6.02a Work done: concept and definition

1 \includegraphics[max width=\textwidth, alt={}, center]{9fbb63e3-4017-461e-9110-500be2c20778-2_122_803_255_671} A block is pushed along a horizontal floor by a force of magnitude 45 N acting at an angle of \(14 ^ { \circ }\) to the horizontal (see diagram). Find the work done by the force in moving the block a distance of 25 m .

2 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_385_389_918_879} A block \(B\) lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N , making angles of \(\alpha\) and \(\beta\) respectively with the \(x\)-direction, act on \(B\) as shown in the diagram, and \(B\) is moving in the \(x\)-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).

1. Find the total work done by the forces shown in the diagram when \(B\) has moved a distance of 20 m .
2. Given that the coefficient of friction between the block and the plane is \(\frac { 5 } { 8 }\), find the weight of the block.

3 A cyclist exerts a constant driving force of magnitude \(F \mathrm {~N}\) while moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 36 } { 325 }\). A constant resistance to motion of 32 N acts on the cyclist. The total weight of the cyclist and his bicycle is 780 N . The cyclist's acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(F\). The cyclist's speed is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill.
2. Find how far up the hill the cyclist travels before coming to rest.

5 A lorry of mass 15000 kg climbs from the bottom to the top of a straight hill, of length 1440 m , at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 16 m above the level of the bottom of the hill. The resistance to motion is constant and equal to 1800 N .

1. Find the work done by the driving force. On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point \(P\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and is now equal to 1600 N . The work done by the lorry's engine from the top of the hill to the point \(P\) is 5030 kJ .
2. Find the distance from the top of the hill to the point \(P\).

2 A box of mass 25 kg is pulled in a straight line along a horizontal floor. The box starts from rest at a point \(A\) and has a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it reaches a point \(B\). The distance \(A B\) is 15 m . The pulling force has magnitude 220 N and acts at an angle of \(\alpha ^ { \circ }\) above the horizontal. The work done against the resistance to motion acting on the box, as the box moves from \(A\) to \(B\), is 3000 J . Find the value of \(\alpha\).

6 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-3_526_519_902_813} Particles \(A\) of mass 0.4 kg and \(B\) of mass 1.6 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 1.2 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the work done on \(B\) by the tension in the string, as \(B\) moves to the floor. When particle \(B\) reaches the floor it remains at rest. Particle \(A\) continues to move upwards.
2. Find the greatest height above the floor reached by particle \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-3_246_1006_1781_571} A block of mass 60 kg is pulled up a hill in the line of greatest slope by a force of magnitude 50 N acting at an angle \(\alpha ^ { \circ }\) above the hill. The block passes through points \(A\) and \(B\) with speeds \(8.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram). The distance \(A B\) is 250 m and \(B\) is 17.5 m above the level of \(A\). The resistance to motion of the block is 6 N . Find the value of \(\alpha\).
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3 A train of mass 200000 kg moves on a horizontal straight track. It passes through a point \(A\) with speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and later it passes through a point \(B\). The power of the train's engine at \(B\) is 1.2 times the power of the train's engine at \(A\). The driving force of the train's engine at \(B\) is 0.96 times the driving force of the train's engine at \(A\).

1. Show that the speed of the train at \(B\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. For the motion from \(A\) to \(B\), find the work done by the train's engine given that the work done against the resistance to the train's motion is \(2.3 \times 10 ^ { 6 } \mathrm {~J}\).

1 A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.

1. Find the work done by the weightlifter.
2. Given that the time taken to raise the mass is 1.2 s , find the average power developed by the weightlifter.

2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.

1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
2. Find the distance that the particle travels along the ground before it comes to rest.

7 A car of mass 1600 kg moves with constant power 14 kW as it travels along a straight horizontal road. The car takes 25 s to travel between two points \(A\) and \(B\) on the road.

1. Find the work done by the car's engine while the car travels from \(A\) to \(B\). The resistance to the car's motion is constant and equal to 235 N . The car has accelerations at \(A\) and \(B\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) respectively. Find
2. the gain in kinetic energy by the car in moving from \(A\) to \(B\),
3. the distance \(A B\). {www.cie.org.uk} after the live examination series. }

6 A block of mass 25 kg is pulled along horizontal ground by a force of magnitude 50 N inclined at \(10 ^ { \circ }\) above the horizontal. The block starts from rest and travels a distance of 20 m . There is a constant resistance force of magnitude 30 N opposing motion.

1. Find the work done by the pulling force.
2. Use an energy method to find the speed of the block when it has moved a distance of 20 m .
3. Find the greatest power exerted by the 50 N force. \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_236_1027_2161_566} After the block has travelled the 20 m , it comes to a plane inclined at \(5 ^ { \circ }\) to the horizontal. The force of 50 N is now inclined at an angle of \(10 ^ { \circ }\) to the plane and pulls the block directly up the plane (see diagram). The resistance force remains 30 N .
4. Find the time it takes for the block to come to rest from the instant when it reaches the foot of the inclined plane.
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1 A crane is used to raise a block of mass 50 kg vertically upwards at constant speed through a height of 3.5 m . There is a constant resistance to motion of 25 N .

1. Find the work done by the crane.
2. Given that the time taken to raise the block is 2 s , find the power of the crane.

3 A roller-coaster car (including passengers) has a mass of 840 kg . The roller-coaster ride includes a section where the car climbs a straight ramp of length 8 m inclined at \(30 ^ { \circ }\) above the horizontal. The car then immediately descends another ramp of length 10 m inclined at \(20 ^ { \circ }\) below the horizontal. The resistance to motion acting on the car is 640 N throughout the motion.

1. Find the total work done against the resistance force as the car ascends the first ramp and descends the second ramp.
2. The speed of the car at the bottom of the first ramp is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Use an energy method to find the speed of the car when it reaches the bottom of the second ramp.

3 A van of mass 2500 kg descends a hill of length 0.4 km inclined at \(4 ^ { \circ }\) to the horizontal. There is a constant resistance to motion of 600 N and the speed of the van increases from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it descends the hill. Find the work done by the van's engine as it descends the hill.

2 A train of mass 150000 kg ascends a straight slope inclined at \(\alpha ^ { \circ }\) to the horizontal with a constant driving force of 16000 N . At a point \(A\) on the slope the speed of the train is \(45 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Point \(B\) on the slope is 500 m beyond \(A\). At \(B\) the speed of the train is \(42 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a resistance force acting on the train and the train does \(4 \times 10 ^ { 6 } \mathrm {~J}\) of work against this resistance force between \(A\) and \(B\). Find the value of \(\alpha\).

3 A small block \(B\) of mass 0.25 kg is released from rest at a point \(O\) on a smooth horizontal surface. After its release the velocity of \(B\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when its displacement is \(x \mathrm {~m}\) from \(O\). The force acting on \(B\) has magnitude \(\left( 2 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and is directed horizontally away from \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 8\).
2. Find the velocity of \(B\) when \(x = 1.5\). An extra force acts on \(B\) after \(x = 1.5\). It is given that, when \(x > 1.5\), $$v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 1.2 x ^ { 2 } + 6 - 3 x$$
3. Find the magnitude of this extra force and state the direction in which it acts.

2. \begin{figure}[h] \end{figure} A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 7 }\). The points \(A\) and \(B\) are on a line of greatest slope of the ramp with \(A B = 2.5 \mathrm {~m}\) and \(B\) above \(A\), as shown in Figure 1. A package of mass 2 kg is projected up the ramp from \(A\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and first comes to instantaneous rest at \(B\). The coefficient of friction between the package and the ramp is \(\mu\). The package is modelled as a particle. Use the work-energy principle to find the value of \(\mu\).
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2. A car of mass 600 kg tows a trailer of mass 200 kg up a hill along a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The trailer is attached to the car by a light inextensible towbar. The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude 150 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 300 N . When the engine of the car is working at a constant rate of \(P \mathrm {~kW}\) the car and the trailer have a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the value of \(P\). Later, at the instant when the car and the trailer are travelling up the hill with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. When the towbar breaks the trailer is at the point \(X\). The trailer continues to travel up the hill before coming to instantaneous rest at the point \(Y\). The resistance to the motion of the trailer from non-gravitational forces is again modelled as a constant force of magnitude 300 N .
2. Use the work-energy principle to find the distance \(X Y\).
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5. \begin{figure}[h] \end{figure} A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The points \(A\) and \(B\) are on a line of greatest slope of the ramp, with \(A B = 2.5 \mathrm {~m}\) and \(B\) above \(A\), as shown in Figure 2. A package of mass 1.5 kg is projected up the ramp from \(A\) with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and first comes to instantaneous rest at \(B\). The coefficient of friction between the package and the ramp is \(\frac { 2 } { 7 }\) The package is modelled as a particle.

1. Find the work done against friction as the package moves from \(A\) to \(B\).
2. Use the work-energy principle to find the value of \(U\). After coming to instantaneous rest at \(B\), the package slides back down the slope.
3. Use the work-energy principle to find the speed of the package at the instant it returns to \(A\).

6. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Particle \(A\) is held at rest at the point \(X\) on a fixed rough ramp that is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The string passes over a small smooth pulley \(P\) that is fixed at the top of the ramp. Particle \(B\) hangs vertically below \(P\), 2 m above the ground, as shown in Figure 4. The particles are released from rest with the string taut so that \(A\) moves up the ramp and the section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the ramp. The coefficient of friction between \(A\) and the ramp is \(\frac { 3 } { 8 }\) Air resistance is ignored.

1. Find the potential energy lost by the system as \(A\) moves 2 m up the ramp.
2. Find the work done against friction as \(A\) moves 2 m up the ramp. When \(B\) hits the ground, \(B\) is brought to rest by the impact and does not rebound and \(A\) continues to move up the ramp.
3. Use the work-energy principle to find the speed of \(B\) at the instant before it hits the ground. Particle \(A\) comes to instantaneous rest at the point \(Y\) on the ramp, where \(X Y = ( 2 + d ) \mathrm { m }\).
4. Use the work-energy principle to find the value of \(d\).

8. \begin{figure}[h] \end{figure} Figure 4 shows a rough ramp fixed to horizontal ground.
The ramp is inclined at angle \(\alpha\) to the ground, where \(\tan \alpha = \frac { 1 } { 6 }\) The point \(A\) is on the ground at the bottom of the ramp.
The point \(B\) is at the top of the ramp.
The line \(A B\) is a line of greatest slope of the ramp and \(A B = 4 \mathrm {~m}\).
A particle \(P\) of mass 3 kg is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) directly towards \(B\).
The coefficient of friction between the particle and the ramp is \(\frac { 3 } { 4 }\)

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that at the instant \(P\) reaches the point \(B\), the speed of \(P\) is \(5 \mathrm {~ms} ^ { - 1 }\)
2. use the work-energy principle to find the value of \(U\). The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the ground at the point \(C\).
3. Find the horizontal distance from \(B\) to \(C\).

1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A particle \(P\) of mass \(m\) is held at rest at a point \(A\) on the plane.
The particle is then projected with speed \(u\) up a line of greatest slope of the plane and comes to instantaneous rest at the point \(B\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 7 }\)

1. Show that the magnitude of the frictional force acting on the particle, as it moves from \(A\) to \(B\), is \(\frac { 4 m g } { 35 }\) Given that \(u = \sqrt { 10 a g }\), use the work-energy principle
2. to find \(A B\) in terms of \(a\),
3. to find, in terms of \(a\) and \(g\), the speed of \(P\) when it returns to \(A\).

4. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).

2 A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. When P has moved 12 m , its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that friction is the only non-gravitational resistive force acting on P , find

1. the work done against friction as the speed of \(P\) increases from \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. the coefficient of friction between the particle and the plane.