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Work, energy and power

Publisher Summary

This chapter discusses the concept of work, energy, and power. If a body moves as a result of a force being applied to it, the force is said to do work on the body. The amount of work done is the product of the applied force and the distance. The unit of work is the joule, J, which is defined as the amount of work done when a force of one Newton acts for a distance of one meter in the direction of the force. Energy is the capacity, or ability, to do work. Energy is also measured in joule and it expended when work is done. There are several forms of energy, such as mechanical energy, chemical energy, heat or thermal energy, nuclear energy, electrical energy, light energy, and sound energy. Energy can be converted from one form to another. On the other hand, power is a measure of the rate at which work is done or at which energy is converted from one form to another. The unit of power is the watt, W, where one watt is equal to one joule per second. The watt is a small unit for many purposes and a larger unit called the kilowatt, kW, is used, where one kW = 1000 W.

1. Fuel, such as oil, coal, gas or petrol, when burnt, produces heat. Heat is a form of energy and may be used, for example, to boil water or to raise steam. Thus fuel is useful since it is a convenient method of storing energy, that is, fuel is a source of energy.

(i) If a body moves as a result of a force being applied to it, the force is said to do work on the body. The amount of work done is the product of the applied force and the distance, i.e.

$Work done = force x distance moved in the direction of the force$ $Work done = force x distance moved in the direction of the force$

(ii) The unit of work is the joule, J, which is defined as the amount of work done when a force of 1 Newton acts for a distance of 1 metre in the direction of the force.

$Thus, 1 J = l Nm .$ $Thus, 1 J = l Nm .$

3. If a graph is plotted of experimental values of force (on the vertical axis) against distance moved (on the horizontal axis) a force-distance graph or work diagram is produced. The area under the graph represents the work done.

For example, a constant force of 20 N used to raise a load a height of 8 m may be represented on a force-distance graph as shown in Figure 37.1(a). The area under the graph shown shaded, represents the work done.

$Hence, work done = 20 N x 8 m = 160 J$ $Hence, work done = 20 N x 8 m = 160 J$

Similarly, a spring extended by 20 mm by a force of 500 N may be represented by the work diagram shown in Figure 37.1(b).

$\begin{array}{l} Work done=shaded area = \frac{1}{2} base×height \\ = \frac{1}{2} ( 2 0 \times 10^{-3}) m \times 500 N = 5 J \end{array}$ $\begin{array}{l} Work done=shaded area = \frac{1}{2} base×height \\ = \frac{1}{2} ( 2 0 \times 10^{-3}) m \times 500 N = 5 J \end{array}$

4. Energy is the capacity, or ability, to do work. The unit of energy is the joule, the same as for work. Energy is expended when work is done.

5. There are several forms of energy and these include:

(i) Mechanical energy;

(ii) Heat or thermal energy;

(iii) Electrical energy;

(iv) Chemical energy;

(v) Nuclear energy;

(vi) Light energy;

(vii) Sound energy.

6. Energy may be converted from one form to another. The principle of conservation of energy states that the total amount of energy remains the same in such conversions, i.e. energy cannot be created or destroyed. Some examples of energy conversions include:

(i) Mechanical energy is converted to electrical energy by a generator.

(ii) Electrical energy is converted to mechanical energy by a motor.

(iii) Heat energy is converted to mechanical energy by a steam engine.

(iv) Mechanical energy is converted to heat energy by friction.

(v) Heat energy is converted to electrical energy by a solar cell.

(vi) Electrical energy is converted to heat energy by an electric fire.

(vii) Heat energy is converted to chemical energy by living plants.

(vii) Chemical energy is converted to heat energy by burning fuels.

(ix) Heat energy is converted to electrical energy by a thermocouple.

(x) Chemical energy is converted to electrical energy by batteries.

(xi) Electrical energy is converted to light energy by a light bulb.

(xii) Sound energy is converted to electrical energy by a microphone.

(xiii) Electrical energy is converted to chemical energy by electrolysis.

7. Efficiency is defined as the ratio of the useful output energy to the input energy. The symbol for efficiency is η (Greek letter eta).

$Hence, efficiency, η = \frac{useful output energy}{input energy}$ $Hence, efficiency, η = \frac{useful output energy}{input energy}$

Efficiency has no units and is often stated as a percentage. A perfect machine would have an efficiency of 100%. However, all machines have an efficiency lower than this due to friction and other losses. Thus, if the input energy to a motor is 1000 J and the output energy is 800 J then the efficiency is

$\frac{800}{1000} \times 100 %, i . e . 80 %$ $\frac{800}{1000} \times 100 %, i . e . 80 %$

8. Power is a measure of the rate at which work is done or at which energy is converted from one form to another.

$Power P = \frac{energy used}{time taken} (or P = \frac{work done}{time taken})$ $Power P = \frac{energy used}{time taken} (or P = \frac{work done}{time taken})$

The unit of power is the watt, W, where 1 watt is equal to 1 joule per second. The watt is a small unit for many purposes and a larger unit called the kilowatt, kW, is used, where 1 kW = 1000 W. The power output of a motor which does 120 kJ of work is 30 s is thus given by

$P = \frac{120}{30} \frac{kJ}{s} =4 kW .$ $P = \frac{120}{30} \frac{kJ}{s} =4 kW .$

(For electrical power, see page 14.)

9. Since, work done = force × distance

$\begin{array}{l} Then, power = \frac{work done}{time taken} = \frac{force \times distance}{time taken} \\ = force \times \frac{distance}{time taken} \\ However, \frac{distance}{time taken} = velocity . \\ Hence, power = force \times velocity . \end{array}$ $\begin{array}{l} Then, power = \frac{work done}{time taken} = \frac{force \times distance}{time taken} \\ = force \times \frac{distance}{time taken} \\ However, \frac{distance}{time taken} = velocity . \\ Hence, power = force \times velocity . \end{array}$

Thus, for example, if a lorry is travelling at a constant speed of 72 km/h and the force resisting motion is 800 N, then the tractive power necessary to keep the lorry moving at this speed is given by:

$\begin{array}{l} power = force \times velocity = (800 N) (\frac{72}{3.6} m/s) = 16000 \frac{Nm}{s} \\ = 16000 J / s =16 kW \end{array}$ $\begin{array}{l} power = force \times velocity = (800 N) (\frac{72}{3.6} m/s) = 16000 \frac{Nm}{s} \\ = 16000 J / s =16 kW \end{array}$

10.

(i) Mechanical engineering is concerned principally with two kinds of energy, these being potential energy and kinetic energy.

(ii) Potential energy is energy due to the position of a body. The force exerted on a mass of m kg is mg N (where g = 9.81 N/kg, the earth’s gravitational field). When the mass is lifted vertically through a height h m above some datum level, the work done is given by: force × distance = (mg) (h) J. This work done is stored as potential energy in the mass.

Hence potential energy = mg h joules (the potential energy at the datum level being taken as zero).

(iii) Kinetic energy is the energy due to the motion of a body. Suppose a resultant force F acts on an object of mass m originally at rest and accelerates it to a velocity v in a distance s.

$Work done = force \times distance = F_{s} = (m a) (s),$ $Work done = force \times distance = F_{s} = (m a) (s),$

where a is the acceleration.

$\begin{array}{l} However, υ^{2} = 2 a s . from which a = \frac{υ^{2}}{2 s} \\ Hence, work done = m (\frac{υ^{2}}{2 s}) s = \frac{1}{2} m υ^{2} \end{array}$ $\begin{array}{l} However, υ^{2} = 2 a s . from which a = \frac{υ^{2}}{2 s} \\ Hence, work done = m (\frac{υ^{2}}{2 s}) s = \frac{1}{2} m υ^{2} \end{array}$

This energy is called the kinetic energy of the mass m,

$i .e . kinetic energy= \frac{1}{3} {m v}^{2} joules .$ $i .e . kinetic energy= \frac{1}{3} {m v}^{2} joules .$

For example, at the instant of striking, a hammer of mass 30 kg has a velocity of 15 m/s. The kinetic energy in the hammer is given by:

$Kinetic energy = \frac{1}{2} m v_{2} = \frac{1}{2} (30 k g) {(15 m / s)}^{2} = 3375 J .$ $Kinetic energy = \frac{1}{2} m v_{2} = \frac{1}{2} (30 k g) {(15 m / s)}^{2} = 3375 J .$

11.

(i) Energy may be converted from one form to another. The principle of conservation of energy states that the total amount of energy remains the same in such conversions, i.e. energy cannot be created or destroyed.

(ii) In mechanics, the potential energy possessed by a body is frequently converted into kinetic energy, and vice versa. When a mass is falling freely, its potential energy decreases as it loses height, and its kinetic energy increases as its velocity increases. Ignoring air frictional losses, at all times:

$potential energy + kinetic energy = a constant .$ $potential energy + kinetic energy = a constant .$

(iii) If friction is present, then work is done overcoming the resistance due to friction and this is dissipated as heat. Then,

$\begin{array}{l} potential energy + = final energy \\ kinetic energy + work done overcoming \\ frictional resistance . \end{array}$ $\begin{array}{l} potential energy + = final energy \\ kinetic energy + work done overcoming \\ frictional resistance . \end{array}$

(iv) Kinetic energy is not always conserved in collisions. Collisions in which kinetic energy is conserved (i.e. stays the same) are called elastic collisions, and those in which it is not conserved are termed inelastic collisions.

Kinetic energy of rotation

12.

(i) The tangential velocity v of a particle of mass m moving at an angular velocity ω rad/s at a radius r metres (see Figure 37.2) is given by v = ωr m/s.

(ii) The kinetic energy of a particle of mass m is given by:

$kinetic energy = \frac{1}{2} m υ^{2} = \frac{1}{2} m {(ω r)}^{2} = \frac{1}{2} m ω^{2} r joules .$ $kinetic energy = \frac{1}{2} m υ^{2} = \frac{1}{2} m {(ω r)}^{2} = \frac{1}{2} m ω^{2} r joules .$

(iii) The total kinetic energy of a system of masses rotating at different radii about a fixed axis but with the same angular velocity to, as shown in Figure 37.3, is given by:

$total kinetic energy = \frac{1}{2} m_{1} ω^{2} r_{1}^{2} + \frac{1}{2} m_{2} ω^{2} r_{2}^{2} + \frac{1}{2} m_{3} ω^{2} r_{3}^{2}$ $total kinetic energy = \frac{1}{2} m_{1} ω^{2} r_{1}^{2} + \frac{1}{2} m_{2} ω^{2} r_{2}^{2} + \frac{1}{2} m_{3} ω^{2} r_{3}^{2}$

$= (m_{1} r_{1}^{2} + m_{2} r_{2}^{2} + m_{3} r_{3}^{2}) \frac{ω^{2}}{2}$ $= (m_{1} r_{1}^{2} + m_{2} r_{2}^{2} + m_{3} r_{3}^{2}) \frac{ω^{2}}{2}$

In general, this may be written as:

$total kinetic energy = (\sum m r^{2}) \frac{ω^{2}}{2} = I \frac{ω^{2}}{2},$ $total kinetic energy = (\sum m r^{2}) \frac{ω^{2}}{2} = I \frac{ω^{2}}{2},$

where I (= Σ mr²) is called the moment of inertia of the system about the axis of rotation.

The moment of inertia of a system is a measure of the amount of work done to give the system an angular velocity of ω rad/s, or the amount of work which can be done by a system turning at ω rad/s.

$In general, total kinetic energy = I \frac{ω^{2}}{2} = M k^{2} \frac{ω^{2}}{2},$ $In general, total kinetic energy = I \frac{ω^{2}}{2} = M k^{2} \frac{ω^{2}}{2},$

where M (= Σ m) is the total mass and k is called the radius of gyration of the system for the given axis.

If all of the mass were concentrated at the radius of gyration it would give the same moment of inertia as the actual system.

Flywheels

13. The function of a flywheel is to restrict fluctuations of speed by absorbing and releasing large quantities of kinetic energy for small speed variations.

To do this they require large moments of inertia and to avoid excessive mass they need to have radii of gyration as large as possible. Most of the mass of a flywheel is usually in its rim.

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Table of Contents for Chapter 37: Work, energy and power

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Publisher Summary

Kinetic energy of rotation

Flywheels

Table of Contents for
Chapter 37: Work, energy and power