6.02b Calculate work: constant force, resolved component

1 One end of a light inextensible string is attached to a block. The string is used to pull the block along a horizontal surface with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The string makes an angle of \(20 ^ { \circ }\) with the horizontal and the tension in the string is 25 N . Find the work done by the tension in a period of 8 seconds.

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 \(C\).

6 A lorry of mass 16000 kg climbs a straight hill \(A B C D\) which makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). For the motion from \(A\) to \(B\), the work done by the driving force of the lorry is 1200 kJ and the resistance to motion is constant and equal to 1240 N . The speed of the lorry is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) and \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\).

1. Find the distance \(A B\). For the motion from \(B\) to \(D\) the gain in potential energy of the lorry is 2400 kJ .
2. Find the distance \(B D\). For the motion from \(B\) to \(D\) the driving force of the lorry is constant and equal to 7200 N . From \(B\) to \(C\) the resistance to motion is constant and equal to 1240 N and from \(C\) to \(D\) the resistance to motion is constant and equal to 1860 N .
3. Given that the speed of the lorry at \(D\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the distance \(B C\).

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 .

6 A car of mass 1250 kg moves from the bottom to the top of a straight hill of length 500 m . The top of the hill is 30 m above the level of the bottom. The power of the car's engine is constant and equal to 30000 W . The car's acceleration is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at the bottom of the hill and is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at the top. The resistance to the car's motion is 1000 N . Find

1. the car's gain in kinetic energy,
2. the work done by the car's engine.

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.

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 \(\alpha\).

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.

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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3. \begin{figure}[h] \end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 7 }\) The line \(A B\) is a line of greatest slope of the ramp, with \(B\) above \(A\) and \(A B = 6 \mathrm {~m}\), as shown in Figure 1. A block \(P\) of mass 2 kg is pushed, with constant speed, in a straight line up the slope from \(A\) to \(B\). The force pushing \(P\) acts parallel to \(A B\). The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 3 }\) The block is modelled as a particle and air resistance is negligible.

1. Use the model to find the total work done in pushing the block from \(A\) to \(B\). The block is now held at \(B\) and released from rest.
2. Use the model and the work-energy principle to find the speed of the block at the instant it reaches \(A\).

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.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(A B = 3 \mathrm {~m}\) with \(B\) above \(A\), as shown in Fig. 1. The speed of \(P\) at \(A\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assuming the plane is smooth,

1. find the speed of \(P\) at \(B\). The plane is now assumed to be rough. At \(A\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle, or otherwise,
2. find the coefficient of friction between \(P\) and the plane.

4. \begin{figure}[h] \end{figure} A box of mass 30 kg is held at rest at point \(A\) on a rough inclined plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. Point \(B\) is 50 m from \(A\) up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from \(A\) to \(B\) by a force acting parallel to \(A B\) and then held at rest at \(B\). The coefficient of friction between the box and the plane is \(\frac { 1 } { 4 }\). Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,

1. find the work done in dragging the box from \(A\) to \(B\). The box is released from rest at the point \(B\) and slides down the slope. Using the workenergy principle, or otherwise,
2. find the speed of the box as it reaches \(A\).
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5. The point \(A\) lies on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 24 } { 25 }\). A particle \(P\) is projected from \(A\), up a line of greatest slope of the plane, with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg and the coefficient of friction between \(P\) and the plane is \(\frac { 5 } { 12 }\). The particle comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 1.5 \mathrm {~m}\). It then moves back down the plane to \(A\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Use the work-energy principle to find the value of \(U\).
3. Find the speed of \(P\) when it returns to \(A\).

2. A particle \(P\) of mass 3 kg moves from point \(A\) to point \(B\) up a line of greatest slope of a fixed rough plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 Given that \(A B = 15 \mathrm {~m}\) and that the speed of \(P\) at \(A\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find

1. the work done against friction as \(P\) moves from \(A\) to \(B\),
2. the speed of \(P\) at \(B\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is released from rest at a point \(A\) on a rough inclined plane and slides down a line of greatest slope. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The point \(B\) is 5 m from \(A\) on the line of greatest slope through \(A\), as shown in Figure 3.

1. Find the potential energy lost by \(P\) as it moves from \(A\) to \(B\). The speed of \(P\) as it reaches \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. 1. Use the work-energy principle to find the magnitude of the constant frictional force acting on \(P\) as it moves from \(A\) to \(B\).
   2. Find the coefficient of friction between \(P\) and the plane. The particle \(P\) is now placed at \(A\) and projected down the plane towards \(B\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the frictional force remains constant,
3. find the speed of \(P\) as it reaches \(B\).

8. The points \(A\) and \(B\) are 10 m apart on a line of greatest slope of a fixed rough inclined plane, with \(A\) above \(B\). The plane is inclined at \(25 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 5 kg is released from rest at \(A\) and slides down the slope. As \(P\) passes \(B\), it is moving with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find, using the work-energy principle, the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the coefficient of friction between the particle and the plane.