In this article, we shall study the concept of energy, types of mechanical energies, and the law of conservation of energy

Energy:

Different types of energy are mechanical energy, sound energy, heat energy, light energy, chemical energy, electrical energy, atomic energy, nuclear energy. Mechanical energy is further classified into kinetic energy and potential energy.

Kinetic Energy:

The energy possessed by the body on account of its motion is called kinetic energy. e.g Energy possessed by flowing water and wind, moving bicycle

Consider a body of mass ‘m’ lying on the smooth horizontal surface, is acted upon by a constant force of magnitude ‘F’ which displaces it through a distance ‘s’ in its own direction. Then the work done by the force is given by

W  =  F .  s     ……….. (1)

By Newton’s second law of motion

F  =  m . a     ……….. (2)

Where ‘a’ is the magnitude of the acceleration in the body.

From equations  (1) and (2)

∴  W  =  m a s  …………… (3)

By equation of motion we have

v² = u²   +  2as

Where u  = magnitude of the initial velocity. In this case u = 0

v =  magnitude of final velocity after covering the distance ‘s’

∴  v² =  2 a s

∴ as =  v²/2

Substituting in equation (3) we get

∴  W  =  mv²/2

∴  W  =  ½mv²

This work is stored as kinetic energy in the body. Thus the kinetic energy of the body is given by

K.E. = ½mv²

This is an expression of the kinetic energy of a body.

Potential Energy:

The energy possessed by a body or a system on account of its position and configuration is called potential energy. e.g. energy possessed by water stored in a dam, in wound spring of a watch

Suppose that body of mass ‘m’ be raised to some height say ‘h’ against the gravitational force which is equal to the weight of the body ‘mg’. Where ‘g’ is an acceleration due to gravity.

As the applied force and the displacement of the body are in the same direction.

Work = Force × Displacement

W = mg × h

∴  W = mgh

This work is stored as the potential energy in the body.

∴  P.E. = mgh

This is an expression for the gravitational potential energy of a body, raised to some height above the earth’s surface.

Units and Dimensions of Energy and that of Work are the Same:

The capacity of a body to do work is called energy. Hence energy is measured in terms of work. Therefore, the units and dimensions of energy and that of work are the same.

kilowatt-hour:

a kilowatt-hour is a unit of measuring energy. This unit is a general unit of energy consumption bills (Electricity bills)

Now Work = Power x time

Hence, 1 kilowatt hour= 1 kilowatt × 1 hour

If the power of 1 kilowatt is used for 1 hour, the work done or energy consumed is said to be 1 kilowatt hour.

1 kWh   = 1kW x 1 hour

= 1000 W x 60 x 60 sec

= 1000 J/s x 3600 s

= 3600000 J

= 3.6 x 10⁶ J

Kinetic Energy is Always Positive:

The kinetic energy of a body is given by the expression. K.E. = ½mv²

The right-hand side contains the term mass ‘m’ which is always positive and a term square of velocity which is also positive. Thus the right-hand side of the expression is always positive. Thus kinetic energy is always positive.

Principle of Conservation of Energy:

The energy cannot be created nor it can be destroyed but can be converted from one form to another. Thus the total energy of the isolated system remains the same.

Energy can be converted from one form to another Examples

• In an electrical bulb, electrical energy is converted into light energy and heat energy.
• When the hammer strikes the nail mechanical energy gets converted into sound energy and heat energy.

Working of Hydroelectric Power Station :

The principle of conservation of energy can be explained by the example of a hydroelectric power station.

Water is stored in the artificial reservoirs created in the mountains by constructing a dam across the river. Thus the kinetic energy of flowing water is converted into potential energy of stored water. This stored water is brought downhill i.e. at the foot of the mountain through pipes. This water is then directed on blades of the wheel of the turbine. Thus the kinetic energy of water is used to rotate the coil in the turbine. Due to rotation of the coil in the magnetic field the kinetic energy gets converted into electrical energy. This energy can further be converted into different forms of energy like sound, heat, light, magnetism, etc.

Principle of Conservation of Mass:

The mass cannot be created nor it can be destroyed but can be converted from one form to another. Thus the total mass of isolated system remains the same.

Einstein’s Mass-Energy Relation:

According to Albert Einstein, the mass and energy are interconvertible and the equivalence between them is given by the relation

E  =  m c²

Where   E = amount of energy

M = Mass

c = speed of light in vacuum

This relation is known as Einstein’s mass-energy relation.

Thus mass and energy are not two different physical quantity or the mass is a form energy.

Examples of Mass-Energy Interconversion:

Phenomenon of pair production :

In the phenomenon of pair production, the energy of gamma rays photons is converted under proper conditions, into a positron-electron pair. Thus here energy gets converted into mass.

Phenomenon of pair annihilation:

In the phenomenon of pair annihilation, a positron and electron under proper conditions combine to form the gamma-ray photon. Thus the particles (mass) are converted into energy.

Note:

Positrons and electrons both are similar particles having the same mass only difference is their charges. Positrons are positively charged while electrons are negatively charged.

Modified Law of Conservation of Mass and Energy:

The total amount of mass and energy in the universe is always constant.

Einstein’s Formula for the Variation of Mass with Velocity:

When the velocity of light is comparable with that of light, then, the mass of the particle in motion is given by

Where m_o = mass of a body at rest.

m = mass of a body when moving with a velocity ‘ v ’

c = velocity of light in vacuum.

This relation is known as Einstein’s formula for the variation of mass with velocity. This relation shows that the mass of a body increases with the increase in its velocity.

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In this article, we shall study the concept of work and power.

Work:

When a force is applied to a body and there is the displacement of the body in the direction of the force or along the direction of the component of force, then work is said to be done by the force.

Work is defined as the product of the force applied and the displacement of the body in the direction of the force. Both force and displacement are vector quantities but work is a scalar quantity.

W   =   F . S

Where W = Work done

F = Force applied

S = Displacement of the body in the direction of the force.

In vector form, the formula can be written as

W = F. S

Dimensions of work are [L²M¹T^-2]

Sign of Work:

• Work can be zero, positive and negative
• When force is applied to the body and there is no displacement of the body, then work done by the force is zero. e.g. Consider a body suspended in the air using thread. The gravitational force pulls the body down but there is no displacement of the body in the direction of gravitational force. In this case, the work done by the gravitational force is zero.
• When applied force and the displacement of the body are perpendicular to each other then the work done by the force is zero. e.g. The moon revolves around the earth in a stable orbit. The earth’s gravitational force acts on it and pulls the moon towards its centre, but the moon moves in the direction perpendicular to the direction of gravitational force. Thus there is no displacement of the moon in the direction of gravitational force. Thus work done by the gravitational force is zero.
• When the displacement of the body is in the direction of force causing the displacement, the work done by the force is positive. e.g. Consider a freely falling body. A gravitational force acts on it and pulls downward. Thus the displacement of the body is in the direction of gravitational force. Hence the work done by the gravitational force is positive.
• When the displacement of the body is in the opposite direction to that of force causing the displacement, the work done by the force is negative. e.g. Consider a body which is being lifted up. The gravitational force pulls the body down, but the body moves up i.e. in the opposite direction to that of gravitational force. Thus work done by the gravitational force is negative.

Unit of Work:

When the applied force and the displacement of the body are in the same direction, work done is given by

Work  =  Force  ×   Displacement

Unit work  =  Unit force  ×   Unit displacement

Definition of Unit Work: Unit work is said to be done when the unit force produces a unit displacement in its own direction.

S.I. unit of work is joule (J): 1 J  =  1 N  ×  1 m

When a force of 1 newton acting on a body produces a displacement of 1 metre in the direction of force, then work done by the force is called 1 joule.

C.G.S. unit of work is erg: 1 erg  =  1 dyne  x   1 cm

When a force of 1 dyne acting on a body produces a displacement of 1 centimetre in the direction of force, then work done by the force is called 1 erg.

Derivation of  Expression for the Work Done by the Force:

Suppose the force produces a displacement in the direction making an angle θ with the direction of the force. The component of the force along the direction of displacement is F.Cos θ.

Now, Work done = Component of force in the direction of displacement × displacement

∴  W = (F.Cos θ)(s)

∴  W = F. s. Cos θ

W = F. S

Thus the work done is a scalar product of force and displacement. Thus work done is a scalar quantity.

When the displacement is  in the direction of the force

In such a case, θ = 0°

W = F. S . Cos 0°

W = F. S (1)

W = F. S

Thus when the displacement is in the direction of force the work done, is

equal to the product of magnitudes of force and the displacement.

When the displacement is perpendicular to force

In such a case, θ = 90°

W = F. s . Cos 90°

W = F. s (0)

W = 0

Thus when the displacement is perpendicular to the direction of force the work done is zero.

Work done by the Gravitational Force of the Earth on the Moon:

Work is defined as the product of the force applied and the displacement of the body in the direction of the force. The moon revolves around the earth in a stable circular orbit. The earth’s gravitational force acts on it and pulls the moon towards its centre, but the moon moves in the direction perpendicular to the direction of gravitational force. Thus there is no displacement of the moon in the direction of gravitational force. Thus work done by the gravitational force is zero.

In this case, θ = 90°

W = F. s . Cos 90°

W = F. s (0)

W = 0

Power:

The rate at which work is done is called power. As work and time are scalar quantity power is also a scalar quantity.

P = W / t

Unit of Power:

By the definition of power, unit of power = unit of work / unit of time = 1 J /  1s = 1 W

In S.I. system of units the unit of power is watt. Its symbol is ‘W’. Thus the power is said to be 1 watt if the rate of doing work is 1 joule per second.

In C.G.S. system of units the unit of power erg/s. Thus the power is said to be 1 erg/s if the rate of doing work is 1 erg per second.

But in practice, the unit power may be used with some prefixes.

1 kW  =  1000 W

1 MW = 1000000 W

1 horsepower = 746 W

Relation Between the Power and Velocity of a Body:

Suppose a force F acts a body which causes a displacement of s in the direction of the force in the body in ‘t’ seconds.

Then work don is given by W  =  F  .  S

By definition of power  P = W / t

∴   P = F  .  S/ t

∴   P = F  .  (S/ t)

∴  P = F  . v

Where v is the magnitude of the instantaneous velocity.

Thus the power is the product of magnitudes of the force acting on the body and velocity of the body.

Horsepower:

A horsepower is a unit of power used in the engineering. Its symbol is hp. Its relation with watt is as follows

1 horsepower (hp) = 746 watts

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In this article, we shall study two types of products of vectors: a) Scalar product and b) Vector product

Scalar Product of Two Vectors:

The scalar or dot product of two vectors is defined as the product of magnitudes of the two vectors and the cosine of the angles between them.

If  a and  b are two vectors and θ is the angle between the two vectors then by the definition scalar product of two vectors

a · b = a b cos θ

Where  a = magnitude of a and b = magnitude of vector  b

Characteristics of the Scalar Product:

• The scalar product of two vectors is always a pure number i.e. the scalar product is always a scalar.
• The scalar product of two vectors is commutative. i.e. a · b =  b · a

a · b = a b cos θ = b a cos θ = b · a

• Scalar product obeys the distributive law of multiplication. i.e.  a ·( b +  c) = a · b  +  a · c
•  a and b  are two vectors perpendicular to each other, if and only if   a · b =  b · a  = 0
•  a and b  are two vectors parallel to each other, if and only if   a · b =  b · a  = ab
• If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then their scalar product is given by

  a · b =(a_x b_x + a_y b_y + a_z b_z )

• The scalar product of two vectors may be zero or positive or negative.

Scalar Products of Standard Unit Vectors:

i · i = i × i cos 0° = 1 × 1× 1 = 1, Similarly,  We have j · j = 1  and  k · k = 1

i · j = i · j = i × j cos 90° = 1 × 1× 0 = 0, Similarly, We have j · k = k · j = o and k · i = i · k = 0

Expression for Scalar Product of Two Vectors:

If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then the scalar product is given by

a · b  = ( a_x i + a_y j + a_z k ) · (b_x i + b_y j + b_z k )

∴ a · b  =  a_x i · (b_x i + b_y j + b_z k ) + a_y j · (b_x i + b_y j + b_z k )  + a_z k · (b_x i + b_y j + b_z k )

∴ a · b  =  a_xb_x  i ·i + a_xb_y  i ·j +  a_xb_z   i ·k  +  a_y b_x  j ·i+ a_y b_y j ·j + a_y b_z j ·k

+ a_zb_x  k ·i + a_zb_y k ·j   +  a_zb_z k ·k

∴ a · b  =  a_xb_x (1) + a_xb_y (0) +  a_xb_z (0) +  a_y b_x(0) +  a_y b_y (1) + a_y b_z (0)

+ a_zb_x  (0) + a_zb_y (0)  +  a_zb_z (1)

∴ a · b  =  a_xb_x  +  a_y b_y   +  a_zb_z 

This is an expression for scalar product of two vectors.

Examples of Scalar Product of Two Vectors:

• Work done is defined as scalar product as W = F · s, Where F is a force and s  is a displacement produced by the force
• Power is defined as a scalar product as P = F · v, Where F  is a force and v  is a velocity.

Notes:

• if two vectors are perpendicular to each other then θ = 90° , thus cos θ  = cos 90° = 0 Hence  a · b  =  ab cos 90° = ab(0) = 0
• if two vectors are parallel  to each other then θ = 0° , thus cos θ  = cos 0° = 1 Hence a · b = ab cos 0° = ab(1) = ab
• if two vectors are equal  to each other then θ = 0° , thus cos θ  = cos 0° = 1 Hence  a · a  =  aa cos 0° = aa(1) = a²

Vector Product of Two Vectors:

The vector or cross product of two vectors is a vector whose magnitude is equal to the product of the magnitudes of the two vectors and the sine of the angle between the two vectors.

If a and  b are two vectors and θ is the angle between the two vectors then by the definition  of the vector  product of two vectors

a × b = a b sin θ n

Where  a = magnitude of a and b = magnitude of vector  b

n = unit vector perpendicular to the plane of a and  b

The direction of n is given by right-hand thumb rule

Characteristics of the Vector product:

• Vector product two vectors is always a  vector.
• The Vector product of two vectors is noncommutative. i.e.  a · b ≠  b · a but a · b = –  b · a
• vector product obeys the distributive law of multiplication. i.e.  a ×( b +  c) = a  ×  b  +  a ×  c
• If a · b = 0 and a ≠ o, b ≠ o then the two vectors are parallel to each other.
• If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then their vector product is given by

Direction of Vector Product By Right Hand Thumb or Grip Rule:

Hold the right hand along the first vector such that the fingers are parallel to the plane of the vectors and the curled fingers are along the angular direction in which we have to move to the second vector then the outstretched thumb indicates the direction of the vector obtained by vector product of two vectors.

Vector Products of Standard Unit Vectors:

i × i = i × i sin 0° = 1 × 1× 0 = 1, Similarly,  We have j × j = 0  and  k × k = 0

Using the reference circle for vector product

i × j = k ,  j × i = – k  ; j × k = i ,  k × j = – i  ;  k × i = j ,  i × k = – j  ;

Expression for the Vector Product of Two Vectors:

If  a =  a_x i + a_y j + a_z k and b = b_x i + b_y j + b_z k are the two vectors, then the scalar product is given by

a × b  = ( a_x i + a_y j + a_z k ) × (b_x i + b_y j + b_z k )

∴ a × b  =  a_x i × (b_x i + b_y j + b_z k ) + a_y j × (b_x i + b_y j + b_z k )  + a_z k × (b_x i + b_y j + b_z k )

∴ a × b  =  a_xb_x  i × i + a_xb_y  i × j +  a_xb_z   i × k  +  a_y b_x  j × i+ a_y b_y j × j + a_y b_z j × k

+ a_zb_x  k × i + a_zb_y k × j   +  a_zb_z k × k

∴ a × b  =  a_xb_x (0) + a_xb_y (k) +  a_xb_z (- j ) +  a_y b_x(- k) +  a_y b_y (0) + a_y b_z (i)

+ a_zb_x  (j) + a_zb_y (- i)  +  a_zb_z (0)

∴ a × b  =  a_xb_y k  –  a_xb_z j –  a_y b_xk +  a_y b_z i + a_zb_x  j – a_zb_y i

∴ a × b  =  a_y b_z i  – a_zb_y i  –  a_xb_z j + a_zb_x  j    +  a_xb_y k   –  a_y b_xk

∴ a × b  =  i( a_y b_z – a_zb_y) –   j(a_xb_z + a_zb_x)    +  k(a_xb_y    –  a_y b_x )

This is an expression for vector product of two vectors.

Examples of Vector Product of Two Vectors:

• Torque acting on the rotating body is defined as vector product τ = r × F, Where F is a force and r  is a position vector of the point of action of the force.
• If a and b are the adjacent sides of a parallelogram, then the area of the parallelogram is given as A =  | a × b|
• If a and b are the adjacent sides of a triangle, then the area of the triangle is given as A =  ½| a × b|

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