# 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 =v ^{2} dm +mv dv**

Now, we know that

**m ^{2} = m_{0}^{2} c^{2} /c^{2} –v^{2}**

** M ^{2} c^{2} -m^{2}v^{2} = m_{0}^{2} c^{2}**

On differentiation both sides, we get,

2m x dm x c^{2}– [2m x dm x v^{2} + 2v x dv x m^{2}] = 0

ð 2 m x dm x c^{2} = 2m [ dm x v^{2} + v x m x dv]

c^{2} dm = v^{2} dm + v m dv

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

D _{K} = c^{2} 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 } c^{2} dm

k =c^{2} (m m_{0})

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 + m_{0} c^{2} = c^{2} (m- m_{0}) + m_{0} c^{2} = m c^{2}

Hence, [E=m c^{2} ]

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.