6.02a Work done: concept and definition

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.

5. A car of mass 1000 kg is towing a trailer of mass 1500 kg along a straight horizontal road. The tow-bar joining the car to the trailer is modelled as a light rod parallel to the road. The total resistance to motion of the car is modelled as having constant magnitude 750 N . The total resistance to motion of the trailer is modelled as of magnitude \(R\) newtons, where \(R\) is a constant. When the engine of the car is working at a rate of 50 kW , the car and the trailer travel at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(R = 1250\). When travelling at \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the driver of the car disengages the engine and applies the brakes. The brakes provide a constant braking force of magnitude 1500 N to the car. The resisting forces of magnitude 750 N and 1250 N are assumed to remain unchanged. Calculate
2. the deceleration of the car while braking,
3. the thrust in the tow-bar while braking,
4. the work done, in kJ , by the braking force in bringing the car and the trailer to rest.
5. Suggest how the modelling assumption that the resistances to motion are constant could be refined to be more realistic.

4. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The particle \(B\) hangs freely below \(P\), as shown in Figure 2. The particles are released from rest with the string taut and the section of the string from \(A\) to \(P\) parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 5 } { 8 }\). When each particle has moved a distance \(h , B\) has not reached the ground and \(A\) has not reached \(P\).

1. Find an expression for the potential energy lost by the system when each particle has moved a distance \(h\). When each particle has moved a distance \(h\), they are moving with speed \(v\). Using the workenergy principle,
2. find an expression for \(v ^ { 2 }\), giving your answer in the form \(k g h\), where \(k\) is a number.

2. A bullet of mass 6 grams passes horizontally through a fixed, vertical board. After the bullet has travelled 2 cm through the board its speed is reduced from \(400 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The board exerts a constant resistive force on the bullet. Find, to 3 significant figures, the magnitude of this resistive force.
(5)

2. A particle of mass 4 kg is moving along the horizontal \(x\)-axis under the action of a single force which acts in the positive \(x\)-direction. At time \(t\) seconds the force has magnitude \(\left( 1 + 3 t ^ { \frac { 1 } { 2 } } \right) \mathrm { N }\).
When \(t = 0\) the particle has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Find the work done by the force in the interval \(0 \leqslant t \leqslant 4\)

2. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the positive \(x\)-direction. The only force on \(P\) is a force of magnitude \(\left( 2 t + \frac { 1 } { 2 } \right) \mathrm { N }\) acting in the direction of \(x\) increasing, where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0\), \(P\) is at rest at \(O\).

1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance \(O A\).

7. At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The distance travelled down the chute by each brick is 8 m . A brick of mass 3 kg is released from rest at the top of the chute. When it reaches the bottom of the chute, its speed is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the potential energy lost by the brick in moving down the chute.
2. By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.
3. Hence find the coefficient of friction between the brick and the chute. Another brick of mass 3 kg slides down the chute. This brick is given an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the chute.
4. Find the speed of this brick when it reaches the bottom of the chute.

1 A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N . The bullet enters the barrier with speed \(320 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). You may assume that the motion takes place in a horizontal straight line. Find

1. the magnitude of the impulse that acts on the bullet,
2. the thickness of the barrier,
3. the time taken for the bullet to pass through the barrier.

2. A car of mass 800 kg is moving on a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving up the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(3 P\) watts. When the car is moving down the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(P\) watts.

1. Find
   1. the value of \(P\),
   2. the value of \(R\).
      (6) When the car is moving up the road at \(12.5 \mathrm {~ms} ^ { - 1 }\) the engine is switched off and the car comes to rest, without braking, in a distance \(d\) metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude \(R\) newtons.
2. Use the work-energy principle to find the value of \(d\).

6 A cyclist is travelling along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg . There is a constant resistance force of magnitude 32 N to the cyclist's motion. At an instant when she is travelling at \(7 \mathrm {~ms} ^ { - 1 }\), her acceleration is \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Find the power output of the cyclist.
2. Find the steady speed that the cyclist can maintain if her power output and the resistance force are both unchanged. \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-08_2718_35_141_2012} \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-09_2724_35_136_20} The cyclist later descends a straight hill of length 32.2 m , inclined at an angle of \(\sin ^ { - 1 } \left( \frac { 1 } { 20 } \right)\) to the horizontal. Her power output is now 120 W , and the resistance force now has variable magnitude such that the work done against this force in descending the hill is 1128 J . The time taken to descend the hill is 4 s .
3. Given that the speed of the cyclist at the top of the hill is \(7.5 \mathrm {~ms} ^ { - 1 }\), find her speed at the bottom of the hill.

4 A skier of mass 80 kg is pulled up a slope which makes an angle of \(20 ^ { \circ }\) with the horizontal. The skier is subject to a constant frictional force of magnitude 70 N . The speed of the skier increases from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(A\) to \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\), and the distance \(A B\) is 25 m .

1. By modelling the skier as a small object, calculate the work done by the pulling force as the skier moves from \(A\) to \(B\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-2_451_1019_1425_603} It is given that the pulling force has constant magnitude \(P \mathrm {~N}\), and that it acts at a constant angle of \(30 ^ { \circ }\) above the slope (see diagram). Calculate \(P\).

4 A block of mass 25 kg is dragged 30 m up a slope inclined at \(5 ^ { \circ }\) to the horizontal by a rope inclined at \(20 ^ { \circ }\) to the slope. The tension in the rope is 100 N and the resistance to the motion of the block is 70 N . The block is initially at rest. Calculate

1. the work done by the tension in the rope,
2. the change in the potential energy of the block,
3. the speed of the block after it has moved 30 m up the slope.

6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5 ^ { \circ }\) to the horizontal. At a certain point \(P\) on the hill the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
2. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW , correct to 3 significant figures.
3. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\).

2 A particle \(P\) of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is \(2.1 \mathrm {~ms} ^ { - 1 } P\) is at a point \(A\) on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on \(P\).

1. Show that as a result of the impulse \(P\) starts moving up the plane with a speed of \(7.5 \mathrm {~ms} ^ { - 1 }\). While still moving up the plane, \(P\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) at a point \(B\) where \(A B = 4.2 \mathrm {~m}\). The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The frictional force exerted by the plane on \(P\) is modelled as constant.
2. Calculate the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Hence find the magnitude of the frictional force acting on \(P\). \(P\) first comes to instantaneous rest at point \(C\) on the plane.
4. Calculate \(A C\).

2 A hockey puck of mass 0.2 kg is sliding down a rough slope which is inclined at \(10 ^ { \circ }\) to the horizontal. At the instant that its velocity is \(14 \mathrm {~ms} ^ { - 1 }\) directly down the slope it is hit by a hockey stick. Immediately after it is hit its velocity is \(24 \mathrm {~ms} ^ { - 1 }\) directly up the slope.

1. Find the magnitude of the impulse exerted by the hockey stick on the puck. After it has been hit, the puck first comes to instantaneous rest when it has travelled 15 m up the slope. While the puck is moving up the slope, the resistance to its motion has constant magnitude \(R \mathrm {~N}\).
2. Use an energy method to determine the value of \(R\).

3 A crate of mass 45 kg is sliding with a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) in a straight line across a smooth horizontal floor. One end of a light inextensible rope is attached to the crate. At a certain instant a builder takes the other end of the rope and starts to pull, applying a constant force of 80 N for 5 seconds. While the builder is pulling the crate, the rope makes a constant angle of \(40 ^ { \circ }\) above the horizontal. Both the rope and the velocity of the crate lie in the same vertical plane (see diagram).
It may be assumed that there is no resistance to the motion of the crate.

1. Determine the work done by the builder in pulling the crate.
2. 1. Find the kinetic energy of the crate at the instant when the builder stops pulling the crate.
   2. Explain why the answers to part (a) and part (b)(i) are not equal.
3. Find the average power developed by the builder in pulling the crate.
4. Calculate the total impulse exerted on the crate by the builder.

7 An engineer is modelling the motion of a particle \(P\) of mass 0.5 kg in a wind tunnel. \(P\) is modelled as travelling in a straight line. The point \(O\) is a fixed point within the wind tunnel. The displacement of \(P\) from \(O\) at time \(t\) seconds is \(x\) metres, for \(t \geqslant 0\). You are given that \(x \geqslant 0\) for all \(t \geqslant 0\) and that \(P\) does not reach the end of the wind tunnel.
If \(t \geqslant 0\), then \(P\) is subject to three forces which are modelled in the following way.

• The first force has a magnitude of \(5 ( t + 1 ) \cosh t \mathrm {~N}\) and acts in the positive \(x\)-direction.
• The second force has a magnitude of \(0.5 x \mathrm {~N}\) and acts towards \(O\).
• The third force has a magnitude of \(\left| \frac { d x } { d t } \right| \mathrm { N }\) and acts in the direction of motion of the particle.
  1. 1. Show that the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t\) is \(6 - 5 t\).
     2. Use your answer to part (a)(ii) to show that the term in \(t ^ { 2 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(- 3 t ^ { 2 }\).
     3. By differentiating the differential equation in part (a) with respect to \(t\), show that the term in \(t ^ { 3 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(0.5 t ^ { 3 }\). You are given that the complete Maclaurin series for the function f is valid for all values of \(t \geqslant 0\).
        After 0.25 seconds \(P\) has travelled 1.43 m towards the origin.
  2. 1. By using the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\), evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 0.25\).
     2. Explain why it might not be sensible to use the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\) to evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 10\).

3 A particle \(Q\) of mass \(m \mathrm {~kg}\) is acted on by a single force so that it moves with constant acceleration \(\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\). Initially \(Q\) is at the point \(O\) and is moving with velocity \(\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }\). After \(Q\) has been moving for 5 seconds it reaches the point \(A\).

1. Use the equation \(\mathbf { v . v } = \mathbf { u . u } + 2 \mathbf { a x }\) to show that at \(A\) the kinetic energy of \(Q\) is 37 m J .
2. 1. Show that the power initially generated by the force is - 8 mW .
   2. The power in part (b)(i) is negative. Explain what this means about the initial motion of \(Q\).
3. 1. Find the time at which the power generated by the force is instantaneously zero.
   2. Find the minimum kinetic energy of \(Q\) in terms of \(m\).

2 The coordinates of two points, \(A\) and \(B\), are \(( - 1,6 )\) and \(( 5,12 )\) respectively, where the units of the coordinate axes are metres. A particle \(P\) moves from \(A\) to \(B\) under the action of several forces. The force \(\mathbf { F } = 7 \mathbf { i } - 2 \mathbf { j } \mathbf { N }\) is one of the forces acting on \(P\).

1. Calculate the work done by \(\mathbf { F }\) on \(P\) as \(P\) moves from \(A\) to \(B\). At the instant when \(P\) reaches \(B\) its velocity is \(- \mathbf { i } - 5 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
2. Find the power generated by \(\mathbf { F }\) at the instant that \(P\) reaches \(B\). One end of a light elastic string was attached to the origin of the coordinate system and the other to \(P\) when \(P\) was at \(A\), before it moved to \(B\). The natural length of the string is 8 m and its modulus of elasticity is 24 N .
3. At the instant that \(P\) reaches \(B\), find the following.

3 A particle \(P\) of mass 6 kg moves in a straight line under the action of a single force of magnitude \(F N\) which acts in the direction of motion of \(P\).
At time \(t\) seconds, where \(t \geqslant 0 , F\) is given by \(\mathrm { F } = \frac { 1 } { 5 - 4 \mathrm { e } ^ { - \mathrm { t } ^ { 2 } } }\).
When \(t = 0\), the speed of \(P\) is \(1.9 \mathrm {~ms} ^ { - 1 }\).

1. Find the impulse of the force over the period \(0 \leqslant t \leqslant 2\).
2. Find the speed of \(P\) at the instant when \(t = 2\).
3. Find the work done by the force on \(P\) over the period \(0 \leqslant t \leqslant 2\).

1 A force of \(\binom { 2 } { 10 } \mathrm {~N}\) is the only horizontal force acting on a particle \(P\) of mass 1.25 kg as it moves in a horizontal plane. Initially \(P\) is at the origin, \(O\), and 5 seconds later it is at the point \(A ( 50,140 )\). The units of the coordinate system are metres.

1. Calculate the work done by the force during these 5 seconds.
2. Calculate the average power generated by the force during these 5 seconds. The speed of \(P\) at \(O\) is \(10 \mathrm {~ms} ^ { - 1 }\).
3. Calculate the speed of \(P\) at \(A\).

4 A block of mass 20 kg is pulled by a light, horizontal string over a rough, horizontal plane. During 6 seconds, the work done against resistances is 510 J and the speed of the block increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the power of the pulling force. The block is now put on a rough plane that is at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The frictional resistance to sliding is \(11 g \mathrm {~N}\). A light string parallel to the plane is connected to the block. The string passes over a smooth pulley and is connected to a freely hanging sphere of mass \(m \mathrm {~kg}\), as shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-6_348_855_847_605} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} In parts (ii) and (iii), the sphere is pulled downwards and then released when travelling at a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards. The block never reaches the pulley.
2. Suppose that \(m = 5\) and that after the sphere is released the block moves \(x \mathrm {~m}\) up the plane before coming to rest.
   (A) Find an expression in terms of \(x\) for the change in gravitational potential energy of the system, stating whether this is a gain or a loss.
   (B) Find an expression in terms of \(x\) for the work done against friction.
   (C) Making use of your answers to parts (A) and (B), find the value of \(x\).
3. Suppose instead that \(m = 15\). Calculate the speed of the sphere when it has fallen a distance 0.5 m from its point of release.

2 One way to load a box into a van is to push the box so that it slides up a ramp. Some removal men are experimenting with the use of different ramps to load a box of mass 80 kg . \begin{figure}[h] \end{figure} Fig. 2 shows the general situation. The ramps are all uniformly rough with coefficient of friction 0.4 between the ramp and the box. The men push parallel to the ramp. As the box moves from one end of the ramp to the other it travels a vertical distance of 1.25 m .

1. Find the limiting frictional force between the ramp and the box in terms of \(\theta\).
2. From rest at the bottom, the box is pushed up the ramp and left at rest at the top. Show that the work done against friction is \(\frac { 392 } { \tan \theta } \mathrm {~J}\).
3. Calculate the gain in the gravitational potential energy of the box when it is raised from the ground to the floor of the van. For the rest of the question take \(\theta = 35 ^ { \circ }\).
4. Calculate the power required to slide the box up the ramp at a steady speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. The box is given an initial speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the ramp and then slides down without anyone pushing it. Determine whether it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while it is on the ramp.