5.1 Energy: Conservation, Work, Power & Efficiency

A small mass is placed at point P on the inside surface of a smooth hemisphere. It is then released from rest. When it reaches the lowest point T, its speed is 4.0 m s^–1.

The diagram (not to scale) shows the speed of the mass at other points Q, R and S as it slides down. Air resistance is negligible.

The mass loses potential energy E in falling from P to T.

At which point has the mass lost potential energy   ?

A wind turbine has blades that sweep an area of 2000 m². It converts the power available in the wind to electrical power with an efficiency of 50 %.

 What is the electrical power generated if the wind speed is 10 m s^–1? (The density of air is 1.3 kg m^–3)

The figure shows a mass m on an inclined plane connected by an inextensible string over a smooth pulley to a larger mass M. When the mass M falls through a distance x, the mass m moves up a vertical distance y at a constant speed.

 The friction between m and the inclined plane is F.

What amount of energy is converted to heat?

The diagram shows a wheel of circumference 0.30 m. A rope is fastened at one end to a force meter. The rope passes over the wheel and supports a freely hanging load of 100 N. The wheel is driven by an electric motor at a constant rate of 50 revolutions per second.

 When the wheel is turning at this rate, the force meter reads 20 N.

What is the output power of the motor?

The figure below shows two blocks P and Q, initially stationary and connected to each other by a taut string passing over a smooth pulley. The system is released from rest and block Q moves vertically downwards from position A to B.

Which of the following statements is correct?

A boy on a boat P pulls on a rope with a constant force F over a duration of time t. The other end of the rope is either tied to an anchor on the pier (Diagram a) or a freely floating boat Q (Diagram b) of equal mass as shown in the figures shown.

W_a and W_b represent the total work done by the boy during the time t and P_a and P_b represent the average power output of the boy in diagrams (a) and (b) respectively.

 Which of the following statements is correct?

What is the power required to give a body of mass m a forward acceleration a when it is moving with velocity v up a frictionless track inclined at an angle θ to the horizontal?

A ball of mass m falls freely from rest. When it has reached a speed v, it strikes a vertical spring.

 The spring is compressed by a distance y before the ball moves upwards again.

Assume that all energy the ball loses becomes elastic potential energy in the spring.

 What is the average force exerted by the spring during its compression?

Trains supply coal to a power station. The table below gives quantities describing the operation of the power station.

Which expression gives efficiency of the power station?

Water flows from a lake into a turbine that is a vertical distance of 90 m below the lake, as shown

The mass flow rate of the water is 2400 kg min^–1. The turbine has an efficiency of 75 %.

 What is the output power of the turbine?

Initially, four identical uniform blocks, each of mass m and thickness h, are spread on a table.

How much work is done on the blocks in stacking them on top of one another?

The first column in the table gives four examples of work being done. The second column gives more detail of the action.

 Which row is not correct?

The data below are taken from a test of a petrol engine for a motor car.

Which expression will evaluate the efficiency of the engine?

A constant force F, acting on a car of mass m, moves the car up a slope through a distance s at constant velocity v. The angle of the slope to the horizontal is α.

Which expression gives the efficiency of the process?

A weight W hangs from a trolley that runs along a rail. The trolley moves horizontally through a distance p and simultaneously raises the weight through a height q. 

As a result, the weight moves through a distance r from X to Y. It starts and finishes at rest.

 How much work is done on the weight during this process?

A bungee jumper has 24 kJ of gravitational potential energy at the top of his jump. He is attached to an elastic rope which starts to stretch after a short time of free fall. Assume that energy loss through air resistance is negligible.

What are the possible values when the jumper is half-way down?

The diagram shows the design of a water wheel which drives a generator to produce electrical energy. The flow rate of the water is 200 kg s^–1. The generator supplies a current of 32 A at a voltage of 230 V.

Ignoring any changes in kinetic energy of the water, what is the efficiency of the system?

A conveyor belt is driven at velocity V by a motor. Sand drops vertically on to the belt at a rate of .

What is the additional power needed to keep the conveyor belt moving at a steady speed when the sand starts to fall on it ?

The force resisting the motion of a car is taken as being proportional to the square of the car's speed. The magnitude of the force at a speed of  is 800 N.

What effective power is required from the car's engine to maintain a steady speed of ?

A mass is raised vertically. In a time t, the increase in its gravitational potential energy is and the increase i its kinetic energy is .

what is the average power input to the mass?