6.02i Conservation of energy: mechanical energy principle

1 A ball is thrown vertically upwards from ground level with an initial speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has a mass of 0.6 kg . Assume that the only force acting on the ball after it is thrown is its weight.

1. Calculate the initial kinetic energy of the ball.
2. By using conservation of energy, find the maximum height above ground level reached by the ball.
3. By using conservation of energy, find the kinetic energy and the speed of the ball when it is at a height of 3 m above ground level.
4. State one modelling assumption which has been made.

6 A light elastic string has one end attached to a point \(A\) fixed on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle of mass 6 kg . The elastic string has natural length 4 metres and modulus of elasticity 300 newtons. The particle is pulled down the plane in the direction of the line of greatest slope through \(A\). The particle is released from rest when it is 5.5 metres from \(A\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-4_314_713_1900_660}

1. Calculate the elastic potential energy of the string when the particle is 5.5 metres from the point \(A\).
2. Show that the speed of the particle when the string becomes slack is \(3.66 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
3. Show that the particle will not reach point \(A\) in the subsequent motion.

7 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end. The particle is moving in a vertical circle, centre \(O\). When the particle is at \(B\), vertically above \(O\), the string is taut and the particle is moving with speed \(3 \sqrt { a g }\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-5_422_399_497_778}

1. Find, in terms of \(g\) and \(a\), the speed of the particle at the lowest point, \(A\), of its path.
2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(A\).

2 A particle is placed on a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The particle, of mass 4 kg , is released from rest at a point \(A\) and travels down the plane, passing through a point \(B\). The distance \(A B\) is 5 m . \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-04_371_693_500_680}

1. Find the potential energy lost as the particle moves from point \(A\) to point \(B\).
2. Hence write down the kinetic energy of the particle when it reaches point \(B\).
3. Hence find the speed of the particle when it reaches point \(B\).

6 A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A small bead, of mass \(m\), is attached to the other end of the string. The bead is moving in a vertical circle, centre \(O\). When the bead is at \(B\), vertically below \(O\), the string is taut and the bead is moving with speed \(5 v\). \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_536_554_502_774}

1. The speed of the bead at the highest point of its path is \(3 v\). Find \(v\) in terms of \(a\) and \(g\).
2. Find the ratio of the greatest tension to the least tension in the string, as the bead travels around its circular path.
   \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-14_1261_1709_1446_153}

7

1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
2. A block, of mass 4 kg , is attached to one end of a light elastic string. The string has natural length 2 m and modulus of elasticity 196 N . The other end of the string is attached to a fixed point \(O\).
   1. A second block, of mass 3 kg , is attached to the 4 kg block and the system hangs in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-16_374_291_890_877} Find the extension in the string.
   2. The block of mass 3 kg becomes detached from the 4 kg block and falls to the ground. The 4 kg block now begins to move vertically upwards. Find the extension of the string when the 4 kg block is next at rest.
   3. Find the extension of the string when the speed of the 4 kg block is a maximum.
      (3 marks)
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-18_2486_1714_221_153}
      \includegraphics[max width=\textwidth, alt={}]{9d039ec3-fd0a-40ae-9afe-7627439081df-19_2486_1714_221_153}

1 A plane is dropping packets of aid as it flies over a flooded village. The speed of a packet when it leaves the plane is \(60 \mathrm {~ms} ^ { - 1 }\). The packet has mass 25 kg . The packet falls a vertical distance of 34 metres to reach the ground.

1. Calculate the kinetic energy of the packet when it leaves the plane.
2. Calculate the potential energy lost by the packet as it falls to the ground.
3. Assume that the effect of air resistance on the packet as it falls can be neglected.
   1. Find the kinetic energy of the packet when it reaches the ground.
   2. Hence find the speed of the packet when it reaches the ground.

8 An elastic string has one end attached to a point \(O\) fixed on a rough horizontal surface. The other end of the string is attached to a particle of mass 2 kg . The elastic string has natural length 0.8 metres and modulus of elasticity 32 newtons. The particle is pulled so that it is at the point \(A\), on the surface, 3 metres from the point \(O\).

1. Calculate the elastic potential energy when the particle is at the point \(A\).
2. The particle is released from rest at the point \(A\) and moves in a straight line towards \(O\). The particle is next at rest at the point \(B\). The distance \(A B\) is 5 metres. \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-6_179_1055_877_497} Find the frictional force acting on the particle as it moves along the surface.
3. Show that the particle does not remain at rest at the point \(B\).
4. The particle next comes to rest at a point \(C\) with the string slack. Find the distance \(B C\).
5. Hence, or otherwise, find the total distance travelled by the particle after it is released from the point \(A\).

1 Tim is playing cricket. He hits a ball at a point \(A\). The speed of the ball immediately after being hit is \(11 \mathrm {~ms} ^ { - 1 }\). The ball strikes a tree at a point \(B\). The height of \(B\) is 5 metres above the height of \(A\).
The ball is to be modelled as a particle of mass 0.16 kg being acted upon only by gravity.

1. Calculate the initial kinetic energy of the ball.
2. Calculate the potential energy gained by the ball as it moves from the point \(A\) to the point \(B\).
3. 1. Find the kinetic energy of the ball immediately before it strikes the tree.
   2. Hence find the speed of the ball immediately before it strikes the tree.

7 A small ball, of mass 3 kg , is suspended from a fixed point \(O\) by a light inextensible string of length 1.2 m . Initially, the string is taut and the ball is at the point \(P\), vertically below \(O\). The ball is then set into motion with an initial horizontal velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball moves in a vertical circle, centre \(O\). The point \(A\), on the circle, is such that angle \(A O P\) is \(25 ^ { \circ }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-5_663_702_660_701}

1. Find the speed of the ball at the point \(A\).
2. Find the tension in the string when the ball is at the point \(A\).

8

1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
2. A particle, of mass 5 kg , is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(O\). The string has natural length 1.6 m and modulus of elasticity 392 N .
   1. Find the extension of the string when the particle hangs in equilibrium.
   2. The particle is pulled down to a point \(A\), which is 2.2 m below the point \(O\). Calculate the elastic potential energy in the string.
   3. The particle is released when it is at rest at the point \(A\). Calculate the distance of the particle from the point \(A\) when its speed first reaches \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A ball of mass 0.6 kg is thrown vertically upwards from ground level with an initial speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the initial kinetic energy of the ball.
2. Assuming that no resistance forces act on the ball, use an energy method to find the maximum height reached by the ball.
3. An experiment is conducted to confirm the maximum height for the ball calculated in part (b). In this experiment the ball rises to a height of only 8 metres.
   1. Find the work done against the air resistance force that acts on the ball as it moves.
   2. Assuming that the air resistance force is constant, find its magnitude.
4. Explain why it is not realistic to model the air resistance as a constant force.

4 A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(P\) vertically below \(O\). The particle is then set into motion with a horizontal velocity \(U\) so that it moves in a complete vertical circle with centre \(O\). The point \(Q\) on the circle is such that \(\angle P O Q = 60 ^ { \circ }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_566_540_1797_751}

1. Find, in terms of \(g , l\) and \(U\), the speed of the particle at \(Q\).
2. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle is at \(Q\).
3. Find, in terms of \(g , l , m\) and \(U\), the tension in the string when the particle returns to \(P\).
   (2 marks)

1 A hot air balloon moves vertically upwards with a constant velocity. When the balloon is at a height of 30 metres above ground level, a box of mass 5 kg is released from the balloon. After the box is released, it initially moves vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the initial kinetic energy of the box.
2. Show that the kinetic energy of the box when it hits the ground is 1720 J .
3. Hence find the speed of the box when it hits the ground.
4. State two modelling assumptions which you have made.

5 A bead of mass \(m\) moves on a smooth circular ring of radius \(a\) which is fixed in a vertical plane, as shown in the diagram. Its speed at \(A\), the highest point of its path, is \(v\) and its speed at \(B\), the lowest point of its path, is \(7 v\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-4_419_317_484_842}

1. Show that \(v = \sqrt { \frac { a g } { 12 } }\).
2. Find the reaction of the ring on the bead, in terms of \(m\) and \(g\), when the bead is at \(A\).

6 An elastic string has one end attached to a point \(O\), fixed on a horizontal table. The other end of the string is attached to a particle of mass 5 kilograms. The elastic string has natural length 2 metres and modulus of elasticity 200 newtons. The particle is pulled so that it is 2.5 metres from the point \(O\) and it is then released from rest on the table.

1. Calculate the elastic potential energy when the particle is 2.5 m from the point \(O\).
2. If the table is smooth, show that the speed of the particle when the string becomes slack is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The table is, in fact, rough and the coefficient of friction between the particle and the table is 0.4 . Find the speed of the particle when the string becomes slack.

2 John is at the top of a cliff, looking out over the sea. He throws a rock, of mass 3 kg , horizontally with a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The rock falls a vertical distance of 51 metres to reach the surface of the sea.

1. Calculate the kinetic energy of the rock when it is thrown.
2. Calculate the potential energy lost by the rock when it reaches the surface of the sea.
3. 1. Find the kinetic energy of the rock when it reaches the surface of the sea.
   2. Hence find the speed of the rock when it reaches the surface of the sea.
4. State one modelling assumption which has been made.
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-05_2484_1709_223_153}

8 A particle is attached to one end of a light inextensible string of length 3 metres. The other end of the string is attached to a fixed point \(O\). The particle is set into motion horizontally at point \(P\) with speed \(v\), so that it describes part of a vertical circle whose centre is \(O\). The point \(P\) is vertically below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-16_510_334_493_861} The particle first comes momentarily to rest at the point \(Q\), where \(O Q\) makes an angle of \(15 ^ { \circ }\) to the vertical.

1. Find the value of \(v\).
2. When the particle is at rest at the point \(Q\), the tension in the string is 22 newtons. Find the mass of the particle.
   
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-17_2484_1709_223_153}

1 In an Olympic diving competition, Kim, who has mass 58 kg , dives from a fixed platform, 10 metres above the surface of the pool. She leaves the platform with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assume that Kim's weight is the only force that acts on her after she leaves the platform. Kim is to be modelled as a particle which is initially 1 metre above the platform.

1. Calculate Kim's initial kinetic energy.
2. By using conservation of energy, find Kim's speed when she is 6 metres below the platform.

9 At a theme park, a light elastic rope is used to bring a carriage to rest at the end of a ride. The carriage has mass 200 kg and is travelling at \(8 \mathrm {~ms} ^ { - 1 }\) when the elastic rope is attached to the carriage as it passes over a point \(O\). The other end of the elastic rope is fixed to the point \(O\). The carriage then moves along a horizontal surface until it is brought to rest. The elastic rope is then detached so that the carriage remains at rest. The elastic rope has natural length 6 m and modulus of elasticity 1800 N . The rope, once taut, remains horizontal throughout the motion.

1. Calculate the elastic potential energy of the rope when the carriage is 10 m from \(O\).
   (3 marks)
2. A student's simple model assumes that there are no resistance forces acting on the carriage so that it is brought to rest by the elastic rope alone. Find the distance of the carriage from \(O\) when it is brought to rest.
3. The student improves the model by also including a constant resistance force of 800 N which acts while the carriage is in motion. Find the distance of the carriage from \(O\) when it is brought to rest.
   (8 marks)

1 Alan, of mass 76 kg , performed a ski jump. He took off at the point \(A\) at the end of the ski run with a speed of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and landed at the point \(B\). The level of the point \(B\) is 31 metres vertically below the level of the point \(A\), as shown in the diagram. Assume that his weight is the only force that acted on Alan during the jump. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-2_581_914_664_571}

1. Calculate the kinetic energy of Alan when he was at the point \(A\).
2. Calculate the potential energy lost by Alan during the jump as he moved from the point \(A\) to the point \(B\).
3. 1. Find the kinetic energy of Alan when he reached the point \(B\).
   2. Hence find the speed of Alan when he reached the point \(B\).

6 Simon, a small child of mass 22 kg , is on a swing. He is swinging freely through an angle of \(18 ^ { \circ }\) on both sides of the vertical. Model Simon as a particle, \(P\), of mass 22 kg , attached to a fixed point, \(Q\), by a light inextensible rope of length 2.4 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-5_700_310_466_849}

1. Find Simon's maximum speed as he swings.
2. Calculate the tension in the rope when Simon's speed is a maximum.

8 Zoë carries out an experiment with a block, which she places on the horizontal surface of an ice rink. She attaches one end of a light elastic string to a fixed point, \(A\), on a vertical wall at the edge of the ice rink at the height of the surface of the ice rink. The block, of mass 0.4 kg , is attached to the other end of the string. The string has natural length 5 m and modulus of elasticity 120 N . The block is modelled as a particle which is placed on the surface of the ice rink at a point \(B\), where \(A B\) is perpendicular to the wall and of length 5.5 m . \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-6_499_1429_813_333} The block is set into motion at the point \(B\) with speed \(9 \mathrm {~ms} ^ { - 1 }\) directly towards the point \(A\). The string remains horizontal throughout the motion.

1. Initially, Zoë assumes that the surface of the ice rink is smooth. Using this assumption, find the speed of the block when it reaches the point \(A\).
2. Zoë now assumes that friction acts on the block. The coefficient of friction between the block and the surface of the ice rink is \(\mu\).
   1. Find, in terms of \(g\) and \(\mu\), the speed of the block when it reaches the point \(A\).
   2. The block rebounds from the wall in the direction of the point \(B\). The speed of the block immediately after the rebound is half of the speed with which it hit the wall. Find \(\mu\) if the block comes to rest just as it reaches the point \(B\).

2 Carol, a circus performer, is on a swing. She jumps off the swing and lands in a safety net. When Carol leaves the swing, she has a speed of \(7 \mathrm {~ms} ^ { - 1 }\) and she is at a height of 8 metres above the safety net. Carol is to be modelled as a particle of mass 52 kg being acted upon only by gravity.

1. Find the kinetic energy of Carol when she leaves the swing.
2. Show that the kinetic energy of Carol when she hits the net is 5350 J , correct to three significant figures.
3. Find the speed of Carol when she hits the net.

8 A bead, of mass \(m\), moves on a smooth circular ring, of radius \(a\) and centre \(O\), which is fixed in a vertical plane. At \(P\), the highest point on the ring, the speed of the bead is \(2 u\); at \(Q\), the lowest point on the ring, the speed of the bead is \(5 u\).

1. Show that \(u = \sqrt { \frac { 4 a g } { 21 } }\).
   (4 marks)
2. \(\quad S\) is a point on the ring so that angle \(P O S\) is \(60 ^ { \circ }\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-4_600_540_657_760} Find, in terms of \(m\) and \(g\), the magnitude of the reaction of the ring on the bead when the bead is at \(S\).