg are attached to opposite ends of a light inextensible string. The string passes over a small smooth pulley fixed at the top of a smooth plane, which is inclined at angle \(\theta\), where \(\sin \theta = 0.8\). Particle \(A\) is held at rest at the bottom of the plane and \(B\) hangs at a height of 3.24 m above the level of the bottom of the plane (see diagram). \(A\) is released from rest and the particles start to move.

1. Show that the loss of potential energy of the system, when \(B\) reaches the level of the bottom of the plane, is 23.328 J .
2. Hence find the speed of the particles when \(B\) reaches the level of the bottom of the plane.

6 A particle \(P\) of mass 0.2 kg is released from rest at a point 7.2 m above the surface of the liquid in a container. \(P\) falls through the air and into the liquid. There is no air resistance and there is no instantaneous change of speed as \(P\) enters the liquid. When \(P\) is at a distance of 0.8 m below the surface of the liquid, \(P\) 's speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only force on \(P\) due to the liquid is a constant resistance to motion of magnitude \(R \mathrm {~N}\).

1. Find the deceleration of \(P\) while it is falling through the liquid, and hence find the value of \(R\). The depth of the liquid in the container is \(3.6 \mathrm {~m} . P\) is taken from the container and attached to one end of a light inextensible string. \(P\) is placed at the bottom of the container and then pulled vertically upwards with constant acceleration. The resistance to motion of \(R \mathrm {~N}\) continues to act. The particle reaches the surface 4 s after leaving the bottom of the container.
2. Find the tension in the string.

4 A small ball of mass 0.4 kg is released from rest at a point 5 m above horizontal ground. At the instant the ball hits the ground it loses 12.8 J of kinetic energy and starts to move upwards.

1. Show that the greatest height above the ground that the ball reaches after hitting the ground is 1.8 m .
2. Find the time taken for the ball's motion from its release until reaching this greatest height.

5 A lorry of mass 16000 kg travels at constant speed from the bottom, \(O\), to the top, \(A\), of a straight hill. The distance \(O A\) is 1200 m and \(A\) is 18 m above the level of \(O\). The driving force of the lorry is constant and equal to 4500 N .

1. Find the work done against the resistance to the motion of the lorry. On reaching \(A\) the lorry continues along a straight horizontal road against a constant resistance of 2000 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\) on the road. The distance \(A B\) is 2400 m .
2. Use an energy method to find \(F\), where \(F \mathrm {~N}\) is the average value of the driving force of the lorry while moving from \(A\) to \(B\).
3. Given that the driving force at \(A\) is 1280 N greater than \(F \mathrm {~N}\) and that the driving force at \(B\) is 1280 N less than \(F \mathrm {~N}\), show that the power developed by the lorry's engine is the same at \(B\) as it is at \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-4_342_1257_255_445} A smooth inclined plane of length 160 cm is fixed with one end at a height of 40 cm above the other end, which is on horizontal ground. Particles \(P\) and \(Q\), of masses 0.76 kg and 0.49 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(P\) is held at rest on the same line of greatest slope as the pulley and \(Q\) hangs vertically below the pulley at a height of 30 cm above the ground (see diagram). \(P\) is released from rest. It starts to move up the plane and does not reach the pulley. Find

1. the acceleration of the particles and the tension in the string before \(Q\) reaches the ground,
2. the speed with which \(Q\) reaches the ground,
3. the total distance travelled by \(P\) before it comes to instantaneous rest.

3 A block of weight 6.1 N slides down a slope inclined at \(\tan ^ { - 1 } \left( \frac { 11 } { 60 } \right)\) to the horizontal. The coefficient of friction between the block and the slope is \(\frac { 1 } { 4 }\). The block passes through a point \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find how far the block moves from \(A\) before it comes to rest.

4 A lorry of mass 14000 kg moves along a road starting from rest at a point \(O\). It reaches a point \(A\), and then continues to a point \(B\) which it reaches with a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The part \(O A\) of the road is straight and horizontal and has length 400 m . The part \(A B\) of the road is straight and is inclined downwards at an angle of \(\theta ^ { \circ }\) to the horizontal and has length 300 m .

1. For the motion from \(O\) to \(B\), find the gain in kinetic energy of the lorry and express its loss in potential energy in terms of \(\theta\). The resistance to the motion of the lorry is 4800 N and the work done by the driving force of the lorry from \(O\) to \(B\) is 5000 kJ .
2. Find the value of \(\theta\).

3 A plane is inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 8 } \right)\) to the horizontal. \(A\) and \(B\) are two points on the same line of greatest slope with \(A\) higher than \(B\). The distance \(A B\) is 12 m . A small object \(P\) of mass 8 kg is released from rest at \(A\) and slides down the plane, passing through \(B\) with speed \(4.5 \mathrm {~ms} ^ { - 1 }\). For the motion of \(P\) from \(A\) to \(B\), find

1. the increase in kinetic energy of \(P\) and the decrease in potential energy of \(P\),
2. the magnitude of the constant resisting force that opposes the motion of \(P\).

6 Two particles of masses 1.3 kg and 0.7 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The particles are held at the same vertical height with the string taut. The distance of each particle above a horizontal plane is 2 m , and the distance of each particle below the pulley is 4 m . The particles are released from rest.

1. Find
   1. the tension in the string before the particle of mass 1.3 kg reaches the plane,
   2. the time taken for the particle of mass 1.3 kg to reach the plane.
   3. Find the greatest height of the particle of mass 0.7 kg above the plane.

1 A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of \(20 ^ { \circ }\) below the horizontal. There is a constant resistance to motion of 15 N . The wheelbarrow moves a distance of 12 m from rest.

1. Find the work done by the man.
2. Find the speed attained by the wheelbarrow after 12 m .

1 A man has mass 80 kg . He runs along a horizontal road against a constant resistance force of magnitude \(P \mathrm {~N}\). The total work done by the man in increasing his speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while running a distance of 60 metres is 1200 J . Find the value of \(P\).

5 A particle \(P\) moves in a straight line from a fixed point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) at time \(t \mathrm {~s}\) is given by $$v = t ^ { 2 } - 8 t + 12 \quad \text { for } 0 \leqslant t \leqslant 8$$

1. Find the minimum velocity of \(P\).
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 8\). \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-12_401_1102_260_520} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is held on a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely 0.5 m above horizontal ground (see diagram). The particles are released from rest with both sections of the string taut.
3. Given that the system is in equilibrium, find \(\theta\).
4. It is given instead that \(\theta = 20\). In the subsequent motion particle \(A\) does not reach \(P\) and \(B\) remains at rest after reaching the ground.
   1. Find the tension in the string and the acceleration of the system.
   2. Find the speed of \(A\) at the instant \(B\) reaches the ground.
   3. Use an energy method to find the total distance \(A\) moves up the plane before coming to instantaneous rest.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 A particle of mass 18 kg is on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is projected up a line of greatest slope of the plane with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that the plane is smooth, use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
2. Given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.25 , find the speed of the particle as it returns to its starting point.

7 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-4_601_515_699_815} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held on the horizontal floor and particle \(A\) hangs in equilibrium. Particle \(B\) is released and each particle starts to move vertically with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(a\). Particle \(A\) hits the floor 1.2 s after it starts to move, and does not rebound upwards.
2. Show that \(A\) hits the floor with a speed of \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the gain in gravitational potential energy by \(B\), from leaving the floor until reaching its greatest height.

5 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-3_515_789_995_676} The diagram shows the vertical cross-section \(L M N\) of a fixed smooth surface. \(M\) is the lowest point of the cross-section. \(L\) is 2.45 m above the level of \(M\), and \(N\) is 1.2 m above the level of \(M\). A particle of mass 0.5 kg is released from rest at \(L\) and moves on the surface until it leaves it at \(N\). Find

1. the greatest speed of the particle,
2. the kinetic energy of the particle at \(N\). The particle is now projected from \(N\), with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface towards \(M\).
3. Find the least value of \(v\) for which the particle will reach \(L\).

7 A particle of mass \(m \mathrm {~kg}\) moves up a line of greatest slope of a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The frictional and normal components of the contact force on the particle have magnitudes \(F \mathrm {~N}\) and \(R \mathrm {~N}\) respectively. The particle passes through the point \(P\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it reaches its highest point on the plane.

1. Show that \(R = 9.336 m\) and \(F = 1.416 m\), each correct to 4 significant figures.
2. Find the coefficient of friction between the particle and the plane. After the particle reaches its highest point it starts to move down the plane.
3. Find the speed with which the particle returns to \(P\).

2 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_609_967_536_589} A smooth narrow tube \(A E\) has two straight parts, \(A B\) and \(D E\), and a curved part \(B C D\). The part \(A B\) is vertical with \(A\) above \(B\), and \(D E\) is horizontal. \(C\) is the lowest point of the tube and is 0.65 m below the level of \(D E\). A particle is released from rest at \(A\) and travels through the tube, leaving it at \(E\) with speed \(6 \mathrm {~ms} ^ { - 1 }\) (see diagram). Find

1. the height of \(A\) above the level of \(D E\),
2. the maximum speed of the particle.

6 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-3_476_1305_1519_420} A smooth slide \(A B\) is fixed so that its highest point \(A\) is 3 m above horizontal ground. \(B\) is \(h \mathrm {~m}\) above the ground. A particle \(P\) of mass 0.2 kg is released from rest at a point on the slide. The particle moves down the slide and, after passing \(B\), continues moving until it hits the ground (see diagram). The speed of \(P\) at \(B\) is \(v _ { B }\) and the speed at which \(P\) hits the ground is \(v _ { G }\).

1. In the case that \(P\) is released at \(A\), it is given that the kinetic energy of \(P\) at \(B\) is 1.6 J . Find
   1. the value of \(h\),
   2. the kinetic energy of the particle immediately before it reaches the ground,
   3. the ratio \(v _ { G } : v _ { B }\).
   4. In the case that \(P\) is released at the point \(X\) of the slide, which is \(H \mathrm {~m}\) above the ground (see diagram), it is given that \(v _ { G } : v _ { B } = 2.55\). Find the value of \(H\) correct to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-4_384_679_258_733} Particles \(P\) and \(Q\), of masses 0.2 kg and 0.5 kg respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(P\) hangs freely and \(Q\) is in contact with the table. A force of magnitude 3.2 N acts on \(Q\), upwards and away from the pulley, at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).
      1. The system is in limiting equilibrium with \(P\) about to move upwards. Find the coefficient of friction between \(Q\) and the table. The force of magnitude 3.2 N is now removed and \(P\) starts to move downwards.
      2. Find the acceleration of the particles and the tension in the string.

4 A block of mass 20 kg is pulled from the bottom to the top of a slope. The slope has length 10 m and is inclined at \(4.5 ^ { \circ }\) to the horizontal. The speed of the block is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the slope and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the slope.

1. Find the loss of kinetic energy and the gain in potential energy of the block.
2. Given that the work done against the resistance to motion is 50 J , find the work done by the pulling force acting on the block.
3. Given also that the pulling force is constant and acts at an angle of \(15 ^ { \circ }\) upwards from the slope, find its magnitude.

2 \includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_301_1267_616_440} The diagram shows the vertical cross-section \(A B C\) of a fixed surface. \(A B\) is a curve and \(B C\) is a horizontal straight line. The part of the surface containing \(A B\) is smooth and the part containing \(B C\) is rough. \(A\) is at a height of 1.8 m above \(B C\). A particle of mass 0.5 kg is released from rest at \(A\) and travels along the surface to \(C\).

1. Find the speed of the particle at \(B\).
2. Given that the particle reaches \(C\) with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the work done against the resistance to motion as the particle moves from \(B\) to \(C\).