**This article is formulated according to the 6th chapter in the NCERT Syllabus of 11th Std Physics OR 1st PUC Physics.**

## 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 π΄β and π΅β denoted by π΄ β π΅β is given by π¨β β π©β = π¨π© ππ¨π¬ π½ Where π β angle between the two vectors **π΄**** β ****π΅** = (ππππππ‘π’ππ ππ π΄) Γ (ππππππ‘π’ππ ππ π΅β πππππ π΄ )

## Properties of scalar product of two vectors

**i. Dot product is commutative.**

**π΄**** β ****π΅** = π΄π΅ cos π

**π΅**** β ****π΄** = π΅π΄ cos π = π΄π΅ cos π

**π΄**** β ****π΅** = **π΅**** β ****π΄**

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

**π΄**** **β (**π΅** +**πΆ**) = **π΄**(**π΅** + **πΆ**) cos π = π΄π΅ cos π + π΄πΆ cos π = **π΄** β **π΅** + **π΄** β **πΆ**

**π΄** β (**π΅** + **πΆ**) = **π΄** β **π΅** + **π΄** β **πΆ**

**iii. Dot product of two parallel vectors.**

Let **π΄** and **π΅** be two parallel vectors. The angle between two parallel vectors is zero.

**π΄** β **π΅** = AB cos 0

**π΄** β **π΅** = π΄π΅ (cos 0 = 1)

**iv. Dot product of two equal vectors. Let π΄ and π΅β be two equal vectors. The angle between two equal vectors is zero.**

**π΄** β **π΄** = π΄π΄ cos 0 = π΄^{2}

Similarly πΜβ πΜ= 1 Γ 1 Γ cos 0 = 1 πΜβ πΜ= 1 Γ 1 Γ cos 0 = 1 πΜ β πΜ = 1 Γ 1 Γ cos 0 = 1

**v. Dot product of perpendicular vectors. If two vector π΄ and π΅β are perpendicular, then angle between them is π = 90Β°**

**π΄**** β ****π΅** = π΄π΅ cos 90Β°

**π΄**** β ****π΅** = 0

Similarly

πΜβ πΜ= 1 Γ 1 Γ cos 90Β° = 0

πΜβ πΜ = 1 Γ 1 Γ cos 90Β° = 0

πΜ β πΜ= 1 Γ 1 Γ cos 90Β° = 0

**vi. Dot product of two anti-parallel vectors.**

If **π΄** and **π΅** are two anti-parallel vectors, angle between them is 180Β°.

**π΄**** β ****π΅** = π΄π΅ cos 180Β°

**π΄**** β ****π΅** = βπ΄π΅ (cos 180Β° = β1)

Dot product of two vectors in terms of their components

π΄ = π΄_{π₯}πΜ+ π΄_{π¦}πΜ+ π΄_{π§}πΜ and π΅ = π΅_{π₯}πΜ+ π΅_{π¦}πΜ+ π΅_{π§}πΜ

**π΄**** β ****π΅** = (π΄_{π₯}πΜ+ π΄_{π¦}πΜ+ π΄_{π§}πΜ) β (π΅_{π₯}πΜ+ π΅_{π¦}πΜ+ π΅_{π§}πΜ) = π΄_{π₯}π΅_{π₯} (πΜβ πΜ) + π΄_{π₯}π΅_{π¦} (πΜβ πΜ) + π΄_{π₯}π΅_{π§}(πΜβ πΜ) + π΄_{π¦}π΅_{π₯} (πΜβ πΜ) + π΄_{π¦}π΅_{π¦} (πΜβ πΜ) + π΄_{π¦}π΅_{π§}(πΜβ πΜ) +π΄_{π§}π΅_{π₯}(πΜ β π)Μ + π΄_{π§}π΅_{π¦}(πΜ β πΜ) + π΄_{π§}π΅_{π§}(πΜ β πΜ)

But πΜβ πΜ= πΜβ πΜ= πΜ β πΜ = 1 and πΜβ πΜ= πΜβ πΜ = πΜ β πΜ= 0

π¨β β π©β = π¨_{π}π©_{π} + π¨_{π}π©_{π} + π¨_{π}π©_{π}

## Determination of angle between vectors

Let π be the angle between π΄ and π΅, then **π΄**** β ****π΅** = |π΄||π΅| cos π

cos π = (**π΄**** β ****π΅**)/|π΄||π΅| = (π΄_{π₯}π΅_{π₯} + π΄_{π¦}π΅_{π¦} + π΄_{π§}π΅_{π§})/(β(π΄_{π₯}^{2} + π΄_{π¦}^{2} + π΄_{π§}^{2})β(π΅_{π₯}^{2} + π΅_{π¦}^{2} + π΅_{π§}^{2}))

π½ = πππ^{β}^{π}[(π¨_{π}π©_{π} + π¨_{π}π©_{π} + π¨_{π}π©_{π})/(β(π¨_{π}^{π} + π¨_{π}^{π} + π¨_{π}^{π})β(π©_{π}^{π} + π©_{π}^{π} + π©_{π}^{π}))]

## 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 πΉβ be applied on the body such that the direction of the force makes an angle π with the horizontal direction. Let the body is displaced through a distance π horizontally. Then work done is, π = (πΉ cos π)π πΎ = ππ ππ¨π¬ π½ = πβ β π β

## 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. π is less than 90Β°.

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 (π = 90Β°) 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. π is greater than 90Β° up to 180Β°

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 π has velocity π£β. The kinetic energy is, πΎ = (1/2)π(π£β β π£β)

π² = (π/π)ππ^{π}

The SI Unit energy/kinetic energy/potential energy is πππ’ππ (π½). Dimensions are [ML^{2}T^{-2}]

**Note:** Work done, and energy have same dimensions.

## Alternate units of work / Energy

1. In CGS system βergβ 1erg = 10^{-7}J

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

3. Kilowatt hour 1 kWh = 3.6 x 10^{6}J

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, π£^{2} = π£_{0}^{2} + 2ππ₯. By generalizing the equation to three dimensions, we have π£^{2} = π£_{0}^{2} + 2(π β π)

π£^{2} β π£_{0}^{2} = 2(π β π)

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

(1/2)ππ£^{2} β (1/2)ππ£_{0}^{2} = (1/2)π^{2}(2**π** β **π**)

(1/2)ππ£^{2} β (1/2)ππ£_{0}^{2} = π**π** β **π**

πΎ_{π} β πΎ_{π} = **πΉ** β **π**

π²_{π} β π²_{π} = πΎ or βπ² = πΎ

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, **π₯** 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 βπ₯. The width βπ₯ is so small that the force πΉ(π₯) is considered constant over that interval. Then βπ = πΉ(π₯)βπ₯. 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 πΎ = 1/2 ππ£^{2}

Differentiating both sides with respect to π‘,

ππ/ππ‘ = π/ππ‘((1/2)ππ£^{2})

ππ/ππ‘ = (1/2)π(2π£(ππ£/ππ‘)) = π(ππ£/ππ‘)π£

ππ/ππ‘ = πππ£

ππ/ππ‘ = πΉπ£ = πΉ(ππ₯/ππ‘)

When π₯ = π₯_{π}, πΎ = πΎ_{π} & π₯ = π₯_{π}, πΎ = πΎπ

## Discussion of work energy theorem

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

π = (1/2)ππ£^{2} β (1/2)ππ£_{0}^{2} = (1/2)ππ£^{2} β (1/2)ππ£^{2} = 0. (π£ = π£_{0})

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. π = (1/2)π(π£^{2} β π£_{0}^{2})

If π£ > π£_{0}, π£ β π£_{0} = +π£π,

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 π = (1/2)π(π£^{2} β π£_{0}^{2})

If π£ > π£_{0}, π£ β π£_{0} = βπ£π,

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 πΎ = (1/2)ππ£^{2} = (1/2)((π^{2}π£^{2})/π) = (1/2)(π^{2}/π) (ππ£ = π)

## 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 π which is to be raised to a height β above the ground. Work done by the External force is,

π = **πΉ _{π}**

**β**

**β**= πΉ

_{π}β cos π

π = ππβ cos 0

π = ππβ

This work done is stored as potential energy π(β).

π½(π) = πππ

## 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 β

## Expression for Gravitational potential energy

Consider a block of mass π which is to be raised to a height β above the ground.

Work done by the Gravitational force is,

π = **πΉ _{π}** β

**β**= πΉ

_{π}β cos π

π = ππβ cos180Β°

π = βππβ

This work done is stored as gravitational potential energy π(β).

π½(π) = βπππ

Here negative sign indicates that gravitational force acts downwards.

## 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,

π£^{2} = π£_{0}^{2} + 2πβ

π£^{2} = 2πβ

Multiplying both sides with π/2,

(1/2)ππ£^{2} = (π/2) Γ 2πβ

(1/2)ππ£^{2} = ππβ

This shows that the gravitational potential energy of the object at height β 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 πΉ(π₯) can be written as, π(π₯) = βπππ₯ β ππ(π₯)/ππ₯ = βππ

βππ/ππ₯ = πΉ(π₯)

This implies that, πΉ(π₯)ππ₯ = βππ

On integration

3) The change in Gravitational potential energy for a conservative force πΉ(π₯), βπ is equal to the negative of the work done by the force.

βπ = βπΉ(π₯)βπ₯

## Conservative force and Non conservative force

## 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.

Conservative force | Non-conservative force |

Work done depends on initial point and final point | Work done depends on path followed by the body |

Work done is zero around a closed path | Work done is not-zero around a closed path |

Work done is path independent | Work done is path dependent |

## Conservation of Mechanical energy

**Statement:**

βThe total mechanical energy of a system is conserved, if the forces doing the work on it are conservative.β

**Explanation:** Suppose a body undergoes displacement ππ₯ under the action of a conservative force, πΉ. Then from work energy theorem

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

From the above equations we get,

πΎ_{π} β πΎ_{π} = π_{π} β π_{π}

πΎ_{π} + π_{π} = πΎ_{π} + π_{π}

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 (πΎ + π) of the body remains constant.

At point A: Consider a body of mass π having π£_{0} = 0 at a height β from the ground. The kinetic energy is, πΎ = (1/2)ππ£_{0}^{2} = 0 (β΅ π£_{0} = 0)

The potential energy is π = ππβ

Since mechanical energy at π΄, πΎ + π = 0 + ππβ = ππβ β β β> (1)

At point B: Let the body is allowed to fall. It reaches to π΅ travelling a distance π₯ with a velocity π£_{π΅}. Then π΄π΅ = π₯ and π΅πΆ = (β β π₯)

The potential energy is, π = ππ (β β π₯)

The velocity attained by the body is, π£_{π΅}^{2} = π£_{0}^{2} + 2ππ₯

π£_{π΅}^{2} = 2ππ₯

The kinetic energy is

πΎ = (1/2)ππ£_{π΅}^{2}

πΎ = (1/2)π Γ 2ππ₯ = πππ₯

Mechanical energy at π΅ is (K+V) = πππ₯ + ππ(β β π₯) = πππ₯ + ππβ β πππ₯ = ππβ β β β> (2)

At point C: Now the body reaches to the ground at C. Here β = 0, then potential energy, π = ππ Γ 0 = 0. The velocity attained by the body just reaches the point C,

π£_{π}^{2} = π£_{0}^{2} + 2πβ

π£_{πΆ}^{2} = 2πβ

The kinetic energy is πΎ = (1/2)ππ£_{π}^{2}

πΎ = (1/2)π Γ 2πβ

πΎ = ππβ

Mechanical energy at C is (πΎ + π) = ππβ + 0 = ππβ β β β> (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,

πΉ_{π } = βππ₯ where π is spring constant. This force law for spring is called Hookeβs 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 [ML^{0}T^{-2}].

π = π/π

## Spring force is conservative

Work done by spring force,

If the spring is stretched from initial displacement π₯_{π} to final displacement π₯_{π} then,

If the spring is pulled from π₯_{π} and allowed to return to π₯_{π} 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 π₯_{π}. Work done by spring force is,

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

π½(π) = β(π/π)ππ^{2}

ππ¨ππ:

(1) The same is true when the spring is compressed π½(π) = β(1/2)ππ₯_{π}^{2}

(2) The work done by the external force is positive and π(π₯) = (1/2)ππ₯^{2}

## Expression for Kinetic Energy of spring

The total mechanical energy of a spring at any arbitrary point π₯, where π₯ lies between βπ₯_{π} and +π₯_{π} will be given by,

Total mechanical energy = π + πΎ

(1/2)ππ₯_{π}^{2} = (1/2)ππ₯^{2} + πΎ

πΎ = (1/2)ππ₯_{π}^{2} β (1/2)ππ₯^{2}

π² = (π/π)π(π_{π}^{π} β π^{π})

## Expression for maximum speed of spring

When π₯ = 0, πΎ_{πππ₯} = (1/2)ππ₯_{π}^{2}

Further, (1/2)ππ£_{πππ₯}^{2} = (1/2)ππ₯_{π}^{2}

π_{πππ} = (β(π/π))π_{π}

π_{π} has the dimension of [π^{β2}]

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

(1) We have πΎ = (1/2)π(π₯_{π}^{2} β π₯^{2})

When π₯ = 0, πΎ = (1/2)ππ₯_{π}^{2} β Kinetic energy is maximum at mean position

But, π = (1/2)ππ₯^{2} = (1/2)π(0) = 0 β Potential energy is minimum at mean position.

(ii) When π₯ = π₯_{π}, πΎ = (1/2)π(π₯_{π}^{2} β π₯_{π}^{2}) = 0 β Kinetic energy is minimum at extreme position

But, π = (1/2)ππ₯_{π}^{2} β 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, πΈ = ππΆ_{2} Where πΆ β speed of light in vacuum= 3 Γ 10^{8}ππ ^{β1}

For 1ππ of matter, πΈ = 1 Γ (3 Γ 10^{8} ) = 9 Γ 10^{16}π½ 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 Einsteinβs equation and released in the form of two πΎ β rays. π^{β} + π^{+} β πΎ + πΎ

## 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 (βπ). βπ is converted in to energy according to the relation πΈ = (βπ)π^{2}. Energy is released in sun and stars by nuclear fusion. When a less stable heavy nucleus like ^{92}U-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

βEnergy 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.β

## Power

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

π΄π£πππππ πππ€ππ = π€πππ ππππ/π‘πππ π‘ππππ

But work done = energy supplied or consumed. β΄ π_{ππ£} = πΈπππππ¦/π‘πππ π‘πππn

**This chapter will be conitnued in the Part-2 of this article.**