6.02b Calculate work: constant force, resolved component

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 box of mass 5 kg is pulled at a constant speed a distance of 15 m up a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The box moves along a line of greatest slope against a frictional force of 40 N . The force pulling the box is parallel to the line of greatest slope.

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

5 A railway engine of mass 75000 kg is moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.01\). The engine is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine is working at 960 kW . There is a constant force resisting the motion of the engine.

1. Find the resistance force.
   The engine comes to a section of track which is horizontal. At the start of the section the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the power of the engine is now reduced to 900 kW . The resistance to motion is no longer constant, but in the next 60 s the work done against the resistance force is 46500 kJ .
2. Find the speed of the engine at the end of the 60 s .

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 \(v\).

1 \includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-2_300_748_274_708} One end of a light inextensible string is attached to a ring which is threaded on a fixed horizontal bar. The string is used to pull the ring along the bar at a constant speed of \(0.4 \mathrm {~ms} ^ { - 1 }\). The string makes a constant angle of \(30 ^ { \circ }\) with the bar and the tension in the string is 5 N (see diagram). Find the work done by the tension in 10 s .

6 A block of mass 50 kg is pulled up a straight hill and passes through points \(A\) and \(B\) with speeds \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 200 m and \(B\) is 15 m higher than \(A\). For the motion of the block from \(A\) to \(B\), find

1. the loss in kinetic energy of the block,
2. the gain in potential energy of the block. The resistance to motion of the block has magnitude 7.5 N.
3. Find the work done by the pulling force acting on the block. The pulling force acting on the block has constant magnitude 45 N and acts at an angle \(\alpha ^ { \circ }\) upwards from the hill.
4. Find the value of \(\alpha\).

1 A block is pulled for a distance of 50 m along a horizontal floor, by a rope that is inclined at an angle of \(\alpha ^ { \circ }\) to the floor. The tension in the rope is 180 N and the work done by the tension is 8200 J . Find the value of \(\alpha\).

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.

1 A block is pulled in a straight line along horizontal ground by a force of constant magnitude acting at an angle of \(60 ^ { \circ }\) above the horizontal. The work done by the force in moving the block a distance of 5 m is 75 J . Find the magnitude of the force.

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.

1 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-2_262_711_248_717} A ring is threaded on a fixed horizontal bar. The ring is attached to one end of a light inextensible string which is used to pull the ring along the bar at a constant speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string makes a constant angle of \(24 ^ { \circ }\) with the bar and the tension in the string is 6 N (see diagram). Find the work done by the tension in a period of 8 s .

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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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 \(A\).

1 A block \(B\) of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(\theta\) above the horizontal. The normal component of the contact force acting on \(B\) has magnitude 20 N .

1. Show that \(\sin \theta = 0.28\).
2. Find the work done by the pulling force in moving the block a distance of 5 m .

1 One end of a light inextensible string is attached to a block. The string makes an angle of \(60 ^ { \circ }\) above the horizontal and is used to pull the block in a straight line on a horizontal floor with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the string is 8 N . The block starts to move with speed \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 5 s of the block's motion, find

1. the distance travelled,
2. the work done by the tension in the string.

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 \(P\).

4 A particle of mass 1.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 12 } { 5 }\). The coefficient of friction between the particle and the plane is \(\mu\).

1. A force of magnitude 20 N parallel to a line of greatest slope of the plane is applied to the particle and the particle is on the point of moving up the plane. Show that \(\mu = 1.6\).
   The force of magnitude 20 N is now removed.
2. Find the acceleration of the particle.
3. Find the work done against friction during the first 2 s of motion.

3 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_143_611_1050_769} A crate of mass 3 kg is pulled at constant speed along a horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(15 ^ { \circ }\) to the horizontal, as shown in the diagram. Find

1. the work done by the pulling force in moving the crate a distance of 2 m ,
2. the normal component of the contact force on the crate.

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 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.

7 A car of mass 1250 kg travels along a horizontal straight road. The power of the car's engine is constant and equal to 24 kW and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\). The car passes through the point \(A\) on the road with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(R\). The car continues with increasing speed, passing through the point \(B\) on the road with speed \(29.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car subsequently passes through the point \(C\).
2. Find the acceleration of the car at \(B\), giving the answer in \(\mathrm { m } \mathrm { s } ^ { - 2 }\) correct to 3 decimal places.
3. Show that, while the car's speed is increasing, it cannot reach \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Explain why the speed of the car is approximately constant between \(B\) and \(C\).
5. State a value of the approximately constant speed, and the maximum possible error in this value at any point between \(B\) and \(C\). The work done by the car's engine during the motion from \(B\) to \(C\) is 1200 kJ .
6. By assuming the speed of the car is constant from \(B\) to \(C\), find, in either order,
   1. the approximate time taken for the car to travel from \(B\) to \(C\),
   2. an approximation for the distance \(B C\).