4.6 Work, Energy & Power

A cyclist rides along a road up an incline at a steady speed of 5.0 m s^–1. The mass of the rider and bicycle is 70 kg and the bicycle travels 27 m along the road for every 2.0 m gained in height. Neglect energy loss due to frictional forces.

Calculate the power developed by the cyclist in riding up the slope.

The cyclist now travels along a steeper incline at an angle of 6.5º to the horizontal. 

Calculate the power output required from the cyclist on this new plane and discuss

how this value differs from that in part (a).

The cyclist stops pedalling and the bicycle freewheels up the incline for a short time. 

Considering the energy transfers, calculate the distance travelled along the slope from when the cyclist stops pedalling to where the bicycle comes to rest.

The cyclist decides to become a daredevil and use a cycle ramp to perform some stunts on their motorbike instead. They leave the end of the ramp at 38 m s^–1 and travel a horizontal distance of 73 m, measured from the end of the ramp to the maximum height, in 2 s. 

   Show that the maximum height they reach is 5.7 m. 

   You may ignore effects due to friction and air resistance.

A microlight is a small aircraft powered by a petrol engine. 

Figure 1 represents the flight path, AB, of a microlight on a short horizontal training flight. 

Figure 1

On its outward journey, the wind velocity is 9.2 m s^–1 due North and the resultant velocity of the microlight is 15 m s^–1 in a direction θ West of North, so that it travels along AB. 

If the aircraft is pointed due West, show that the angle θ must be 52º.

The work done by the engine if the aircraft travels 20 km on its outward journey is equal to 25.7 MJ. 

Calculate the output power of the aircraft engine for the outward journey.

The microlight aircraft now drops a package of mass 2 kg that is 90 m away on the ground as shown in Figure 2. 

Figure 2

Calculate the gravitational potential energy of the package just before it leaves the aircraft. 

Ignore the effects of air resistance.

The package is carried on a ship with sail systems used to reduce the running costs of cargo ships. The sail and ship’s engines work together to power the ship. 

One of these sails is shown in Figure 3 below, pulling at an angle of 50° to the horizontal. 

Figure 3

When the sail and the engines are operating, the ship is travelling at a steady speed and the force from the engine is 0.33 MN. 

35% of the ship’s power requirement is provided by the wind when the ship is travelling at this speed. 

   Calculate the average tension in the cable.

Figure 1 shows an object at rest at the top of a straight slope which makes a fixed angle with the horizontal at a distance h above the ground. 

Figure 1

The object is released and slides down the slope from A to B with negligible friction. Assume that the potential energy is zero at B. 

Sketch a graph showing: 

• The variation of potential energy along the slope. Label this P.
• The variation of kinetic energy of the object along the slope. Label thisK.
• The variation of kinetic energy along the slope when there is a constant frictional force between the object and the surface. Label this F. 

Explain the important features of the graphs you have drawn.

In a theme park ride, a cage containing passengers falls freely a distance of x m from A to B and travels in a circular arc of radius 25 m from B to C. 

Brakes are applied at C after which the cage with its passengers travels 70 m along an upward sloping ramp and comes to rest at D. The track, together with relevant distances, is shown in the diagram. CD makes an angle of θ with the horizontal. 

Figure 2

The force required for circular motion on a passenger of mass 70 kg at C is 5.4 kN. 

Calculate height x. 

Assume that friction is negligible between A and C.

The total mass of the cage and passengers is 720 kg. The average resistive force exerted by the brakes between C and D is 4.8 kN.

By considering energy changes, calculate the value of θ.

A small ball of mass 40.0 g is held at the edge of a slope with a horizontal distance of 14.0 cm as shown in Figure 1 below. 

Figure 1

The ball is released from rest at A and rolls down the smooth slope AB. 

Calculate the speed of the ball at B, the tip of the end of the slope.

A box of mass 120 g waits below the slope. The ball moves off from B and lands on the box. The box then travels a distance of 18 cm before coming to a stop. 

Assume that the ball lands on the box and sticks to the box without rebounding. 

   Calculate the friction force from the ground that acts on the box as it travels this distance.

Once the ball and box come to rest, they then experience a variable force placed upon them shown by Figure 2. 

Figure 2

The distance is measured from where the ball and box initially came to rest. 

Show that the final velocity of the box after 50 m is 17 m s^–1.

The diagram below shows a possible design for a pumped storage system used to generate electricity. 

Figure 1

Water from the upper reservoir is to fall through a vertical distance of 120 m before reaching a powerplant chamber. The water rotates a turbine in the chamber that drives an electricity generator. After leaving the turbine, the water travels through an exit pipe to a lake.

Show that the maximum possible speed of the water as it arrives at the turbine is about 50 m s⁻¹.

The volume of water flowing into the turbine every second is 4.5 m³. 

Estimate the radius of the intake pipe that is required for the system.

The water leaves the powerplant chamber at a speed of 19 m s⁻¹. 

Calculate the maximum possible power output of the turbine and generator. 

Give an appropriate unit for your answer.

Density of water = 1000 kg m⁻³.

Energy losses are estimated to reduce the output power for the turbine and generator to 60% of the value you calculated in part (c). 

Explain two possible reasons for this energy loss.

A parcel of mass 23 kg drops from a delivery chute onto a conveyor belt as shown in Figure 1. The belt is moving at a steady speed of 1.5 m s⁻¹. 

The parcel lands on the moving belt with negligible speed and initially starts to slip. It takes 0.75 s for the parcel to gain enough speed to stop slipping and move at the same speed as the conveyor belt. 

Figure 1

Calculate the change in kinetic energy of the parcel during the first 0.75 s.

The average horizontal force acting on the parcel during the first 0.75 s is 50 N. 

Calculate the horizontal distance between the parcel and the end of the delivery chute 0.75 s after the parcel lands on the conveyor belt. 

Assume that the parcel does not reach the end of the conveyor belt.

At a later stage the parcel is being raised by another conveyor belt as shown in Figure 2. 

Figure 2

This conveyor belt is angled at 21° to the horizontal and the parcel moves at a steady speed of 1.5 m s⁻¹ without slipping. 

Calculate the rate at which work is done on the parcel.

The efficiency of the angled conveyor belt acceleration system is 66%. 

Calculate the average power input to this system during the acceleration.

An electric wheelchair, powered by a battery, allows the user to move around independently. 

One type of electric wheelchair has a mass of 25 kg. The maximum distance it can travel on level ground is 15 km when carrying a user of mass 75 kg and travelling at its maximum speed of 2.5 m s⁻¹. 

The battery used provides an emf of 12 V and can deliver 9.8 × 10⁴ C as it discharges fully. 

Show that the average power output of the battery during the journey is about 200 W.

During the journey, forces due to friction and air resistance act on the wheelchair and its user. 

Assume that all the energy available in the battery is used to move the wheelchair and its user during the journey. 

Calculate the total mean resistive force that acts on the wheelchair and its user.

Figure 1 shows the wheelchair and its user travelling up a hill. The hill makes an angle of 6.5° to the horizontal. 

Figure 1

Calculate the maximum speed of the wheelchair and its user when travelling up this hill when the power output of the battery is 200 W. 

Assume that the resistive forces due to friction and air resistance are the same as in part (b).

Explain how and why the maximum range of the wheelchair on level ground is affected by 

• The mass of the user
• The speed at which the wheelchair travels

Figure 1 shows a roller coaster car which is accelerated from rest to a top speed of 60 m s^–1 on a horizontal track, A, before ascending the steep part of the track. The roller coaster car then becomes stationary at C, the highest point of the track. 

The total mass of the car and passengers is 7900 kg. 

Figure 1

At B, there is an accelerator placed on the track which provides a force of 5500 N to ensure the rollercoaster car continues to travel at top speed and make it over the top of the track. The angle of the track at B is 28° to the horizontal. The rollercoaster travels at this angle for a distance of 1.5 m. 

Calculate the work done by the accelerator when the rollercoaster travels through position B as shown in Figure 1.

Calculate the maximum height above A that would be reached by the car and passengers if all the kinetic energy could be transferred to gravitational potential energy.

The car does not reach the height calculated in part (b). 

Explain the main reason why the car does not reach this height.

The car reaches point C which is at a height of 155 m above A. 

Calculate the speed that the car would reach when it descends from rest at C to its original height from the ground at D if 85% of its energy at C is converted to kinetic energy.