Work and Energy

1 Lecturer Arshad Ali 0333-6504720

WORK AND ENERGYWork“The scalar product of force and displacement is called work.”Work Done by Constant ForceConsider an object which is being displaced (or pulled) by a constant force at angle to the direction of motion. This force produces a displacement . Under the action of this force the object moves from A to B. then work will be

The quantity ( ) is the component of force in the direction of the displacement .

ResultThus, the work done on a body by constant force is defined as the product of magnitudes of the displacement and the component of the force in the direction of displacement.

← The amount of work done on the pail when a person holding the pail by the force is moving forward.In this case force is being applied perpendicular to the direction of motion of the pail. So the angle between and is 90o

Fig.

= Work = = 0

← The amount of work done on the wall as shown in figure.


Fig.When a wall is pushed by the force, the displacement will remain zero.Since d = 0 therefore.

Work = =Work = 0

Graphical ExplanationIf we plot a graph between and displacement , where is constant force (measure in Newton). In this graph the distance d is taken along x-axis and F is taken along y-axis.

Fig.From the graph we see that, it is straight line showing that force does not change.If the constant force (N) and the displacement (m) are in the same direction then work done is Fd (Joule). The area under the straight line represents the work done by the force.If is not in the direction of displacement, the graph will be plotted between and d. In this fig. we will take the component of F along the direction of d as

Work = d

Important Points about the Work1. Work is scalar quantity2. Work is positive when <3. If = then Work = =04. If > , the work done is said to be negative

e.g., if =180180

= -1 Work = -Fd

5. The unit of work in SI system is N m or Joule.Joule “ The work is said to be one joule if a force of one newton displaces a body through one meter.” Work Done by a Variable Force


Consider a particle moving in xy-plane under the action of variable force, the particle moves from point a to point b.In order to find the work done on the particle moving along the path ab, we divide this path into n small segments having displacements , , , … . The forces acting during theses

displacements are , , … . During each small interval (or displacement) the force is supposed to nearly constant.The work done for displacement is

For the displacement,

Similarly

Fig. aThe total work done is given by

+ + +…++ + +…+

Graphical ExplanationIf we plot a graph between F and d, it will be of the form shown in fig. b.

Fig. bThe value of F at the start of each interval is represented by dotted horizontal lines.The area of the shaded rectangle = Which is the work done during the distance into a large number of vertical strips called rectangle. The length of each strip (or rectangle) is F and width .Sum of areas of all strips = + + +…+

If approaches to zero then exact result for work done is


From the fig. b we see that total area of rectangles approaches the area between the curve and d-axis from a to b as shown in the shaded portion.ResultThus the work done by variable force in moving a particle between two points is equal to the area under the verses d curve between the two points a and b as shown in the fig. c

Fig. cGravitational FieldThe space around the earth in which its gravitational force acts on a body is called gravitational filed.Work done by Gravitational FieldConsider a body of mass m placed inside the gravitational field under the action of the gravitational force. This force is equal to the weight, mg of the body.The body is displaced from A to B with constant velocity. The body reaches the point B by following the different paths as shown in the figure 1. The work done by the gravitational force along the path ABD can be split into two paths as

……………….1

Fig.

= =

=

Put these values in eq. 1

Similarly if we consider the path ACB then……………………..2


Also,

Put these values of and in eq.2

ConclusionThe amount of work done along the paths ADB and ACB is same i.e., -mghNow If the Path is CurvedThis curved path is broken into horizontal and vertical steps as shown in the figure.

Fig.There is no work done along the horizontal steps because mg is perpendicular to the displacement. Now the work done by the force, mg will be only along the vertical displacements.

The net amount of work done for curved path is still –mgh.ResultThus the work done in the gravitational field is independent of the path followed.Conservative FieldThe field in which the work done is independent of the path followed or work done in a closed path is zero is called conservative filed.ExamplesGravitational field, electric field, magnetic filed.Conservative Forces

1. Gravitational Force2. Electric Force


3. Magnetic Force.Non-Conservative FieldThe field in which work done in moving a body between two points depends upon the path followed by the body between the two points.Non-Conservative ForceThe frictional force is a non-conservative force because if an object is moved over a rough surface between two points along different paths, the work done against the frictional force certainly depends on the path followed.Other examples of Non-Conservative Force

1. Air resistance2. Tension in string3. Propulsion force of a motor4. Propulsion force of a rocket5. Normal force or reaction.

Q. Prove that work done along the closed path is zero.AnswerWe can prove the work done along a closed path such as ADBA is zero as shown in fig.1

Fig.1

From the fig. we see that

Similarly we can prove for a closed path such as ACBA is zero

ResultThus the work done in moving a body along a closed path is zero.Such field is called conservative field or closed field. Also gravitational field is a conservative field.Power“The rate of doing work is called Power” OR“Power is the measures of the rate at which work is being done”Explanation


If work is done in a time interval , then the average power during the interval Δt is defined as

avΔWPΔt

If work is expressed as a function of time, the instantaneous power P at any instant is defined as




Where is the work done in short interval of time

following the instant t.Relation between Power and VelocityIt is sometimes, convenient to express power in term of constant force acting on an object moving at constant velocity .Example


When the propeller of a motor boat causes the water to exert a constant force on the boat, it


moves with a constant velocity . The power delivered by the motor at ay instant is then given by



So

Since Hence

Units of PowerThe SI unit of power is a watt, defined as one joule of work done in one second.Some times, e.g., in electrical measurements, the unit of work is expressed as watt second. However a commercial unit of electrical energy is kilowatt-hoursKilowatt hour “One kilowatt hour is the work done in one hour by an agency whose power is one kilowatt.”Therefore,

1KWh=1000W×3600sec1KWh=3.6×106=3.6MJ

Energy“Energy of a body is its capacity to do work.”Basic Types of EnergyThere are two basic types of energy

1. Kinetic Energy(K.E.)2. Potential Energy (P.E.)The kinetic energy is possessed by a body due to its motion and is given by the formula.

Where m is the mass of the body moving with velocity . The potential energy is possessed by a body because of its position in a force filed e.g., gravitational filed or because of its constrained state. The potential energy due to gravitational filed near the surface of the earth at a height h is given by the formula.

P.E. = mghThis is called gravitational potential energy. The gravitational P.E. is always determined relative to some arbitrary position which is assigned the value of zero P.E. The earth’s surface or a point at infinity from the earth can be choose as zero reference level of gravitational P.E. In the present case this reference level is the surface of the earth as position of zero P.E.The energy stored in a compressed spring is the potential energy possessed by the spring due to its compressed or stretched state. This form of energy is called elastic energy.Work-Energy PrincipleIt states as follows “Work done on the body equals change in its kinetic energy”.ExpressionWhenever work is done on a body, it increases its energy. e.g., a body of mass m is moving with velocity vi. A force acting through a distance d increases the velocity v f, then from equation of motion


Or


From 2nd law of motion


Multiplying eq. 1 and 2 Where the left hand side of the equation gives the work done on the body and right hand side gives

the increase or change in kinetic energy of the body.If a body is raised up from the earth’s surface, the work done changes the gravitational potential energy. Similarly if a spring is compressed, the work done on it equals the increase in its elastic potential energy.Absolute Potential EnergyThe absolute potential energy of an object at a certain position is “the work done by the gravitational force in displacing the object from that position to infinity where the force of gravity becomes zero.”ExplanationThe relation for the calculation of work done by the gravitational force or potential energy (mgh) is true only near the surface of the earth where the gravitational force is nearly constant. But if the body is displaced through a large distance in space from, let , a point 1 to N (as in fig) in the gravitational filed, then the gravitational force will not remain constant, since it varies inversely to the square of the distance.

Fig.


In order to overcome this difficult, we divide the distance between point 1 and N into small steps each of length


so that the value of force remains constant for each small step. Hence the total work done can be calculated by adding the work done during the all these steps. If r1 and r2 are the distances of point 1 and 2 respectively, from the centre O of the earth, the work done during the first step i.e., displacing a body from point 1 to point 2 can be calculated asThe distance between the centre of this step and the centre of the earth will be

If

then Hence

So

…….. (1)The gravitational force F at the centre of this step is

…….. (2)

Where m = Mass of an objectM = Mass of the earthG = Gravitational Constant

Squaring eq. 1, we have



As , so this term can be neglected as compared to

Hence Substituting the value of




Hence eq. 2 becomes

…….. (8)

As this force is assumed to be constant during the interval , so the work done is



The negative sign indicates that work has to be done on the body form point 1 to 2 because displacement is opposite to the gravitational force.

Or Similarly the work done during the second step in which the body is displaced from point 2 to point 3

is

And similarly the work done in the last step is

Hence, the total work done in displacing the body from point 1 to N is calculated by adding up the work done during all these steps.




On simplification, we get

If the point N is situated at an infinite distance from the earth, so

, then Hence

Therefore the general expression for the gravitational potential energy of a body situated at distance r from the earth is

This is also known as the absolute value of gravitational potential energy of a body at a distance r from the centre of the earth.

1. Note that when r increases, the gravitational force does negative work and U increase i.e., becomes less negative.

2. When r decreases the body falls toward the earth, work is positive and P.E. decrease i.e., it becomes more negative.

3. U is zero when (r = ), the mass m is infinitely away from the earth, this is quite a different choice from making U = 0 at some arbitrary position. But we know that the choice of zero point is arbitrary and that only difference of P.E., from one point to another is significant, so negative value of U should not be too alarming.

4. The absolute potential on the surface of the earth is formed by putting r = R (radium of the earth).


Absolute Potential Energy = The negative sign shows that the earth’s gravitational filed for mass m is attraction.

5. The above expression gives the work or the energy required to take the body out of the earth’s gravitational field, where its potential energy with respect to earth is zero.

Escape Velocity“The minimum initial velocity of a body thrown vertically upward, from surface of earth, due to which it crosses the filed of earth, is called escape velocity.”ExplanationIt is our daily life experience that an object projected upward comes back to the ground after rising to a certain height. This is due to the force of gravity acting downward. With increased initial velocity, the object rises to the greater height before coming back. If we go on increasing the initial velocity of the object, a stage comes when it will not return to the ground. It will escape out of the influence of gravity. The initial velocity of an object with which it goes out of the earth’s gravitational filed, is known as escape velocity.ExpressionThe escape velocity corresponds to the initial kinetic energy gained by the body, which carries it to an infinite distance from the surface of the earth.

………………1

We know that work done in lifting a body from earth’s surface to an infinite distance is equal to the absolute potential energy of the body at earth’s surface.




…………….2Where M and R are the mass and radius of the earth respectively. The body will escape out of the gravitational filed if the initial K.E. of the body is equal to the absolute P.E., thenFrom eq. 1 and eq. 2

Or Or As

hence, The value of comes out to be approximately 11km/sec.

Inter-Conversion of K.E. & P.E.Consider a body of mass m at rest, at a height h above the surface of the earth as shown in fig. We release the body and as it falls, K.E. and P.E. associated with it interchanges.Let us calculate P.E. and K.E. at the position B when the body has fallen to a distance x, ignoring air friction.

And

Fig.

Velocity , at B can be calculated from the relation

As vf = vB, vi = 0 and s = x, then

Total energy at B Thus At Point C


At position C, just before the body strikes the earth, P.E. = 0 and where



can be find out by the following expression

i.e., Thus at point C, kinetic energy is equal to the original value of the potential energy of the body.

Actually when the body falls, its velocity increases i.e., the body is being accelerated under the action of gravity. The increases in velocity results in the increase in its kinetic energy. On the other hand, as the body falls, its height decreases and hence its potential energy also decreases.


Where and are velocities of the body at the heights and respectively.


This result is true only when frictional force is not considered.

Fig.If we assume that a frictional force f is present during the downward motion, then a part of P.E. is used in doing work against friction equal to fh. The remaining P.E. = mgh – fh is converted to K.E.Hence

Or Thus Conservation of Energy

“Energy can not be destroyed. It can be transformed from one kind into another, but the total amount of energy remains constant.”ExplanationThe kinetic and potential energies are both different form of the same basic quantity, i.e., mechanical energy. Total mechanical energy of a body is the sum of the kinetic energy and potential energy. As we know in a falling body, potential energy may change into kinetic energy and vice versa, but the total energy remains constant.Mathematically,

Total Energy = K.E. + P.E. = ConstantThis is one of the basic laws of physics. We daily observe many energy transformations from one form to another. Some forms such as electrical and chemical energy, are more easily transferred than others, such as heat. Ultimately all energy transfers result in heating of the environment and energy is wasted. For example, the potential energy of the falling object changes to K.E. but on striking the ground, K.E. changes to heat and sound. If it seems in an energy transfer that some energy has disappeared, the lost energy is often converted into heat. This appears to be fate of all available energies and is one reason why new sources of useful energy have to be developed?Non Conventional SourcesThere are the energy sources not very common these days. However, it is expected that these sources will contribute substantially to the world energy demand of the future. Some of these are introduced briefly here.Energy from TidesOne very novel example of obtaining energy from gravitational field is the energy obtained from tides. Gravitational force of the moon gives rise to tides in the sea. The tides raise the water in the sea roughly twice a day. If the water at high tide is trapped in a basin by constructing a dam, then it is possible to use this as a source of energy. The dam is filled at high tide and water is released in a controlled way at low tide to drive the turbines. At the next high tide the dam is filled again and the in rushing water also drives turbines and generates electricity.

Fig.Energy from WavesThe tidal movement and the winds blowing across the surface of the ocean produce strong water waves. Their energy can be utilized to generate electricity. A method of harnessing wave energy is to use large floats which move up and down with the waves. One such a device invented by Professor Salter is known as Salter’s duck. It consists of two parts

1. Duck Float


2. Balance FloatThe wave energy makes duck float move relative to the balance float. The relative motion of the duck float is then used to run electricity generators.

Fig.

Solar EnergyThe earth receives huge amount of energy directly form the sun each day.Solar ConstantSolar energy at normal incidence outside the earth’s atmosphere is about 1.4KWm-2 which is referred as solar constant.Solar Energy reaching EarthWhile passing through the atmosphere, the total energy is reduced due to reflection, scattering and absorption by dust particles, water vaporous and other gases. On a clear day at noon, the intensity of the solar energy reaching the earth’s surface is about 1KWm-2.Utilization

1. This energy can be used directly to heat water using large solar reflectors and thermal absorbers or be converted to electricity,

2. In one method the flat plate collectors are used for heating water. A typical detector is shown in the Fig. It has a blackened surface which absorbs energy directly from solar radiation. Cold water passes over the surface and is heated up to about 70oC.

3. Much higher temperature can be achieved by concentrating solar radiation on to a small surface area by using huge reflectors (mirrors) or lenses to produced steam for running a turbine.

Fig.4. The other methods are the direct conversion of sunlight into electricity through the use of

semi conductor devices called solar cells known as photo voltaic cells. Solar cells are thin wafers made from silicon. Electrons in the silicon gain energy from sunlight ot create a voltage. The voltage produced by a single voltaic cell is very low. In order to get sufficient high voltage for practicle use, a large number of such cells are connected in series forming a solar cell panel.

5. for cloudy days and nights, electric energy can be stored during sun light in Nickel cadmium batteries by connecting then to solar panels. Theses batteries can then provide power to electrical appliance at bights or on cloudy days.

6. solar cells are although expensive but last a long time and have low running cost. Solar cells are used to power satellites having large solar panels which are kept faring the sun. other examples of the use of solar cells are remote ground bases weather stations and rain forest communication systems. Solar calculators are also in use now a day.

Energy form BiomassBiomass is a potential source of renewable energy. This includes all the organic materials such as crop residue, natural vegetation, trees, animal dung and sewage. Biomass energy or bio conversion refers to the use of this material as fuel or its conversion into fuels.Methods for Conversion of Biomass into FuelThere are many methods used for the conversion of biomass into fuels. But the most common are

1. Direct combustion2. Fermentation


1. Direct CombustionDirect combustion method is usually applied to get energy from waste product commonly known as solid waste.2. FermentationBio fuel such as ethanol (alcohol) is a replacement of gasoline. It is obtained by fermentation of biomass using enzymes and by decomposition through bacterial action in the absence of ari.The rotting of biomass in a closed tank called a digester produces biogas which can be piped out to use for cooking and heating.

Fig.

The waster product of the process is a good organic fertilizer. Thus production of biogas provides us energy source and also solves the problem of organic waste disposal.

Energy from Waste ProductsWaste products like wood waste, crop residue and particularly municipal solid waste can be used to get energy by direct conversion. It is probably the most commonly used conversion process in which waste material is burnt in a confined container. Heat produced in this way is directly utilized in the boils to produce steam that can sun turbine generator.Geothermal EnergyDefinitionThis is the heat energy extracted from inside the earth in the form of hot water or steam. Heat within the earth is generated by the following processes.

1. Radioactive DecayThe energy heating the rocks is constantly being released by the decay of radioactive element.2. Residual Heat of the EarthAt some places hot igneous rocks, usually within 1.Km of the earth’s surface, are in a molten and partly molten state. They conduct heat energy from the earth’s interior which is still very hot. The temperature of theses rocks is about 200oC or more.3. Compression of MaterialThe compression of material deep inside the earth also cases generation of heat energy.

Fig.

i. In some places water beneath the ground is in contact with hot rocks and is raised to high temperature and pressure. It comes to the surface as hot springs, geysers, or steam vents. The steam can be directed to turbine of electric generators.

ii. Where water is not present and hot rocks are not very deep, the water is pumped down through tem which returns as steam as shown in this fig.

iii. An interesting phenomenon of geothermal energy is a geyser. It is a hot spring that discharges steam and hot water, intermittently releasing an exposure column to air. Most geysers erupt at irregular intervals. They usually occur in volcanic regions. Extraction of geothermal energy often occurs closer to geyser rights. This extraction seriously disturbs geyser system by reducing heat flow and aquifer pressure. Acquirer is a layer of rock holding water that allows water to pre-locate through it with pressure.


Short QuestionsQ.1 A person holds a bag of groceries while standing still, talking to a friend. A car is stationery with its engine running. From the stand point of work, how are these two situations similar?Ans. A man holding a bag and standing still, and a car is stationery with its engine running, have zero value of displacements. In both of these two cases work done is zero. In this respect both the cases are similar.Q.2 Calculate the work done in kilo joules in lifting a mass of 10kg (at a steady velocity) through a vertical height of 10m.Ans. The work done against gravity is

W = mghW = 10 × 9.8 × 10

W = 980 JQ.3 A force F acts through a distance L. the force is then increased to 3F, and then acts through a further distance of 2L. Draw the work diagram to scale.Ans. The work diagram is shown in the graph.

3F 2F

F

L 2L 3L

Q.4 In which case is more work done? When a 50kg bag of books is lifted through 50cm, or when a 50kg crate is pushed through 2m across the floor with a force of 50N?Ans. In first case In Second case

W1 = 50 × 9.8 × 0.5 W2 = 2 × 50W1 = 245 J W2 = 100 J

This shows that work done in the first case is larger.Q.5 An object has 1J of potential energy. Explain what it means.Ans. An object having 1J potential energy means that it has capacity to do work, 1 J work.Q.6 A ball of mass m is held at a height h1 above a table. The table top is at a height h2 above the floor. One student says that the ball has potential energy mgh1 but another says that it is mg (h1+h2). Who is correct?Ans. The P.E. of the ball of mass m, with respect to the table top = mgh1

The P.E. of the ball of mass m, with respect to the ground = mg (h1+ h2)Therefore both the students are correct, provided they give the reference point.

Q.7 When a rocket re-enters the atmosphere, its nose cone becomes very hot. Where does this heat energy come from?Ans. When the rocket re-enters the atmosphere, it has very high velocity and therefore, has very large K.E. The K.E. is converted into frictional work which appears as heat energy. So nose cone of the rocket becomes hot.Q.8 What sort of energy is in the following?

a. Compressed spring

b. Water in a high dam


c. A moving car

Ans. a. In a compressed spring, the energy stored is elastic P.E.b. When the water stored in a high dam there is gravitational P.E.c. A moving car has K.E.

Q.9 A girl drops a cup from a certain height, which breaks into pieces. What energy changes are involved?Ans. In falling the cup its P.E. is converted into K.E. and when it strikes the floor, its K.E. is converted into heat and sound.Q.10 A boy uses a catapult to throw a stone which accidentally smashes a green house window. List the possible energy changes.Ans.

The possible energy changes are

i. P.E. of stone into K.E.

ii. A part of K.E. of stone is converted into heat energy and sound energy, and a part is converted into K.E. of glass pieces.


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Work and Energy