# WORK, POWER, AND ENERGY – PART 1

## Multiplication of vectors

There are two ways of multiplying vectors. One way known as the scalar product gives a scalar from two vectors. The other known as the vector product gives a new vector from the two vectors.

## The Scalar product (Dot product)

The Scalar product of two vectors is defined as the product of the magnitude of the first vector and the component of second vector in the direction of first vector. The scalar product or dot product of any two vectors 𝐴⃗ and 𝐵⃗ denoted by 𝐴 ∙ 𝐵⃗ is given by 𝑨⃗ ∙ 𝑩⃗ = 𝑨𝑩 𝐜𝐨𝐬 𝜽 Where 𝜃 → angle between the two vectors 𝐴𝐵 = (𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝐴) × (𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝐵⃗ 𝑎𝑙𝑜𝑛𝑔 𝐴 )

## Properties of scalar product of two vectors

i. Dot product is commutative.

𝐴𝐵 = 𝐴𝐵 cos 𝜃

𝐵𝐴 = 𝐵𝐴 cos 𝜃 = 𝐴𝐵 cos 𝜃

𝐴𝐵 = 𝐵𝐴

ii. Dot product is distributive over the addition of vectors.

𝐴 ∙ (𝐵 +𝐶) = 𝐴(𝐵 + 𝐶) cos 𝜃 = 𝐴𝐵 cos 𝜃 + 𝐴𝐶 cos 𝜃 = 𝐴𝐵 + 𝐴𝐶

𝐴 ∙ (𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶

iii. Dot product of two parallel vectors.

Let 𝐴 and 𝐵 be two parallel vectors. The angle between two parallel vectors is zero.

𝐴𝐵 = AB cos 0

𝐴𝐵 = 𝐴𝐵 (cos 0 = 1)

iv. Dot product of two equal vectors. Let 𝐴 and 𝐵⃗ be two equal vectors. The angle between two equal vectors is zero.

𝐴𝐴 = 𝐴𝐴 cos 0 = 𝐴2

Similarly 𝑖̂∙ 𝑖̂= 1 × 1 × cos 0 = 1 𝑗̂∙ 𝑗̂= 1 × 1 × cos 0 = 1 𝑘̂ ∙ 𝑘̂ = 1 × 1 × cos 0 = 1

v. Dot product of perpendicular vectors. If two vector 𝐴 and 𝐵⃗ are perpendicular, then angle between them is 𝜃 = 90°

𝐴𝐵 = 𝐴𝐵 cos 90°

𝐴𝐵 = 0

Similarly

𝑖̂∙ 𝑗̂= 1 × 1 × cos 90° = 0

𝑗̂∙ 𝑘̂ = 1 × 1 × cos 90° = 0

𝑘̂ ∙ 𝑖̂= 1 × 1 × cos 90° = 0

vi. Dot product of two anti-parallel vectors.

If 𝐴 and 𝐵 are two anti-parallel vectors, angle between them is 180°.

𝐴𝐵 = 𝐴𝐵 cos 180°

𝐴𝐵 = −𝐴𝐵 (cos 180° = −1)

Dot product of two vectors in terms of their components

𝐴 = 𝐴𝑥𝑖̂+ 𝐴𝑦𝑗̂+ 𝐴𝑧𝑘̂ and 𝐵 = 𝐵𝑥𝑖̂+ 𝐵𝑦𝑗̂+ 𝐵𝑧𝑘̂

𝐴𝐵 = (𝐴𝑥𝑖̂+ 𝐴𝑦𝑗̂+ 𝐴𝑧𝑘̂) ∙ (𝐵𝑥𝑖̂+ 𝐵𝑦𝑗̂+ 𝐵𝑧𝑘̂) = 𝐴𝑥𝐵𝑥 (𝑖̂∙ 𝑖̂) + 𝐴𝑥𝐵𝑦 (𝑖̂∙ 𝑗̂) + 𝐴𝑥𝐵𝑧(𝑖̂∙ 𝑘̂) + 𝐴𝑦𝐵𝑥 (𝑗̂∙ 𝑖̂) + 𝐴𝑦𝐵𝑦 (𝑗̂∙ 𝑗̂) + 𝐴𝑦𝐵𝑧(𝑗̂∙ 𝑘̂) +𝐴𝑧𝐵𝑥(𝑘̂ ∙ 𝑖)̂ + 𝐴𝑧𝐵𝑦(𝑘̂ ∙ 𝑗̂) + 𝐴𝑧𝐵𝑧(𝑘̂ ∙ 𝑘̂)

But 𝑖̂∙ 𝑖̂= 𝑗̂∙ 𝑗̂= 𝑘̂ ∙ 𝑘̂ = 1 and 𝑖̂∙ 𝑗̂= 𝑗̂∙ 𝑘̂ = 𝑘̂ ∙ 𝑖̂= 0

𝑨⃗ ∙ 𝑩⃗ = 𝑨𝒙𝑩𝒙 + 𝑨𝒚𝑩𝒚 + 𝑨𝒛𝑩𝒛

## Determination of angle between vectors

Let 𝜃 be the angle between 𝐴 and 𝐵, then 𝐴𝐵 = |𝐴||𝐵| cos 𝜃

cos 𝜃 = (𝐴𝐵)/|𝐴||𝐵| = (𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦 + 𝐴𝑧𝐵𝑧)/(√(𝐴𝑥2 + 𝐴𝑦2 + 𝐴𝑧2)√(𝐵𝑥2 + 𝐵𝑦2 + 𝐵𝑧2))

𝜽 = 𝒄𝒐𝒔𝟏[(𝑨𝒙𝑩𝒙 + 𝑨𝒚𝑩𝒚 + 𝑨𝒛𝑩𝒛)/(√(𝑨𝒙𝟐 + 𝑨𝒚𝟐 + 𝑨𝒛𝟐)√(𝑩𝒙𝟐 + 𝑩𝒚𝟐 + 𝑩𝒛𝟐))]

## Work

Work is said to be done when a force applied on a body displaces the body through a certain distance in the direction of applied force. Work done by a constant force: Let the force 𝐹⃗ be applied on the body such that the direction of the force makes an angle 𝜃 with the horizontal direction. Let the body is displaced through a distance 𝑑 horizontally. Then work done is, 𝑊 = (𝐹 cos 𝜃)𝑑 𝑾 = 𝑭𝒅𝐜𝐨𝐬 𝜽 = 𝑭⃗ ∙ 𝒅⃗

## Definition

The work done by the force is defined to be the product of component of the force in the direction of the displacement and the magnitude of this displacement. The SI unit of work is joule (J). Work is a scalar.

## Nature of work done

(i) Positive work: The force and the displacement are in the same direction. i.e. 𝜃 is less than 90°.

Ex: (1) When a spring is stretched, force acting on the spring and the displacement are in same direction. (2) When a lawn roller is pulled or pushed by applying a force then the work done is positive.

(ii) Zero work: If displacement is zero, or if the force is zero or if the force and the displacement are mutually perpendicular (𝜃 = 90°) then the work done is zero.

Ex: (1) A man holding a mass of 50 kg on his head then work done is zero, because d= 0. (2) A particle moving on a smooth surface which is not acted upon by a horizontal force. (3) A man holding a suitcase on his head and moves on a horizontal road. Here force on the suitcase is upward and displacement is along Horizontal. (4) A particle moving in a circle with constant speed the centripetal force is always perpendicular to the displacement.

(iii) Negative work: The force and displacement are in opposite direction. i.e. 𝜃 is greater than 90° up to 180°

Ex: (1) When a body of mass m is raised upwards from the ground through a height h. (2) When breaks are applied to stop a moving car the work done by the breaking force is negative.

## Energy

The energy of a body is its capacity or ability for doing work. There are many forms of energy such as mechanical energy heat energy, light energy, electrical energy etc. Mechanical energy has two types, (i) Kinetic energy and (ii) Potential energy.

## Kinetic energy

The energy possessed by a body by virtue of its motion is called kinetic energy. Ex: Flowing water, moving vehicles, A bullet fired from a gun. Etc.

## Expression for Kinetic energy

Consider an object of mass 𝑚 has velocity 𝑣⃗. The kinetic energy is, 𝐾 = (1/2)𝑚(𝑣⃗ ∙ 𝑣⃗)

𝑲 = (𝟏/𝟐)𝒎𝒗𝟐

The SI Unit energy/kinetic energy/potential energy is 𝑗𝑜𝑢𝑙𝑒 (𝐽). Dimensions are [ML2T-2]

Note: Work done, and energy have same dimensions.

## Alternate units of work / Energy

1. In CGS system ‘erg’ 1erg = 10-7J

2. Electron volt 1eV = 1.6 x 10-19J

3. Kilowatt hour 1 kWh = 3.6 x 106J

4. Calorie 1 Cal = 4.186J

## Work energy Theorem for constant force

Statement:

“The change in kinetic energy of a particle is equal to the work done on it by the net force.”

Proof: One of the equations of motion for rectilinear motion is, 𝑣2 = 𝑣02 + 2𝑎𝑥. By generalizing the equation to three dimensions, we have 𝑣2 = 𝑣02 + 2(𝑎 ∙ 𝑑)

𝑣2 − 𝑣02 = 2(𝑎 ∙ 𝑑)

Multiplying both sides by (1/2)𝑚, we have,

(1/2)𝑚𝑣2 – (1/2)𝑚𝑣02 = (1/2)𝑚2(2𝑎𝑑)

(1/2)𝑚𝑣2 – (1/2)𝑚𝑣02 = 𝑚𝑎𝑑

𝐾𝑓 − 𝐾𝑖 = 𝐹𝑑

𝑲𝒇 − 𝑲𝒊 = 𝑾 or ∆𝑲 = 𝑾

Let us consider a variable force (Whose magnitude changes continuously) acting on a body. Let the body be displaced in the direction of applied force. The graph of variable force, F(x) and displacement, 𝑥 of the body is as shown. To calculate the work done, divide the total displacement of the body into a number of small intervals each of width ∆𝑥. The width ∆𝑥 is so small that the force 𝐹(𝑥) is considered constant over that interval. Then ∆𝑊 = 𝐹(𝑥)∆𝑥. Total work done is given by,

If displacements are allowed to approach zero, then number of strips increases infinitely, and the sum approaches a definite value.

Note: Work = Area under the force & displacement graph

## Work energy theorem for variable force

We know that 𝐾 = 1/2 𝑚𝑣2

Differentiating both sides with respect to 𝑡,

𝑑𝑘/𝑑𝑡 = 𝑑/𝑑𝑡((1/2)𝑚𝑣2)

𝑑𝑘/𝑑𝑡 = (1/2)𝑚(2𝑣(𝑑𝑣/𝑑𝑡)) = 𝑚(𝑑𝑣/𝑑𝑡)𝑣

𝑑𝑘/𝑑𝑡 = 𝑚𝑎𝑣

𝑑𝑘/𝑑𝑡 = 𝐹𝑣 = 𝐹(𝑑𝑥/𝑑𝑡)

When 𝑥 = 𝑥𝑖, 𝐾 = 𝐾𝑖 & 𝑥 = 𝑥𝑓, 𝐾 = 𝐾𝑓

## Discussion of work energy theorem

1. Work done by a Force is zero, if there is no change in the speed of a body.

𝑊 = (1/2)𝑚𝑣2 – (1/2)𝑚𝑣02 = (1/2)𝑚𝑣2 – (1/2)𝑚𝑣2 = 0. (𝑣 = 𝑣0)

Ex: When a body moves in a circular path with constant speed, there is no change in kinetic energy of the body.

2. Work done by a Force is positive, if there is increase in the velocity of the particle. 𝑊 = (1/2)𝑚(𝑣2 − 𝑣02)

If 𝑣 > 𝑣0, 𝑣 − 𝑣0 = +𝑣𝑒,

W = positive

Ex: When a particle is dropped from the top of the building the velocity of the particle increases.

3. Work done by a force is negative, if there is decrease in the speed of the particle 𝑊 = (1/2)𝑚(𝑣2 − 𝑣02)

If 𝑣 > 𝑣0, 𝑣 − 𝑣0 = −𝑣𝑒,

W = negative

Ex: When particle is projected upwards, the speed of the particle decreases, work done is negative.

## Relation between kinetic energy and Linear momentum

We have 𝐾 = (1/2)𝑚𝑣2 = (1/2)((𝑚2𝑣2)/𝑚) = (1/2)(𝑃2/𝑚) (𝑚𝑣 = 𝑃)

## The Concept of Potential energy

The energy possessed by a body by virtue of its position or configuration (shape) is called potential energy. Here position refers to the height above the surface of earth and configuration refers to arrangement/shape of the body. Potential energy is a stored energy when work is done on that body.

Ex:

1. An object lifted to a certain height from the surface of the earth has potential energy at the position.

2. A stretched bow and arrow system has potential energy.

3. A wound spring of a watch has potential energy.

4. An apple/mango hanging from the branch of a tree has a potential energy.

## Expression for Potential energy

Consider a block of mass 𝑚 which is to be raised to a height ℎ above the ground. Work done by the External force is,

𝑊 = 𝐹𝑒 = 𝐹𝑒ℎ cos 𝜃

𝑊 = 𝑚𝑔ℎ cos 0

𝑊 = 𝑚𝑔ℎ

This work done is stored as potential energy 𝑉(ℎ).

𝑽(𝒉) = 𝒎𝒈𝒉

## Gravitational Potential energy

Work done by the gravitational force on the body when it is raised to a certain height is known as gravitational potential energy and denoted by V(h) as function of the height ℎ

## Expression for Gravitational potential energy

Consider a block of mass 𝑚 which is to be raised to a height ℎ above the ground.

Work done by the Gravitational force is,

𝑊 = 𝐹𝑔 = 𝐹𝑔ℎ cos 𝜃

𝑊 = 𝑚𝑔ℎ cos180°

𝑊 = −𝑚𝑔ℎ

This work done is stored as gravitational potential energy 𝑉(ℎ).

𝑽(𝒉) = −𝒎𝒈𝒉

## Types of Potential energy

(i) Gravitational potential energy: The energy passed by a body by virtue of its position.

(ii) Elastic potential energy: The energy possessed by a body by virtue of its deformed shape.

Note:

1) When the block comes down with an increasing speed, just it hits the ground; its speed is given by the equation,

𝑣2 = 𝑣02 + 2𝑔ℎ

𝑣2 = 2𝑔ℎ

Multiplying both sides with 𝑚/2,

(1/2)𝑚𝑣2 = (𝑚/2) × 2𝑔ℎ

(1/2)𝑚𝑣2 = 𝑚𝑔ℎ

This shows that the gravitational potential energy of the object at height ℎ when the object is released manifests itself as kinetic energy of the object on reaching the ground.

2) Mathematically the potential energy is defined if the force 𝐹(𝑥) can be written as, 𝑉(𝑥) = −𝑚𝑔𝑥 ⇒ 𝑑𝑉(𝑥)/𝑑𝑥 = −𝑚𝑔

−𝑑𝑉/𝑑𝑥 = 𝐹(𝑥)

This implies that, 𝐹(𝑥)𝑑𝑥 = −𝑑𝑉

On integration

3) The change in Gravitational potential energy for a conservative force 𝐹(𝑥), ∆𝑉 is equal to the negative of the work done by the force.

∆𝑉 = −𝐹(𝑥)∆𝑥

## Conservative force

If the amount of work done by or against a force depends only on the initial and final positions of a body and not on the path followed by the body then such a force is called a conservative force. Work done by the conservative force on a body around a closed path is zero.

Ex: Gravitational force, spring force and electrostatic force are conservative forces.

## Non conservative force

It the amount of work done against a force depends on the path followed by a body then the force is said to be non-conservative force. Work done by a non-conservative force on a body around a closed path is not zero.

Ex: Frictional force and viscous force are non-conservative forces.

## Conservation of Mechanical energy

Statement:

“The total mechanical energy of a system is conserved, if the forces doing the work on it are conservative.”

Explanation: Suppose a body undergoes displacement 𝑑𝑥 under the action of a conservative force, 𝐹. Then from work energy theorem

The potential energy 𝑉(ℎ) is defined by the force F can be written as,

From the above equations we get,

𝐾𝑓 − 𝐾𝑖 = 𝑉𝑖 − 𝑉𝑓

𝐾𝑖 + 𝑉𝑖 = 𝐾𝑓 + 𝑉𝑓

Thus, Initial mechanical energy of a system is equal to final mechanical energy of system.

## Illustration for conservation of mechanical energy

In case of freely falling body mechanical energy (𝐾 + 𝑉) of the body remains constant.

At point A: Consider a body of mass 𝑚 having 𝑣0 = 0 at a height ℎ from the ground. The kinetic energy is, 𝐾 = (1/2)𝑚𝑣02 = 0 (∵ 𝑣0 = 0)

The potential energy is 𝑉 = 𝑚𝑔ℎ

Since mechanical energy at 𝐴, 𝐾 + 𝑉 = 0 + 𝑚𝑔ℎ = 𝑚𝑔ℎ – − −> (1)

At point B: Let the body is allowed to fall. It reaches to 𝐵 travelling a distance 𝑥 with a velocity 𝑣𝐵. Then 𝐴𝐵 = 𝑥 and 𝐵𝐶 = (ℎ − 𝑥)

The potential energy is, 𝑉 = 𝑚𝑔 (ℎ − 𝑥)

The velocity attained by the body is, 𝑣𝐵2 = 𝑣02 + 2𝑔𝑥

𝑣𝐵2 = 2𝑔𝑥

The kinetic energy is

𝐾 = (1/2)𝑚𝑣𝐵2

𝐾 = (1/2)𝑚 × 2𝑔𝑥 = 𝑚𝑔𝑥

Mechanical energy at 𝐵 is (K+V) = 𝑚𝑔𝑥 + 𝑚𝑔(ℎ − 𝑥) = 𝑚𝑔𝑥 + 𝑚𝑔ℎ − 𝑚𝑔𝑥 = 𝑚𝑔ℎ − − −> (2)

At point C: Now the body reaches to the ground at C. Here ℎ = 0, then potential energy, 𝑉 = 𝑚𝑔 × 0 = 0. The velocity attained by the body just reaches the point C,

𝑣𝑐2 = 𝑣02 + 2𝑔ℎ

𝑣𝐶2 = 2𝑔ℎ

The kinetic energy is 𝐾 = (1/2)𝑚𝑣𝑐2

𝐾 = (1/2)𝑚 × 2𝑔ℎ

𝐾 = 𝑚𝑔ℎ

Mechanical energy at C is (𝐾 + 𝑉) = 𝑚𝑔ℎ + 0 = 𝑚𝑔ℎ – − −> (3)

From the equations (1), (2) and (3), it is clear that, the mechanical energy of a body during the free fall of a body under the action of gravity remains constant.

## Spring force

When a spring is compressed or stretched, the spring force is given by,

𝐹𝑠 = −𝑘𝑥 where 𝑘 is spring constant. This force law for spring is called Hooke’s law. The spring force is an example for a variable force which is conservative.

## Definition of spring constant

Force constant or spring constant is the restoring force per unit displacement of the spring. Its SI unit is Nm-1. Dimensional formula is [ML0T-2].

𝒌 = 𝑭/𝒙

## Spring force is conservative

Work done by spring force,

If the spring is stretched from initial displacement 𝑥𝑖 to final displacement 𝑥𝑓 then,

If the spring is pulled from 𝑥𝑖 and allowed to return to 𝑥𝑖 then,

Thus spring force is Conservative.

## Potential energy of a spring

Work done by the spring force when it is compressed or stretched is stored as energy called Potential energy.

## Expression for Potential energy of a spring

Let a block attached to a light (mass less) spring and resting on a smooth horizontal surface. Let the spring is stretched through a displacement 𝑥𝑚. Work done by spring force is,

The work done is stored as the potential energy of the stretched string.

𝑽(𝒙) = −(𝟏/𝟐)𝒌𝒙2

𝐍𝐨𝐭𝐞:

(1) The same is true when the spring is compressed 𝑽(𝒙) = −(1/2)𝑘𝑥𝑐2

(2) The work done by the external force is positive and 𝑉(𝑥) = (1/2)𝑘𝑥2

## Expression for Kinetic Energy of spring

The total mechanical energy of a spring at any arbitrary point 𝑥, where 𝑥 lies between −𝑥𝑚 and +𝑥𝑚 will be given by,

Total mechanical energy = 𝑉 + 𝐾

(1/2)𝑘𝑥𝑚2 = (1/2)𝑘𝑥2 + 𝐾

𝐾 = (1/2)𝑘𝑥𝑚2 – (1/2)𝑘𝑥2

𝑲 = (𝟏/𝟐)𝒌(𝒙𝒎𝟐 − 𝒙𝟐)

## Expression for maximum speed of spring

When 𝑥 = 0, 𝐾𝑚𝑎𝑥 = (1/2)𝑘𝑥𝑚2

Further, (1/2)𝑚𝑣𝑚𝑎𝑥2 = (1/2)𝑘𝑥𝑚2

𝒗𝒎𝒂𝒙 = (√(𝒌/𝒎))𝒙𝒎

𝒌𝒎 has the dimension of [𝑇−2]

## Discussion of variation of Potential energy and kinetic energy during elongation and compression of spring

(1) We have 𝐾 = (1/2)𝑘(𝑥𝑚2 − 𝑥2)

When 𝑥 = 0, 𝐾 = (1/2)𝑘𝑥𝑚2 ⇒ Kinetic energy is maximum at mean position

But, 𝑉 = (1/2)𝑘𝑥2 = (1/2)𝑘(0) = 0 ⇒ Potential energy is minimum at mean position.

(ii) When 𝑥 = 𝑥𝑚, 𝐾 = (1/2)𝑘(𝑥𝑚2 − 𝑥𝑚2) = 0 ⇒ Kinetic energy is minimum at extreme position

But, 𝑉 = (1/2)𝑘𝑥𝑚2 ⇒ Potential energy is maximum at extreme position

## Graphical representation of Variation of Mechanical energy of spring

The total mechanical energy can be graphically represented as shown. Kinetic energy is maximum at normal position and potential energy is zero or total energy at normal position is purely kinetic and at extreme ends the total energy is purely potential energy. Various forms of energy: We have discussed one form of energy. i.e., mechanical energy which is the sum of K and V. The other forms of energy are discussed below.

## Heat or Thermal energy

In mechanics we say that kinetic energy is lost due to the frictional force, but the work done by the friction is not lost, but is transferred as heat energy. This raises the internal energy of the system. An object possesses heat energy due to the disorderly motion of the molecules of the object.

## Chemical energy

Chemical energy arises from the fact that the molecules participating in the chemical reaction have different binding energies. Chemical energy is released during the chemical reaction. If the total energy of the reactants is more than the products of the reaction, heat is released and the reaction is said to be an exothermic reaction. If the total energy of the reactant is less than the product of the reaction, heat is absorbed and the reaction is endothermic.

## Electrical energy

The flow of charges constitutes the electrical energy. Work is done to move the electric charge from one point to another point in the electric field. This work done appears as the electrical energy.

## The equivalence of mass and energy

The mass of an isolated system is conserved during the physical and chemical process. Albert Einstein showed that mass and energy are inter convertible. That is mass can be converted into energy and energy can be converted into mass according to the relation, 𝐸 = 𝑚𝐶2 Where 𝐶 → speed of light in vacuum= 3 × 108𝑚𝑠−1

For 1𝑘𝑔 of matter, 𝐸 = 1 × (3 × 108 ) = 9 × 1016𝐽 released.

Ex:

## Annihilation of matter

When an electron and a positron combine with each other they destroy each other, and mass of electron and position is converted into energy according to Einstein’s equation and released in the form of two 𝛾 – rays. 𝑒 + 𝑒+ → 𝛾 + 𝛾

## Pair production

This process is the reverse of annihilation of matter and energy is converted into mater.

## Nuclear Energy

The energy required to hold the nucleons in the nucleus is called nuclear energy. When light nuclei combine (fuse) to form a relatively heavy nucleus is called nuclear fusion. In this case mass of heavy nucleus is less than the sum of the masses of the reactants. This mass difference is called mass defect (∆𝑚). ∆𝑚 is converted in to energy according to the relation 𝐸 = (∆𝑚)𝑐2. Energy is released in sun and stars by nuclear fusion. When a less stable heavy nucleus like 92U-235 breaks up into two stable nuclei this reaction is called nuclear fission. In this case the final mass is less than the initial mass and mass difference translates into energy. By the method of controlled nuclear fission electricity is generated and uncontrolled nuclear fission is employed in making of nuclear weapons such as atom bombs and nuclear bombs.

## Principle of conservation of energy

“Energy may be transformed from one form to another but the total energy of the isolated system remains constant energy can neither be created nor destroyed.”

## Power

It is defined as the time rate at which work is done or time rate at which energy is transferred.

𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑤𝑒𝑟 = 𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒/𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛

But work done = energy supplied or consumed. ∴ 𝑃𝑎𝑣 = 𝐸𝑛𝑒𝑟𝑔𝑦/𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒n