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Energy

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Energy - Lesson Summary



The energy of an object is its ability to do work. Energy is the cause and work is its effect. Therefore both work and energy have the same unit, which is joule (J) in the SI system and erg in the CGS system. Energy is  a scalar quantity. Energy exists in many forms.


Forms of  Energy
Mechanical energy (mechanical energy is either in the form of potential energy or kinetic energy or a combination of the both), electrical energy, light energy, thermal energy, nuclear energy and sound energy etc.


Potential Energy
Potential energy is the energy possessed by a body by virtue of its state of rest or deformed state i.e, the energy of an object due to its position or arrangement in a system is called potential energy.

It is further classified into gravitational potential energy (GPE) and elastic potential energy (EPE). GPE is by virtue of height of a body from a reference level. The gravitational potential energy of an object is the work done in raising it from the ground to a certain point against gravity.

It can be expressed as  GPE = mgh (where, m is mass of the body, g is the acceleration due to gravity and h is  the height of the body from the reference level).

Gravitational potential energy, P.E = mgh

If the height, H of a body is considered from the ground, then the gravitational potential energy of the body, P.E = mgH


If the height of the body is raised from a  height, h  to the other height, H the gravitational potential energy of the body is  P.E = mg(H-h).

EPE of a body is by virtue of its stretched state.


Derivation of Equation for P.E
Let the work done on the object against gravity = W

                 Work done, W = force × displacement
                 Work done, W  = mg × h
                 Work done, W = mgh

Since workdone on the object is equal to mgh, an energy equal  to mgh units is gained by the object . This is the potential energy (Ep) of the object.

                              Ep = mgh

Kinetic Energy
Kinetic energy (KE) is the energy possessed by a body by virtue of its motion and is given by,

K.E = ½mv2.

    •  The work done on a body is equal to the change in its kinetic energy.
    •  The kinetic energy of a body is given by  K.E = ½mv2.
    •  Energy can be converted from one form into another.


Derivation of Equation for K.E
The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is

                             v2 - u2 = 2aS
This gives 
                             S = v 2 - u 2 2a

We know F = ma. Thus using above equations, we can write the workdone by the force, F as

                            W = ma × v 2 - u 2 2a
                                  or
                             W = 1 2 m( v 2 - u 2 )

If object is starting from its stationary position, that is, u = 0, then
                            W = 1 2 m v 2

It is clear that the work done is equal to the change in the kinetic energy of an object.

If u = 0, the work done will be  W = 1 2 m v 2 .

Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is Ek =  ½ mv2


The Relation Between Kinetic Energy and Momentum
Momentum is the quantity of motion of a moving body, its magnitude is equal to the producct of its mass and velocity of the body at a particular time.

If mass of the body = m and the velocity = v
its momentum (linear) p = mv
               p = mv
Kinetic energy is defined as the energy possessed by a body because of its motion.
If mass of the boody = m
Velocity = v
Kinetic energy = ½ x mass x velocity2
K.E = ½ mv2
K.E =  (½ mv) x v
but mv = p
K.E = ½ p x v
p = 2K.E/v
Or 
Kinetic Energy = ½ mass x velocity2
⇒ K.E = ½ mv2

On multiplying and dividing the above equation with m
⇒ K.E =  (½ mv )x(v) x m/m
⇒ K.E =  ½ (mv x mv )/m
⇒ K.E =  ½ (mv)2/m  
⇒ K.E =  ½ p2/m


Law of Conservation of Energy
The law of conservation of energy is the fundamental law, law of conservation of energy says that the energy can neither be created nor destroyed, the sum total energy existing in all forms in the universe remains constant. Energy can only be transformed from one form to another.

Principle of Conservation of Mechanical Energy, which states that the energy can neither be created nor destroyed; it can only be transformed from one state to another. Orthe total mechanical energy of a system is conserved if the forces doing the work on it are conservative.

Consider any two points A and B in the path of a body falling freely from a certain height H as in the above figure.

Total mechanical energy at A
M.EA = mgH + ½ mvA2 , here vA = 0
⇒M.EA = mgH


Total mechanical energy at B
M.EB = mg(H - h) +½ mvB2
⇒M.EB = mg(H - h) + ½ m ( u2 +2gh), where u = 0
⇒M.EB = mgH - mgh + ½ m ( 02 +2gh)
⇒M.EB = mgH - mgh + ½ m ×2gh
⇒M.EB = mgH - mgh + mgh
⇒M.EB = mgH

Total mechanical energy at C
As the body reaches the ground its height from the ground becomes zero.

M.EC = mgH + ½ mvC 2,here H = 0
⇒M.EC = 0 +  mvC 2, but  vC 2, = 2gH
⇒M.EC =  ½ m× 2gH
⇒M.EC = mgH

Hence the total mechanical energy at any point in its path is Constant i.e., M.EA = M.EB = M.EC = mgH

According to the Principle of Conservation of Mechanical Energy, we can say that for points A and B, the total mechanical energy is constant in the path travelled by a body under the action of a conservative force, i.e., the total mechanical energy at A is equal to the total mechanical energy at B.

Note
Work done by a conservative force is path independent. It is equal to the difference between the potential energies of the initial and final positions and is completely recoverable.”

As the work done by a conservative force depends on the initial and final position, we can say that work done by a conservative force in a closed path is zero as the initial and final positions in a closed path are the same.


Comparision between  P.E and K.E
The energy possessed by a body or a system due to the motion of the body or of the particles in the system. Kinetic energy of an object is relative to other moving and stationary objects in its immediate environment.


Examples
Flowing water, such as when falling from a waterfall.

SI Unit

Joule (J)

Examples
Water at the top of a waterfall, before the precipice.

SI Unit

Joule (J)

Electrical Energy

Electrical energy commercially is measured in the units of kilowatt hour (kW h). Power is defined as the rate of doing work. Power is measured in watt which is equal to joule per second. Power can also be measured as the product of force and velocity of an object. Energy can be expressed in terms of product of power and time.

1 kW h = 3.6 x 106 J.

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