6.02a Work done: concept and definition

A particle \(P\) has mass 4 kg. It is projected from a point \(A\) up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\frac{2}{5}\). The particle comes to rest instantaneously at the point \(B\) on the plane, where \(AB = 2.5\) m. It then moves back down the plane to \(A\).

1. Find the work done by friction as \(P\) moves from \(A\) to \(B\). [4]

1. Using the work-energy principle, find the speed with which \(P\) is projected from \(A\). [4]

1. Find the speed of \(P\) when it returns to \(A\). [4]

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° to the horizontal. When \(P\) has moved 12 m, its speed is 4 m 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 m s\(^{-1}\) to 4 m s\(^{-1}\), (4)
2. the coefficient of friction between the particle and the plane. (4)

A particle \(P\) of mass \(m\) is held at a point \(A\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{2}{3}\). The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(O\) on the plane, where \(OA = \frac{3}{4}a\). The particle \(P\) is released from rest and comes to rest at a point \(B\), where \(OB < a\). Using the work-energy principle, or otherwise, calculate the distance \(AB\). [6]

\(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors in a horizontal plane. A body of mass 1 kg moves under the action of a constant force \((4\mathbf{i} + 5\mathbf{j})\) N. The body moves from the point \(P\) with position vector \((-3\mathbf{i} - 15\mathbf{j})\) m to the point \(Q\) with position vector \(9\mathbf{i}\) m.

1. Find the work done by the force in moving the body from \(P\) to \(Q\). [5 marks]
2. Given that the body started from rest at \(P\), find its speed when it is at \(Q\). [5 marks]

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

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. [4]

Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.

1. 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]
2. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\). [3]

A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at 15° below the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find

1. the work done by the force, [3]
2. the power with which the force is working. [2]

\(A\) and \(B\) are two points on a line of greatest slope of a plane inclined at \(55°\) to the horizontal. \(A\) is below the level of \(B\) and \(AB = 4\) m. A particle \(P\) of mass 2.5 kg is projected up the plane from \(A\) towards \(B\) and the speed of \(P\) at \(B\) is \(6.7 \text{ m s}^{-1}\). The coefficient of friction between the plane and \(P\) is 0.15. Find

1. the work done against the frictional force as \(P\) moves from \(A\) to \(B\), [3]
2. the initial speed of \(P\) at \(A\). [4]

A cyclist and her bicycle have a combined mass of 80 kg.

1. Initially, the cyclist accelerates from rest to 3 m s\(^{-1}\) against negligible resistances along a horizontal road.
   1. How much energy is gained by the cyclist and bicycle? [2]
   2. The cyclist travels 12 m during this acceleration. What is the average driving force on the bicycle? [2]
2. While exerting no driving force, the cyclist free-wheels down a hill. Her speed increases from 4 m s\(^{-1}\) to 10 m s\(^{-1}\). During this motion, the total work done against friction is 1600 J and the drop in vertical height is \(h\) m. Without assuming that the hill is uniform in either its angle or roughness, calculate \(h\). [5]
3. The cyclist reaches another horizontal stretch of road and there is now a constant resistance to motion of 40 N.
   1. When the power of the driving force on the bicycle is a constant 200 W, what constant speed can the cyclist maintain? [3]
   2. Find the power of the driving force on the bicycle when travelling at a speed of 0.5 m s\(^{-1}\) with an acceleration of 2 m s\(^{-2}\). [5]

A particle \(P\) of mass \(m\) kg is attached to one end of a light elastic string of natural length 0·5 m and modulus of elasticity \(\frac{mg}{2}\) N. The other end of the string is attached to a fixed point \(O\) and \(P\) hangs vertically below \(O\).

1. Find the stretched length of the string when \(P\) rests in equilibrium. [3 marks]
2. Find the elastic potential energy stored in the string in the equilibrium position. [2 marks]

\(P\), which is still attached to the string, is now held at rest at \(O\) and then lowered gently into its equilibrium position.

1. Find the work done by the weight of the particle as it moves from \(O\) to the equilibrium position. [2 marks]
2. Explain the discrepancy between your answers to parts (b) and (c). [1 mark]

At time \(t = 0\), a particle \(P\) of mass \(3\) kg is at rest at the point \(A\) with position vector \((j - 3k)\) m. Two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) then act on the particle \(P\) and it passes through the point \(B\) with position vector \((8i - 3j + 5k)\) m. Given that \(\mathbf{F}_1 = (4i - 2j + 5k)\) N and \(\mathbf{F}_2 = (8i - 4j + 7k)\) N and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only two forces acting on \(P\), find the velocity of \(P\) as it passes through \(B\), giving your answer as a vector. [7]

A particle moves from the point \(A\) with position vector \((3i - j + 3k)\) m to the point \(B\) with position vector \((i - 2j - 4k)\) m under the action of the force \((2i - 3j - k)\) N. Find the work done by the force. [4]

A particle of mass 0.5 kg is at rest at the point with position vector \((2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k})\) m. The particle is then acted upon by two constant forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\). These are the only two forces acting on the particle. Subsequently, the particle passes through the point with position vector \((4\mathbf{i} + 5\mathbf{j} - 5\mathbf{k})\) m with speed 12 m s\(^{-1}\). Given that \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j} - \mathbf{k})\) N, find \(\mathbf{F}_2\). [9]

A particle moves from the point \(A\) with position vector \((3\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m to the point \(B\) with position vector \((\mathbf{i} - 2\mathbf{j} - 4\mathbf{k})\) m under the action of the force \((2\mathbf{i} - 3\mathbf{j} - \mathbf{k})\) N. Find the work done by the force. [4]

The graph shows the resistance force experienced by a cyclist over the first 20 metres of a bicycle ride. \includegraphics{figure_2} Find the work done by the resistance force over the 20 metres of the bicycle ride. Circle your answer. [1 mark] 1600 J \quad 3000 J \quad 3200 J \quad 4000 J

In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) Martin, who is of mass 40 kg, is using a slide. The slide is made of two straight sections \(AB\) and \(BC\). The section \(AB\) has length 15 metres and is at an angle of \(50°\) to the horizontal. The section \(BC\) has length 2 metres and is horizontal. \includegraphics{figure_6} Martin pushes himself from \(A\) down the slide with initial speed \(1\,\text{m}\,\text{s}^{-1}\) He reaches \(B\) with speed \(5\,\text{m}\,\text{s}^{-1}\) Model Martin as a particle.

1. Find the energy lost as Martin slides from \(A\) to \(B\). [4 marks]
2. Assume that a resistance force of constant magnitude acts on Martin while he is moving on the slide.
   1. Show that the magnitude of this resistance force is approximately 270 N [2 marks]
   2. Determine if Martin reaches the point \(C\). [3 marks]

In this question use \(g = 9.8 \text{ m s}^{-2}\) A ball of mass 0.5 kg is projected vertically upwards with a speed of \(10 \text{ m s}^{-1}\)

1. Calculate the initial kinetic energy of the ball. [1 mark]
2. Assuming that the weight is the only force acting on the ball, use an energy method to show that the maximum height reached by the ball is approximately 5.1 m above the point of projection. [2 marks]
3. 1. A student conducts an experiment to verify the accuracy of the result obtained in part (b). They observe that the ball rises to a height of 4.4 m above the point of projection and concludes that this height difference is due to a resistance force, \(R\) newtons. Find the total work done against \(R\) whilst the ball is moving upwards. [2 marks]
   2. Using a model that assumes \(R\) is constant, find the magnitude of \(R\) [2 marks]
   3. Comment on the validity of the model used in part (c)(ii). [1 mark]

A particle \(P\) of mass \(3\) kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = (32\sinh(2t))\mathbf{i} + (32\cosh(2t) - 257)\mathbf{j}.$$

1. 1. By considering kinetic energy, determine the work done by \(\mathbf{F}\) over the interval \(0 \leqslant t \leqslant \ln 2\). [5]
   2. Explain the significance of the sign of the answer to part (a)(i). [1]
2. Determine the rate at which \(\mathbf{F}\) is working at the instant when \(P\) is moving parallel to the \(\mathbf{i}\)-direction. [6]

In this question the box should be modelled as a particle. A box of mass \(m\) kg is placed on a rough slope which makes an angle of \(\alpha\) with the horizontal.

1. Show that the box is on the point of slipping if \(\mu = \tan \alpha\), where \(\mu\) is the coefficient of friction between the box and the slope. [2]

A box of mass 5 kg is pulled up a rough slope which makes an angle of 15° with the horizontal. The box is subject to a constant frictional force of magnitude 3 N. The speed of the box increases from 2 m s\(^{-1}\) at a point A on the slope to 5 m s\(^{-1}\) at a point B on the slope with B higher up the slope than A. The distance AB is 10 m. \includegraphics{figure_6} The pulling force has constant magnitude \(P\) N and acts at a constant angle of 25° above the slope, as shown in the diagram.

1. Use the work–energy principle to determine the value of \(P\). [5]

One end of a rope is attached to a block A of mass 2 kg. The other end of the rope is attached to a second block B of mass 4 kg. Block A is held at rest on a fixed rough ramp inclined at \(30°\) to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P, at a distance \(d\) m above the ground, as shown in the diagram. \includegraphics{figure_7} Block A is more than \(d\) m from P. The blocks are released from rest and A moves up the ramp. The coefficient of friction between A and the ramp is \(\frac{1}{2\sqrt{3}}\). The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.

1. Determine, in terms of \(g\) and \(d\), the work done against friction as A moves \(d\) m up the ramp. [3]
2. Given that the speed of B immediately before it hits the ground is \(1.75 \text{ m s}^{-1}\), use the work–energy principle to determine the value of \(d\). [5]
3. Suggest one improvement, apart from including air resistance, that could be made to the model to make it more realistic. [1]

A car of mass 850 kg is travelling along a straight horizontal road. The power developed by the car is constant and is equal to 18 kW. There is a constant resistance to motion of magnitude 600 N.

1. Find the greatest steady speed at which the car can travel. [2]

Later in the journey, while travelling at a speed of \(15 \text{ m s}^{-1}\), the car comes to the bottom of a straight hill which is inclined at an angle of \(\sin^{-1}\left(\frac{1}{40}\right)\) to the horizontal. The power developed by the car remains constant at 18 kW. The magnitude of the resistance force is no longer constant but changes such that the total work done against the resistance force in ascending the hill is 103 000 J. The car takes 10 seconds to ascend the hill and at the top of the hill the car is travelling at \(18 \text{ m s}^{-1}\).

1. Determine the distance the car travels from the bottom to the top of the hill. [5]

A particle P of mass 5 kg is released from rest at a point O and falls vertically. A resistance of magnitude \(0.05v^2\) N acts vertically upwards on P, where \(v \text{ m s}^{-1}\) is the velocity of P when it has fallen a distance \(x\) m.

1. Show that \(\left(\frac{100v}{980-v^2}\right)\frac{dv}{dx} = 1\). [2]
2. Verify that \(v^2 = 980(1-e^{-0.02x})\). [4]
3. Determine the work done against the resistance while P is falling from O to the point where P's acceleration is \(8.36 \text{ m s}^{-2}\). [5]

A tractor has a mass of 6000 kg. When developing a power of 5 kW, the tractor is travelling at a steady speed of 2.5 m s\(^{-1}\) across a horizontal field.

1. Calculate the magnitude of the resistance to the motion of the tractor. [2]

The tractor comes to horizontal ground where the resistance to motion is different. The power developed by the tractor during the next 10 s has an average value of 8 kW. During this time, the tractor accelerates uniformly from 2.5 m s\(^{-1}\) to 3 m s\(^{-1}\).

1. 1. Show that the work done against the resistance to motion during the 10 s is 71 750 J. [4]
   2. Assuming that the resistance to motion is constant, calculate its value. [3]

The tractor can usually travel up a straight track inclined at an angle \(\alpha\) to the horizontal, where \(\sin\alpha = \frac{1}{20}\), while accelerating uniformly from 3 m s\(^{-1}\) to 3.25 m s\(^{-1}\) over a distance of 100 m against a resistance to motion of constant magnitude of 2000 N. The tractor develops a fault which limits its maximum power to 16kW.

1. Determine whether the tractor could now perform the same motion up the track. [You should assume that the mass of the tractor and the resistance to motion remain the same.] [7]

Geraint is a cyclist competing in a race along the Taff Trail. The Taff Trail is a track that runs from Cardiff Bay to Brecon. The chart below shows the altitude (height above sea level) along the route. \includegraphics{figure_4} Geraint starts from rest at Cardiff Bay and has a speed of \(10\) ms\(^{-1}\) when he crosses the finish line in Brecon. Geraint and his bike have a total mass of \(80\) kg. The resistance to motion may be modelled by a constant force of magnitude \(16\) N.

1. Given that \(1440\) kJ of energy is used in overcoming resistances during the race,
   1. find the length of the track,
   2. calculate the work done by Geraint. [8]
2. The steepest section of the track may be modelled as a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{2}{7}\). \includegraphics{figure_4b} Geraint is capable of producing a maximum power of \(250\) W. Find the maximum speed that Geraint can attain whilst travelling on this section of the track. [5]

Relative to a fixed origin \(O\), the position vector \(\mathbf{r}\) m at time \(t\) s of a particle \(P\), of mass 0.4 kg, is given by $$\mathbf{r} = e^{2t}\mathbf{i} + \sin(2t)\mathbf{j} + \cos(2t)\mathbf{k}.$$

1. Show that the velocity vector \(\mathbf{v}\) and the position vector \(\mathbf{r}\) are never perpendicular to each other. [6]
2. Given that the speed of \(P\) at time \(t\) is \(v\) ms\(^{-1}\), show that $$v^2 = 4e^{4t} + 4.$$ [2]
3. Find the kinetic energy of \(P\) at time \(t\). [1]
4. Calculate the work done by the force acting on \(P\) in the interval \(0 < t < 1\). [2]
5. Determine an expression for the rate at which the force acting on \(P\) is working at time \(t\). [2]

At a demolition site, bricks slide down a straight chute into a container. The chute is rough and is inclined at an angle of \(30°\) 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 \(5\) m s\(^{-1}\).

1. Find the potential energy lost by the brick in moving down the chute.

(2)

1. By using the work-energy principle, or otherwise, find the constant frictional force acting on the brick as it moves down the chute.

(5)

1. Hence find the coefficient of friction between the brick and the chute.

(3) Another brick of mass \(3\) kg slides down the chute. This brick is given an initial speed of \(2\) m s\(^{-1}\) at the top of the chute.

1. Find the speed of this brick when it reaches the bottom of the chute.

(5)