6.02i Conservation of energy: mechanical energy principle

5 A block \(B\) of mass 4 kg is pushed up a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal by a force applied to \(B\), acting in the direction of motion of \(B\). The block passes through points \(P\) and \(Q\) with speeds \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. \(P\) and \(Q\) are 10 m apart with \(P\) below the level of \(Q\).

1. Find the decrease in kinetic energy of the block as it moves from \(P\) to \(Q\).
2. Hence find the work done by the force pushing the block up the slope as the block moves from \(P\) to \(Q\).
3. At the instant the block reaches \(Q\), the force pushing the block up the slope is removed. Find the time taken, after this instant, for the block to return to \(P\).

1 A particle of mass 0.6 kg is projected with a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. Use an energy method to find the speed of the particle after it has moved 15 m down the plane.

5 A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of \(5 ^ { \circ }\) to the horizontal. The car and trailer are connected by a light rigid tow-bar. At the top of the hill the speed of the car and trailer is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at the bottom of the hill their speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. It is given that as the car and trailer descend the hill, the engine of the car does 150000 J of work, and there are no resistance forces. Find the length of the hill.
2. It is given instead that there is a resistance force of 100 N on the trailer, the length of the hill is 200 m , and the acceleration of the car and trailer is constant. Find the tension in the tow-bar between the car and trailer.

2 A particle \(P\) is projected vertically upwards from horizontal ground. \(P\) reaches a maximum height of 45 m . After reaching the ground, \(P\) comes to rest without rebounding.

1. Find the speed at which \(P\) was projected.
2. Find the total time for which the speed of \(P\) is at least \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5 A cyclist is riding along a straight horizontal road. The total mass of the cyclist and her bicycle is 70 kg . At an instant when the cyclist's speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), her acceleration is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant resistance to motion of magnitude 30 N .

1. Find the power developed by the cyclist.
   The cyclist comes to the top of a hill inclined at \(5 ^ { \circ }\) to the horizontal. The cyclist stops pedalling and freewheels down the hill (so that the cyclist is no longer supplying any power). The magnitude of the resistance force remains at 30 N . Over a distance of \(d \mathrm {~m}\), the speed of the cyclist increases from \(6 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
2. Find the change in kinetic energy.
3. Use an energy method to find \(d\). \includegraphics[max width=\textwidth, alt={}, center]{4e555003-16f1-4453-ab25-c50929d4b5b3-10_725_785_260_680} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. \(Q\) lies on a plane \(B C\) which is inclined at \(30 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).

1 A particle of mass 1.6 kg is dropped from a height of 9 m above horizontal ground. The speed of the particle at the instant before hitting the ground is \(12 \mathrm {~ms} ^ { - 1 }\). Find the work done against air resistance.

4 An athlete of mass 84 kg is running along a straight road.

1. Initially the road is horizontal and he runs at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The athlete produces a constant power of 60 W . Find the resistive force which acts on the athlete.
2. The athlete then runs up a 150 m section of the road which is inclined at \(0.8 ^ { \circ }\) to the horizontal. The speed of the athlete at the start of this section of road is \(3 \mathrm {~ms} ^ { - 1 }\) and he now produces a constant driving force of 24 N . The total resistive force which acts on the athlete along this section of road has constant magnitude 13 N . Use an energy method to find the speed of the athlete at the end of the 150 m section of road.

7 \includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-10_525_885_264_625} The diagram shows the vertical cross-section \(X Y Z\) of a rough slide. The section \(Y Z\) is a straight line of length 2 m inclined at an angle of \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The section \(Y Z\) is tangential to the curved section \(X Y\) at \(Y\), and \(X\) is 1.8 m above the level of \(Y\). A child of mass 25 kg slides down the slide, starting from rest at \(X\). The work done by the child against the resistance force in moving from \(X\) to \(Y\) is 50 J .

1. Find the speed of the child at \(Y\).
   It is given that the child comes to rest at \(Z\).
2. Use an energy method to find the coefficient of friction between the child and \(Y Z\), giving your answer as a fraction in its simplest form.
   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 straight slope of length 60 m is inclined at an angle of \(12 ^ { \circ }\) to the horizontal. A bobsled starts at the top of the slope with a speed of \(5 \mathrm {~ms} ^ { - 1 }\). The bobsled slides directly down the slope.

1. It is given that there is no resistance to the bobsled's motion. Find its speed when it reaches the bottom of the slope.
2. It is given instead that the coefficient of friction between the bobsled and the slope is 0.03 . Find the time that it takes for the bobsled to reach the bottom of the slope.

3 \includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-04_519_1018_260_561} The diagram shows the vertical cross-section of a surface. \(A , B\) and \(C\) are three points on the crosssection. The level of \(B\) is \(h \mathrm {~m}\) above the level of \(A\). The level of \(C\) is 0.5 m below the level of \(A\). A particle of mass 0.2 kg is projected up the slope from \(A\) with initial speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The particle remains in contact with the surface as it travels from \(A\) to \(C\).

1. Given that the particle reaches \(B\) with a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there is no resistance force, find \(h\).
2. It is given instead that there is a resistance force and that the particle does 3.1 J of work against the resistance force as it travels from \(A\) to \(C\). Find the speed of the particle when it reaches \(C\).

7 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-12_439_1095_258_525} Two particles \(P\) and \(Q\) of masses 0.5 kg and \(m \mathrm {~kg}\) respectively are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two inclined planes. The particles are initially at rest with \(P\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal and \(Q\) on a plane inclined at \(45 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 0.8 N is applied to \(P\) acting down the plane, causing \(P\) to move down the plane (see diagram).

1. It is given that \(m = 0.3\), and that the plane on which \(Q\) rests is smooth. Find the tension in the string.
2. It is given instead that the plane on which \(Q\) rests is rough, and that after each particle has moved a distance of 1 m , their speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction in this part of the motion is 0.5 J . Use an energy method to find the value of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-14_388_1216_264_461} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with \(A\) on the horizontal plane and \(B\) on the inclined plane, which makes an angle of \(30 ^ { \circ }\) with the horizontal. The string is taut and \(B\) can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to \(B\) acting down the plane (see diagram).

1. Given that both planes are smooth, find the tension in the string and the acceleration of \(B\).
2. It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J . Use an energy method to find the speed of \(B\) when it has moved this distance down the plane. [You should assume that the string is sufficiently long so that \(A\) does not hit the pulley when it moves 0.6 m .]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-04_416_792_260_674} The diagram shows a semi-circular track \(A B C\) of radius 1.8 m which is fixed in a vertical plane. The points \(A\) and \(C\) are at the same horizontal level and the point \(B\) is at the bottom of the track. The section \(A B\) is smooth and the section \(B C\) is rough. A small block is released from rest at \(A\).

1. Show that the speed of the block at \(B\) is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The block comes to instantaneous rest for the first time at a height of 1.2 m above the level of \(B\). The work done against the resistance force during the motion of the block from \(B\) to this point is 4.5 J .
2. Find the mass of the block.

1 \includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-03_471_613_254_766} A metal post is driven vertically into the ground by dropping a heavy object onto it from above. The mass of the object is 120 kg and the mass of the post is 40 kg (see diagram). The object hits the post with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with it after the impact.

1. Calculate the speed with which the combined post and object moves immediately after the impact.
2. There is a constant force resisting the motion of magnitude 4800 N . Calculate the distance the post is driven into the ground.

3 A ball of mass 1.6 kg is released from rest at a point 5 m above horizontal ground. When the ball hits the ground it instantaneously loses 8 J of kinetic energy and starts to move upwards.

1. Use an energy method to find the greatest height that the ball reaches after hitting the ground.
2. Find the total time taken, from the initial release of the ball until it reaches this greatest height.

7 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-12_560_716_258_712} Particles of masses 1.5 kg and 3 kg lie on a plane which is inclined at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The section of the plane from \(A\) to \(B\) is smooth and the section of the plane from \(B\) to \(C\) is rough. The 1.5 kg particle is held at rest at \(A\) and the 3 kg particle is in limiting equilibrium at \(B\). The distance \(A B\) is \(x \mathrm {~m}\) and the distance \(B C\) is 4 m (see diagram).

1. Show that the coefficient of friction between the particle at \(B\) and the plane is 0.75 .
   The 1.5 kg particle is released from rest. In the subsequent motion the two particles collide and coalesce. The time taken for the combined particle to travel from \(B\) to \(C\) is 2 s . The coefficient of friction between the combined particle and the plane is still 0.75 .
2. Find \(x\).
3. Find the total loss of energy of the particles from the time the 1.5 kg particle is released until the combined particle reaches \(C\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 A particle of mass 1.6 kg is projected with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a smooth plane inclined at \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). Use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.

2 \includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-03_510_604_260_769} A machine for driving a nail into a block of wood causes a hammerhead to drop vertically onto the top of a nail. The mass of the hammerhead is 1.2 kg and the mass of the nail is 0.004 kg (see diagram). The hammerhead hits the nail with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and remains in contact with the nail after the impact. The combined hammerhead and nail move immediately after the impact with speed \(40 \mathrm {~ms} ^ { - 1 }\).

1. Calculate \(v\), giving your answer as an exact fraction.
2. The nail is driven 4 cm into the wood. Find the constant force resisting the motion.

4 A car has mass 1600 kg .

1. The car is moving along a straight horizontal road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is subject to a constant resistance of magnitude 480 N . Find, in kW , the rate at which the engine of the car is working.
   The car now moves down a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.09\). The engine of the car is working at a constant rate of 12 kW . The speed of the car is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill. Ten seconds later the car has travelled 280 m down the hill and has speed \(32 \mathrm {~ms} ^ { - 1 }\).
2. Given that the resistance is not constant, use an energy method to find the total work done against the resistance during the ten seconds.

2 A block of mass 20 kg is held at rest at the top of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The block is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of the plane. There is a resistance force acting on the block. As the block moves 2 m down the plane from its point of projection, the work done against this resistance force is 50 J . Find the speed of the block when it has moved 2 m down the plane. \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-04_2716_38_109_2012}

7 \includegraphics[max width=\textwidth, alt={}, center]{87b42689-791c-4f4e-a36e-bfae3191ca11-12_244_668_264_701} Two particles \(A\) and \(B\), of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is placed on a horizontal surface. The string passes over a small smooth pulley \(P\) fixed at the edge of the surface, and \(B\) hangs freely. The horizontal section of the string, \(A P\), is of length 2.5 m (see diagram). The particles are released from rest with both sections of the string taut.

1. Given that the surface is smooth, find the time taken for \(A\) to reach the pulley.
2. It is given instead that the surface is rough and that the speed of \(A\) immediately before it reaches the pulley is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction as \(A\) moves from rest to the pulley is 2 J . Use an energy method to find \(v\).

7 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-4_257_988_267_580} The diagram shows a vertical cross-section \(A B C D\) of a surface. The parts \(A B\) and \(C D\) are straight and have lengths 2.5 m and 5.2 m respectively. \(A D\) is horizontal, and \(A B\) is inclined at \(60 ^ { \circ }\) to the horizontal. The points \(B\) and \(C\) are at the same height above \(A D\). The parts of the surface containing \(A B\) and \(B C\) are smooth. A particle \(P\) is given a velocity of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\), in the direction \(A B\), and it subsequently reaches \(D\). The particle does not lose contact with the surface during this motion.

1. Find the speed of \(P\) at \(B\).
2. Show that the maximum height of the cross-section, above \(A D\), is less than 3.2 m .
3. State briefly why \(P\) 's speed at \(C\) is the same as its speed at \(B\).
4. The frictional force acting on the particle as it travels from \(C\) to \(D\) is 1.4 N . Given that the mass of \(P\) is 0.4 kg , find the speed with which \(P\) reaches \(D\).

4 The top of an inclined plane is at a height of 0.7 m above the bottom. A block of mass 0.2 kg is released from rest at the top of the plane and slides a distance of 2.5 m to the bottom. Find the kinetic energy of the block when it reaches the bottom of the plane in each of the following cases:

1. the plane is smooth,
2. the coefficient of friction between the plane and the block is 0.15 .

4 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_478_1041_269_552} \(O A B C\) is a vertical cross-section of a smooth surface. The straight part \(O A\) has length 2.4 m and makes an angle of \(50 ^ { \circ }\) with the horizontal. \(A\) and \(C\) are at the same horizontal level and \(B\) is the lowest point of the cross-section (see diagram). A particle \(P\) of mass 0.8 kg is released from rest at \(O\) and moves on the surface. \(P\) remains in contact with the surface until it leaves the surface at \(C\). Find

1. the kinetic energy of \(P\) at \(A\),
2. the speed of \(P\) at \(C\). The greatest speed of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the depth of \(B\) below the horizontal through \(A\) and \(C\).

2 An object of mass 8 kg slides down a line of greatest slope of an inclined plane. Its initial speed at the top of the plane is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at the bottom of the plane is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against the resistance to motion of the object is 120 J . Find the height of the top of the plane above the level of the bottom.