XI PHYSICS

[WORK, POWER AND ENERGY] CHAPTER NO. 7

A little concept of vector mathematics is applied here and in which a little mistake may destroy the concept of gravitational mechanics. Also, Energy is a conserved quantity is proved in this chapter. Due to which we can derive Work-Energy Equation which is a very important tool for solving work and energy problems.

M. Affan Khan LECTURER – PHYSICS, AKHSS, K affan_414@live.com https://promotephysics.wordpress.com

Work Work is defined as the product of displacement and the

component of force in the direction of displacement.

Mathematical form,

Consider an object of mass ‘m’ has to move displacement ‘s’ when the force ‘F’ at an angle θ

is applied. Now the component which is responsible for its movement along the direction of

displacement is F∥.

W = F∥S

W = (Fcosθ)S

W = FScosθ

W = F⃗ . S⃗

Types of Work 1) Positive and Maximum work

If the force applied is in the direction of displacement, then the angle between both

vectors will be 00. Then,

W = FScos00

W = FS(1)

W = FS

2) Zero Work

If the force applied is perpendicular to the desired displacement, then the angle

between both vectors will be 900.

W = FScos900

W = FS(0)

W = FS

3) Negative Work

In case of friction the body moves forwards and friction force acts in opposite direction

tries to stop the body then the angle is 1800 between force and displacement, so that

W = FScos1800

W = FScos(−1)

W = −FS

Units of Work:

1) Joule = Newton×meter is S.I unit of work

2) Erg = Dyne×centimeter is C.G.S. unit of work

3) Foot − Pound(lb) is F.P.S unit of work

Conservative Force / Field “A force for which work done around a closed path is zero or is independent of path

between two fixed points, is called conservative force and its field is called as conservative

field.”

E.g. Gravitational field, Electrostatic field etc.

Non-conservative field “A force for which work done around a closed path is not zero or is not independent of path

between two fixed points, is called non-conservative force and its field is called non-

conservative field.”

E.g. force of friction, magnetic force, viscous force

Gravitational Force / Field is conservative To prove that gravitational force/field is conservative we have to consider a closed path.

For this reason, we are considering a closed path in the form of triangle as shown in figure.

If the work done around ABCA is zero, then the force is called conservative force and the

field is called conservative field. Therefore, work done can be written as,

WTOTAL = WA→B + WB→C + WC→A

Work done from A to B:

WA→B = F⃗ . S⃗ 1 = FS1 cos(180 − β)

where, (180 − β) is the angle between force and displacement

WA→B = FS1(−cosβ)

WA→B = −mgS1cosβ

WA→B = −mgh

Work done from B to C:

WB→C = F⃗ . S⃗ 2 = FS2cosα

= mgS2cosα

where, α is the angle between force and displacement

= mgh (S2cosα = h)

Work done from C to A:

WC→A = F⃗ . S⃗ 3 = FS3cos900

WC→A = mgS3(0)

WC→A = 0

Now the total work done can be written as,

WTOTAL = WA→B + WB→C + WC→A

WTOTAL = −mgh + mgh + 0

WTOTAL = 0

A

ℎ

𝛼

B

𝑆2

C

𝑆1

𝛽

𝑆3

Since the total work done around this closed path is zero its mean gravitational force is a

conservative force.

Energy: “Capability of doing work.”

Forms of Energy 1) Mechanical Energy

2) Chemical Energy

3) Electrical Energy

4) Heat Energy

5) Solar Energy

6) Nuclear Energy

and others

Types of Mechanical Energy: a) Kinetic Energy

b) Potential Energy

Kinetic Energy: The energy possessed by a body due to its motion called as “Kinetic Energy”.

Mathematical Expression,

Consider a body of mass ‘m’ is subjected to an external force ‘F’ so that the body covers

some displacement ‘S’ as its velocity from zero to becomes ‘v’ and its acceleration is ‘a’.

Since it is in motion therefore the work “W” is done in the form of kinetic energy “K.E”.

we may write,

2aS = vf2 − vi

2

2aS = v2 − 02

S =v2

2a

And work is given as,

W = F⃗ . S⃗

W = Fscos00

W = (ma)v2

2a

W =m(v2)

2

W =1

2mv2

𝑣𝑖 = 0 𝑣𝑓 = 𝑣

The work stored in the form of Kinetic Energy

W = K. E

Therefore,

K. E =1

2mv2

Work-Energy Equation: Work is also defined as change of energy. Consider an object of mass ‘m’ moving with initial

velocity ‘vi’ subjected to some external force ‘F’ which provides it some acceleration ‘a’ to

change its velocity to ‘vf’. In this time ‘t’ the object is displaced ‘S’ units. So, we may write

2aS = vf2 − vi

2

S =vf

2 − vi2

2a

Then work done is given as,

W = F⃗ . 𝑆

W = Fscos00

W = (ma)vf

2 − vi2

2a

W =m(vf

2 − vi2)

2

W =1

2mvf

2 −1

2mvi

2

W = (K. E)f − (K. E)i

W = ∆K. E

Potential Energy: “The energy stored in a body due to its position is called as Potential Energy.”

Types of Potential Energy:

1. Gravitational Potential Energy

2. Elastic Potential Energy

3. Electrostatic Potential Energy

and many others.

Gravitational Potential Energy: The energy possessed by a body due to its height is called as gravitational potential energy.

P. E = mgh

𝑣𝑖 𝑣𝑓

Explanation:

Suppose an object of mass ‘m’ is subjected to an upward force ‘F’ which is applied to raise

the object to a height ‘h’ from the surface of Earth at a very slow rate. The slow movement

is only possible when the applied force is equal to weight (mg) of the body.

We have work done here as,

W = F⃗ . S⃗

W = FScosθ

W = (mg)(h)cos00

W = mgh

Here the work done is stored in the form of potential energy,

∴ P. E = mgh

Power “The rate of doing work is power.”

OR

“Energy consumed per unit time is called as power.”

Pav =W

t

Instantaneous Power:

Pins = lim∆t→0

W

t

Units

Its S.I unit is Watt (i.e. J/s)

Also, can be written with following prefixes

1 MW = 106 Watts

1 GW = 109 Watts

It has also unit of horse power.

∵ 1 hp = 746 Watts

Another unit is foot-pound/second.

1 ft − lb/s = 1.35 Watts

Since we know that

Power =Work

Time

Therefore,

Work = Power×Time

Or in terms units,

Work = Watt×second

Work = 1kwh = 3.6×106J

Power in terms of Force & Velocity: As we know that

Power =Work

Time

P =W

t

P =F⃗ . S⃗

t

P =FScosθ

t

P = Fcosθ (S

t)

P = Fcosθ(v)

P = Fvcosθ

P = F⃗ . V⃗⃗

Therefore, power is also defined as dot product of Force and Velocity.

Law of Conservation of Energy Statement

“Energy can neither be created nor be destroyed but it can be changed from one form to

another form, therefore, the total energy of the system remains constant.”

Explanation

Consider an object with mass ‘m’ situated at a certain height ‘h’ from the ground level as

shown in figure, if it is allowed to drop from this height then it will fall freely

under the action of gravity.

At Point A

Since at this point the object is at its maximum height ‘h’ and with

initial velocity zero, therefore it has,

P. E = mgh

K. E = 0

E = K. E + P. E

E = 0 + mgh

E = mgh

At Point B

When it will reach at point B, it has covered displacement ‘x’ from its initial position and we

can say according to diagram that it is now located at height ‘h-x’. Also, let us call velocity of

this object at this point as vB.

P. E = mg(h − x)

A

h

B

C

x

K. E =1

2mvB

2

For 𝐯𝐁

2as = vf2 − vi

2

2gx = vB2 − 02

vB2 = 2gx

∴ K. E =1

2m(2gx)

K. E = mgx

Now, the total energy is

E = K. E + P. E

E = mgx + mg(h − x)

E = mgx + mgh − mgx

E = mgh

At Point C

When it will reach ground level, the height from surface of the Earth will be zero, and let us

call this velocity as vc.

P. E = mg(0) = 0

K. E =1

2mvc

2

For 𝐯𝐜

2as = vf2 − vi

2

2gh = vC2 − 02

vc = 2gh

∴ K. E =1

2m(2gh)

K. E = mgh

Now, the total energy at this point will be,

E = K. E + P. E

E = mgh + 0

E = mgh

Hence it is proved that Energy converts from one form to another form and remains

conserved.

We can conclude for this case that

𝐥𝐨𝐬𝐬 𝐨𝐟 𝐩𝐨𝐭𝐞𝐧𝐭𝐢𝐚𝐥 𝐞𝐧𝐞𝐫𝐠𝐲 = 𝐠𝐚𝐢𝐧 𝐨𝐟 𝐤𝐢𝐧𝐞𝐭𝐢𝐜 𝐞𝐧𝐞𝐫𝐠𝐲

If there was air friction present, then the object which was coming downwards had to

perform work done against friction also,

𝐥𝐨𝐬𝐬 𝐨𝐟 𝐩𝐨𝐭𝐞𝐧𝐭𝐢𝐚𝐥 𝐞𝐧𝐞𝐫𝐠𝐲 − 𝐰𝐨𝐫𝐤 𝐝𝐨𝐧𝐞 𝐚𝐠𝐚𝐢𝐧𝐬𝐭 𝐟𝐫𝐢𝐜𝐭𝐢𝐨𝐧 = 𝐠𝐚𝐢𝐧 𝐨𝐟 𝐤𝐢𝐧𝐞𝐭𝐢𝐜 𝐞𝐧𝐞𝐫𝐠𝐲

𝐥𝐨𝐬𝐬 𝐨𝐟 𝐩𝐨𝐭𝐞𝐧𝐭𝐢𝐚𝐥 𝐞𝐧𝐞𝐫𝐠𝐲 = 𝐠𝐚𝐢𝐧 𝐨𝐟 𝐤𝐢𝐧𝐞𝐭𝐢𝐜 𝐞𝐧𝐞𝐫𝐠𝐲 + 𝐰𝐨𝐫𝐤 𝐝𝐨𝐧𝐞 𝐚𝐠𝐚𝐢𝐧𝐬𝐭 𝐟𝐫𝐢𝐜𝐭𝐢𝐨𝐧

𝐦𝐠𝐡 =𝟏

𝟐𝐦𝐯𝟐 + 𝐟𝐡

There are many other cases where we can define Energy Conservation as we did above,

E.g.

1) As we rub our hands it produces heat, then we may write,

𝐦𝐞𝐜𝐡𝐚𝐧𝐢𝐜𝐚𝐥 𝐞𝐧𝐞𝐫𝐠𝐲 = 𝐡𝐞𝐚𝐭 𝐞𝐧𝐞𝐫𝐠𝐲 + 𝐥𝐨𝐬𝐬𝐞𝐬

2) As fuel burns to run engine then,

𝐜𝐡𝐞𝐦𝐢𝐜𝐚𝐥 𝐄𝐧𝐞𝐫𝐠𝐲 = 𝐦𝐞𝐜𝐡𝐚𝐧𝐢𝐜𝐚𝐥 𝐞𝐧𝐞𝐫𝐠𝐲 + 𝐥𝐨𝐬𝐬𝐞𝐬

Absolute Gravitational Potential Energy We measure gravitational potential energy from the surface of Earth (i.e. mgh), which

states that at surface of the Earth h = 0 then P. E = 0, which should not be the case since

the phenomenon of attraction take place from the center of mass, therefore, at surface of

Earth P.E cannot be equal to zero. That’s why to find the absolute gravitational potential

energy let us perform a hypothetical experiment.

Consider an object of mass ‘m’ is placed at certain distance from the center of the Earth,

we’ll be performing work to raise this object from this place to infinity (in this case out of

the gravitational field of Earth). Let us now consider a force ‘F’ is applied on it to raise it to

required height. For simplicity, we can divide this displacement in ‘n’ intervals given that

for all intervals the force is approximately same. As we move from position 1 to 2, the

displacement covered is,

∆r = r2 − r1

And since force is approximately same on each point

since displacement is very small therefore, we can

take average of these two forces

F =F1 + F2

2

F =1

2(GMEm

r12 +

GMEm

r22 )

F =GMEm

2(

1

r12 +

1

r22)

F =GMEm

2(r22 + r1

2

r12r2

2 )

From diagram, we may write, r2 = ∆r + r1

F =GMEm

2((∆r + r1)

2 + r12

r12r2

2 )

F =GMEm

2((∆r)2 + 2∆rr1 + r1

2 + r12

r12r2

2 )

Since ∆r is very small, therefore we can neglect its square term.

F =GMEm

2(2∆rr1 + 2r1

2

r12r2

2 )

F =GMEm

2

(2r1)(∆r + r1)

r12r2

2

F = GMEm(∆r + r1)

r1r22

F = GMEmr2

r1r22

F =GMEm

r1r2

Therefore, work done from 1st to 2nd position can be given as,

W1→2 = F⃗ . ∆r

W1→2 = F∆rcosθ

W1→2 = F∆rcos1800

W1→2 = −F∆r

W1→2 = −GMEm

r1r2(r2 − r1)

W1→2 = −GMEmr2 − r1r1r2

W1→2 = −GMEm(r2

r1r2−

r1r1r2

)

W1→2 = −GMEm(1

r1−

1

r2)

This is the work done when object was displaced from r1 to r2.

Similarly work at other positions will be given as,

W2→3 = −GMEm(1

r2−

1

r3)

W3→4 = −GMEm(1

r3−

1

r4)

.

.

.

Wn−1→n = −GMEm(1

rn−1−

1

rn)

Let us find total work,

WTOTAL = W1→2W2→3W3→4 …Wn−1→n

WTOTAL = (−GMEm(1

r1−

1

r2)) + (−GMEm(

1

r2−

1

r3))

+ (−GMEm(1

r3−

1

r4)) + ⋯+ (−GMEm(

1

rn−1−

1

rn))

WTOTAL = −GMEm(1

r1−

1

r2+

1

r2−

1

r3+

1

r3+

1

r4+. . . +

1

rn→1−

1

rn)

WTOTAL = −GMEm(1

r1−

1

rn)

This work done is stored in the form of Potential Energy

P. E = −GMEm(1

r1−

1

rn)

Since rn is a point which is outside the Earth’s gravitational field, therefore, for absolute

Potential Energy, we may take rn = ∞

(P. E)abs = −GMEm(1

r1−

1

∞)

(P. E)abs = −GMEm(1

r1− 0)

(P. E)abs = −GMEm

r1

For any point r, we may write,

(P. E)abs = −GMEm

r

Since ‘r’ is in denominator which shows that as we go away from the center of Earth, P.E

will decrease and as r approaches ∞, P.E becomes zero. Also, negative sign indicates the

bound condition of objects around Earth that is all objects in the Earth’s gravitational field.

At surface of Earth, 𝐫 = 𝐑𝐄

∴ (P. E) = −GMEm

RE

At some height ‘h’ above Earth’s surface, then 𝐫 = 𝐑𝐄 + 𝐡

(P. E)abs = −GMEm

RE + h

(P. E)abs = −GMEm

RE (1 +hRE

)

(P. E)abs = −GMEm

RE(1 +

h

RE)−1

Applying Binomial Theorem,

(P. E)abs = −GMEm

RE(1 −

h

RE+

2h2

RE2 + ⋯)

Neglecting squares and higher powers we get,

(P. E)abs = −GMEm

RE(1 −

h

RE)