6.02d Mechanical energy: KE and PE concepts

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.

3 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-2_502_661_1219_742} A load of mass 160 kg is pulled vertically upwards, from rest at a fixed point \(O\) on the ground, using a winding drum. The load passes through a point \(A , 20 \mathrm {~m}\) above \(O\), with a speed of \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Find, for the motion from \(O\) to \(A\),

1. the gain in the potential energy of the load,
2. the gain in the kinetic energy of the load. The power output of the winding drum is constant while the load is in motion.
3. Given that the work done against the resistance to motion from \(O\) to \(A\) is 20 kJ and that the time taken for the load to travel from \(O\) to \(A\) is 41.7 s , find the power output of the winding drum.

7 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-3_168_803_1909_671} The frictional force acting on a small block of mass 0.15 kg , while it is moving on a horizontal surface, has magnitude 0.12 N . The block is set in motion from a point \(X\) on the surface, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It hits a vertical wall at a point \(Y\) on the surface 2 s later. The block rebounds from the wall and moves directly towards \(X\) before coming to rest at the point \(Z\) (see diagram). At the instant that the block hits the wall it loses 0.072 J of its kinetic energy. The velocity of the block, in the direction from \(X\) to \(Y\), is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after it leaves \(X\).

1. Find the values of \(v\) when the block arrives at \(Y\) and when it leaves \(Y\), and find also the value of \(t\) when the block comes to rest at \(Z\). Sketch the velocity-time graph.
2. The displacement of the block from \(X\), in the direction from \(X\) to \(Y\), is \(s \mathrm {~m}\) at time \(t \mathrm {~s}\). Sketch the displacement-time graph. Show on your graph the values of \(s\) and \(t\) when the block is at \(Y\) and when it comes to rest at \(Z\).

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.

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.

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 \(P\).

2 A box of mass 25 kg is pulled, at a constant speed, a distance of 36 m up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves up a line of greatest slope against a constant frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope. Find

1. the work done against friction,
2. the change in gravitational potential energy of the box,
3. the work done by the pulling force.

7 A particle of mass 30 kg is on a plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. Starting from rest, the particle is pulled up the plane by a force of magnitude 200 N acting parallel to a line of greatest slope.

1. Given that the plane is smooth, find
   1. the acceleration of the particle,
   2. the change in kinetic energy after the particle has moved 12 m up the plane.
   3. It is given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.12 .
      (a) Find the acceleration of the particle.
      (b) The direction of the force of magnitude 200 N is changed, and the force now acts at an angle of \(10 ^ { \circ }\) above the line of greatest slope. Find the acceleration of the particle.

3 A particle of mass 8 kg is projected with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The motion of the particle is resisted by a constant frictional force of magnitude 15 N . The particle comes to instantaneous rest after travelling a distance \(x \mathrm {~m}\) up the plane.

1. Express the change in gravitational potential energy of the particle in terms of \(x\).
2. Use an energy method to find \(x\).

1 A particle of mass 8 kg is pulled at a constant speed a distance of 20 m up a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal by a force acting along a line of greatest slope.

1. Find the change in gravitational potential energy of the particle.
2. The total work done against gravity and friction is 1146 J . Find the frictional force acting on the particle.

1 A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the work done against air resistance.

4 A car of mass 800 kg is moving up a hill inclined at \(\theta ^ { \circ }\) to the horizontal, where \(\sin \theta = 0.15\). The initial speed of the car is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Twelve seconds later the car has travelled 120 m up the hill and has speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the change in the kinetic energy and the change in gravitational potential energy of the car.
2. The engine of the car is working at a constant rate of 32 kW . Find the total work done against the resistive forces during the twelve seconds.

1 A cyclist has mass 85 kg and rides a bicycle of mass 20 kg . The cyclist rides along a horizontal road against a total resistance force of 40 N . Find the total work done by the cyclist in increasing his speed from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while travelling a distance of 50 m .

4 A lorry of mass 16000 kg climbs from the bottom to the top of a straight hill of length 1000 m at a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 20 m above the level of the bottom of the hill. The driving force of the lorry is constant and equal to 5000 N . Find

1. the gain in gravitational potential energy of the lorry,
2. the work done by the driving force,
3. the work done against the force resisting the motion of the lorry. On reaching the top of the hill the lorry continues along a straight horizontal road against a constant resistance of 1500 N . The driving force of the lorry is not now constant, and the speed of the lorry increases from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill to \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(P\). The distance of \(P\) from the top of the hill is 2000 m .
4. Find the work done by the driving force of the lorry while the lorry travels from the top of the hill to \(P\).

2 A crate of mass 50 kg is dragged along a horizontal floor by a constant force of magnitude 400 N acting at an angle \(\alpha ^ { \circ }\) upwards from the horizontal. The total resistance to motion of the crate has constant magnitude 250 N . The crate starts from rest at the point \(O\) and passes the point \(P\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O P\) is 20 m . For the crate's motion from \(O\) to \(P\), find

1. the increase in kinetic energy of the crate,
2. the work done against the resistance to the motion of the crate,
3. the value of \(\alpha\).

1 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_421_1223_267_461} A box of mass 8 kg is pulled, at constant speed, up a straight path which is inclined at an angle of \(15 ^ { \circ }\) to the horizontal. The pulling force is constant, of magnitude 30 N , and acts upwards at an angle of \(10 ^ { \circ }\) from the path (see diagram). The box passes through the points \(A\) and \(B\), where \(A B = 20 \mathrm {~m}\) and \(B\) is above the level of \(A\). For the motion from \(A\) to \(B\), find

1. the work done by the pulling force,
2. the gain in potential energy of the box,
3. the work done against the resistance to motion of the box.

4 A load of mass 160 kg is lifted vertically by a crane, with constant acceleration. The load starts from rest at the point \(O\). After 7 s , it passes through the point \(A\) with speed \(0.5 \mathrm {~ms} ^ { - 1 }\). By considering energy, find the work done by the crane in moving the load from \(O\) to \(A\).

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.

2 A lorry of mass 15000 kg moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top to the bottom of a straight hill of length 900 m . The top of the hill is 18 m above the level of the bottom of the hill. The total work done by the resistive forces acting on the lorry, including the braking force, is \(4.8 \times 10 ^ { 6 } \mathrm {~J}\). Find

1. the loss in gravitational potential energy of the lorry,
2. the work done by the driving force. On reaching the bottom of the hill the lorry continues along a straight horizontal road against a constant resistance of 1600 N . There is no braking force acting. The speed of the lorry increases from \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill to \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(X\), where \(X\) is 2500 m from the bottom of the hill.
3. By considering energy, find the work done by the driving force of the lorry while it travels from the bottom of the hill to \(X\).

5 A particle of mass 0.8 kg slides down a rough inclined plane along a line of greatest slope \(A B\). The distance \(A B\) is 8 m . The particle starts at \(A\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves with constant acceleration \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of the particle at the instant it reaches \(B\).
2. Given that the work done against the frictional force as the particle moves from \(A\) to \(B\) is 7 J , find the angle of inclination of the plane. When the particle is at the point \(X\) its speed is the same as the average speed for the motion from \(A\) to \(B\).
3. Find the work done by the frictional force for the particle's motion from \(A\) to \(X\).

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 \(C\).

6 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-3_218_1280_1146_431} \(A B\) and \(B C\) are straight roads inclined at \(5 ^ { \circ }\) to the horizontal and \(1 ^ { \circ }\) to the horizontal respectively. \(A\) and \(C\) are at the same horizontal level and \(B\) is 45 m above the level of \(A\) and \(C\) (see diagram, which is not to scale). A car of mass 1200 kg travels from \(A\) to \(C\) passing through \(B\).

1. For the motion from \(A\) to \(B\), the speed of the car is constant and the work done against the resistance to motion is 360 kJ . Find the work done by the car's engine from \(A\) to \(B\). The resistance to motion is constant throughout the whole journey.
2. For the motion from \(B\) to \(C\) the work done by the driving force is 1660 kJ . Given that the speed of the car at \(B\) is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that its speed at \(C\) is \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. The car's driving force immediately after leaving \(B\) is 1.5 times the driving force immediately before reaching \(C\). Find, correct to 2 significant figures, the ratio of the power developed by the car's engine immediately after leaving \(B\) to the power developed immediately before reaching \(C\).