5.1 Energy: Conservation, Work, Power & Efficiency

Engineers around the world are working on ways to provide enough energy to sustain human activity.

Off-shore wind farms use the kinetic energy of strong offshore winds to generate electricity.

Fig. 1.1 shows an offshore wind power installation.

State the definition of power and give the units used to measure power output.

The UK is a world leader in the installation of offshore wind power.

Located in the North Sea off the east coast of England, the Hornsea Phase 2 (Hornsea Two) offshore wind farm was the largest wind farm in the world when it opened in 2022.

It uses 165 turbines rated at 8 MW each, to generate 1.4 GW of energy, enough to supply more than 1.3 million homes. Each turbine has blades of length 81 m.

The power output of a single turbine can be calculated using

where ρ is the density of the air, A is the area swept out by the blades and v is the wind speed.

Calculate the average wind speed at Hornsea Two.

The density of air is 1.225 kg m⁻³.

The Hornsea wind turbines are found to have an efficiency of 40%.

A hybrid electric bike is fitted with both pedals and a battery. The cyclist can use pedals only, pedals and battery, or battery only as they ride. Fig 1.1 shows a cyclist riding a hybrid electric bike up an inclined road.

At one hill the cyclist switches to battery only. A road sign at the base of the hill states that it has an incline of 5%, meaning that for every 100 m travelled horizontally, the height increases by 5 m.

The bike travels at a steady speed of 6.0 m s⁻¹. 

The mass of the rider and bike combined is 72.8 kg. The distance from the base to the top of the hill is 35 m.

Calculate the minimum power required from the battery. Assume that there is no energy loss due to friction.

Near the top of the hill, the cyclist stops pedalling and the bicycle freewheels for 1.6 s until coming to a stop.

The frictional forces which slow down the motion are constant.

Once the cyclist reaches the top, they turn off the motor and head back down to the bottom. At the bottom of the hill, they reach a velocity of 8.8 m s⁻¹.

Fig. 1.1 shows a ride at a water park. Customers are seated in a circular glider which is subject to an accelerating force at point A.

The first part of the ride is horizontal. In this section, the glider is accelerated from rest at point A to 30 m s⁻¹ at point B.

At point B, the track is angled at 30° to the horizontal for 2.5 m.

The total mass of the glider and passengers is 350 kg.

Calculate the work done by the glider against gravity when it travels through point B.

At point C, the glider stops momentarily before sliding down the incline until it lands in the water pool at point D with a speed of 22 m s⁻¹.

Determine the efficiency of the energy transfer between points C and D.

Once the glider reaches point D, the water applies a decelerating force until it comes to a stop after 8.5 m.

Calculate the decelerating force exerted on the glider by the water.   

State any assumptions you made in your calculation.

Pumped hydroelectric systems store large bodies of water behind a dam. When electricity is needed in the grid, the dam is lifted, allowing water to flow from the upper reservoir.

One such power station, known as the Cruachan power station, is shown in Fig. 1.1. This facility has been producing hydroelectric power since 1965.

The vertical distance between the base of the upper reservoir and the turbine is 396 m. This arrangement allows Cruachan power station to produce a power output of 0.62 GW with an efficiency of 76%.

Determine the energy transferred by the water each second in order to achieve this power output.

Water is made to flow out of the upper reservoir through connecting pipes at a rate of 200 m³ per second. The density of water is 1.0 × 10³ kg m⁻³. 

Estimate the depth of the water held in the upper reservoir. State any assumptions made.