6.02b Calculate work: constant force, resolved component

A car has mass \(1250 \text{ kg}\).

1. The car is moving along a straight level road at a constant speed of \(36 \text{ m s}^{-1}\) and is subject to a constant resistance of magnitude \(850 \text{ N}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car travels at a constant speed up a hill and is subject to the same resistance as in part (i). The hill is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta° = 0.1\), and the engine is working at \(63 \text{ kW}\). Find the speed of the car. [3]
3. The car descends the same hill with the engine of the car working at a constant rate of \(20 \text{ kW}\). The resistance is not constant. The initial speed of the car is \(20 \text{ m s}^{-1}\). Eight seconds later the car has speed \(24 \text{ m s}^{-1}\) and has moved \(176 \text{ m}\) down the hill. Use an energy method to find the total work done against the resistance during the eight seconds. [5]

A car of mass \(1400\text{ kg}\) travelling at a speed of \(v\text{ m s}^{-1}\) experiences a resistive force of magnitude \(40v\text{ N}\). The greatest possible constant speed of the car along a straight level road is \(56\text{ m s}^{-1}\).

1. Find, in kW, the greatest possible power of the car's engine. [2]
2. Find the greatest possible acceleration of the car at an instant when its speed on a straight level road is \(32\text{ m s}^{-1}\). [3]
3. The car travels down a hill inclined at an angle of \(\theta°\) to the horizontal at a constant speed of \(50\text{ m s}^{-1}\). The power of the car's engine is \(60\text{ kW}\). Find the value of \(\theta\). [4]

A particle of mass 13 kg is on a rough plane inclined at an angle of \(\theta\) to the horizontal, where \(\tan \theta = \frac{5}{12}\). The coefficient of friction between the particle and the plane is 0.3. A force of magnitude \(T\) N, acting parallel to a line of greatest slope, moves the particle a distance of 2.5 m up the plane at a constant speed. Find the work done by this force. [5]

\includegraphics{figure_7} The diagram shows the vertical cross-section \(PQR\) of a slide. The part \(PQ\) is a straight line of length \(8\) m inclined at angle \(α\) to the horizontal, where \(\sin α = 0.8\). The straight part \(PQ\) is tangential to the curved part \(QR\) at \(Q\), and \(R\) is \(h\) m above the level of \(P\). The straight part \(PQ\) of the slide is rough and the curved part \(QR\) is smooth. A particle of mass \(0.25\) kg is projected with speed \(15\) m s\(^{-1}\) from \(P\) towards \(Q\) and comes to rest at \(R\). The coefficient of friction between the particle and \(PQ\) is \(0.5\).

1. Find the work done by the friction force during the motion of the particle from \(P\) to \(Q\). [4]
2. Hence find the speed of the particle at \(Q\). [4]
3. Find the value of \(h\). [3]

\includegraphics{figure_4} The diagram shows the vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section, and \(A\) is 5 m higher than \(B\). A particle of mass \(0.35\) kg passes through \(A\) with speed \(7 \text{ m s}^{-1}\), moving on the surface towards \(B\).

1. Assuming that there is no resistance to motion, find the speed with which the particle reaches \(B\). [3]
2. Assuming instead that there is a resistance to motion, and that the particle reaches \(B\) with speed \(11 \text{ m s}^{-1}\), find the work done against this resistance as the particle moves from \(A\) to \(B\). [3]

A particle of mass \(0.3\) kg is released from rest above a tank containing water. The particle falls vertically, taking \(0.8\) s to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is \(1.25\) m. The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is \(1.2\) J.

1. Use an energy method to find the speed of the particle when it reaches the bottom of the tank. [4]

When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed \(7\) m s\(^{-1}\). As the particle rises through the water, it experiences a constant resistance force of \(1.8\) N. The particle comes to instantaneous rest \(t\) seconds after it bounces on the bottom of the tank.

1. Find the value of \(t\). [7]

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 constant force acts on a particle of mass 200 grams, moving it 50 cm in a straight line on a rough horizontal surface at a constant speed. The coefficient of friction between the particle and the surface is \(\frac{1}{4}\). Calculate, in J, the work done by the force. [4 marks]

\(\mathbf{i}\) 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 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 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]

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 small bead is threaded on a smooth, straight horizontal wire which passes through the point \(A(-3, 1)\) and the point \(B(2, 5)\) in the \(x\)-\(y\) plane. The bead moves under the action of a horizontal force \(\mathbf{F}\) of magnitude \(8.5\) N whose line of action is parallel to the line with equation \(15x - 8y + 4 = 0\). The unit on both the \(x\) and \(y\) axes has length one metre. Find the work done by \(\mathbf{F}\) as it moves the bead from \(A\) to \(B\). [8]

A bead of mass 0.125 kg is threaded on a smooth straight horizontal wire. The bead moves from rest at the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j} - \mathbf{k})\) m relative to a fixed origin \(O\) to a point with position vector \((3\mathbf{i} - 4\mathbf{j} - \mathbf{k})\) m relative to \(O\) under the action of a force \(\mathbf{F} = (14\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) N. Find

1. the work done by \(\mathbf{F}\) as the bead moves from \(A\) to \(B\), [3]
2. the speed of the bead at \(B\). [2]

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

Use \(g\) as 9.8 m s\(^{-2}\) in this question. A pump is used to pump water out of a pool. The pump raises the water through a vertical distance of 5 metres and then ejects it through a pipe. The pump works at a constant rate of 400 W Over a period of 50 seconds, 300 litres of water are pumped out of the pool and the water is ejected with speed \(v\) m s\(^{-1}\) The mass of 1 litre of water is 1 kg

1. Find the gain in the potential energy of the 300 litres of water. [1 mark]
2. Calculate \(v\) [4 marks]

In this question use \(g = 9.8 \text{ m s}^{-2}\) A lift is used to raise a crate of mass 250 kg The lift exerts an upward force of magnitude \(P\) newtons on the crate. When the crate is at a height of \(x\) metres above its initial position $$P = k(x + 1)(12 - x) + 2450$$ where \(k\) is a constant. The crate is initially at rest, at the point where \(x = 0\)

1. Show that the work done by the upward force as the crate rises to a height of 12 metres is given by $$29400 + 360k$$ [3 marks]
2. The speed of the crate is \(3 \text{ m s}^{-1}\) when it has risen to a height of 12 metres. Find the speed of the crate when it has risen to a height of 15 metres. [5 marks]
3. Find the height of the crate when its speed becomes zero. [2 marks]
4. Air resistance has been ignored. Explain why this is reasonable in this context. [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 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]

A particle \(P\) of mass 0.5 kg is in equilibrium under the action of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\). $$\mathbf{F}_1 = (9\mathbf{i} + 6\mathbf{j} - 12\mathbf{k})\text{N} \quad \text{and} \quad \mathbf{F}_2 = (6\mathbf{i} - 7\mathbf{j} + 3\mathbf{k})\text{N}.$$

1. Find the force \(\mathbf{F}_3\). [2]
2. Forces \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are removed so that \(P\) moves in a straight line \(AB\) under the action of the single force \(\mathbf{F}_1\). The points \(A\) and \(B\) have position vectors \((2\mathbf{i} - 9\mathbf{j} + 7\mathbf{k})\) m and \((8\mathbf{i} - 5\mathbf{j} - \mathbf{k})\) m respectively. The particle \(P\) is initially at rest at \(A\).
   1. Verify that \(\mathbf{F}_1\) acts parallel to the vector \(\overrightarrow{AB}\).
   2. Find the work done by the force \(\mathbf{F}_1\) as the particle moves from \(A\) to \(B\).
   3. By using the work-energy principle, find the speed of \(P\) as it reaches \(B\). [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]