Mass Energy Equivalence Notes for Engineering Physics

Mass Energy Equivalence Notes for Engineering Physics:

Below explained equation gives the universal Mass Energy equivalence. This is another major contribution of Einstein. It signifies the manifestation of energy into the matter and vice-versa. Relativity shows that mass is not any independent quantity, however, it is a form of energy. Its experimental evidences are well known in nuclear fission and fusion reactions.

According to classical mechanics, force F is given by the time derivative of the product of mass (m) and velocity (v), i.e., momentum of the body, and is given by

f = d /dt (mv)

However, according to the theory of relativity, both m and v are variable

So  ,                 f d/dt (mv) =mdv/dt +v dm/dt

When a particle is displaced through a distance ‘dx’ on application of a force F, then the

increase in kinetic energy is given by,

dk =f x dx

= [m dv/dt +v dm/dt ] x dx/dt x dt

=  [m dv /dt +v dm/dt] x v x dt

Dk =v2 dm +mv dv

Now, we know that

Mass Energy Equivalence

 m2 = m02 c2 /c2 –v2

 M2 c2  -m2v2 = m02 c2

On differentiation both sides, we get,

2m x dm x c2– [2m x dm x v2 + 2v x dv x m2] = 0

ð  2 m x dm x c2 = 2m [ dm x v2 + v x m x dv]

c2 dm = v2 dm + v m dv

By comparing equations (2) and (3), we have,

D K = c2 d m … (4)

Now, we can consider that the body initially at rest and on application of force it acquires a velocity v. Also the mass of the body increases from m0 tom. Thereby, the total kinetic energy acquired by the body is given by.

ʃ dk  = ʃ m mo  c2 dm

k =c2 (m m0)

This is the increase in kinetic energy due to the increase in mass.

Now, the total energy of a moving body is the total of kinetic energy due to motion and the energy at rest.

So,                               E = k + m0 c2 = c2 (m- m0) + m0 c2 = m c2

Hence,                         [E=m c2 ]

The conversion of mass and energy and the reverse are also illustrated in nature itself. When a particle collides with an anti-particle there is a mutual annihilation and the total mass is converted into radiant energy. Thus, the conservation of mass takes place into energy. Other way, when a radiant energy comes near a charged nucleus, particle and anti-particle are created. Thus the energy is converted into mass.

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