IEMDaily - Video Lecture Notes

Definition

Mass energy equivalence

The energy

E

equivalent to a mass

m

is given by the equation

E = m c^{2}

Thus, matter can be converted into energy and energy into matter.

Definition

Rest mass energy of a particle

When a particle is at rest, the energy concentrated in it,

E_{0} = m_{0} c^{2}

is called the rest mass energy of the particle.

Definition

Mass of an object moving at high speeds

If a particle moves at a speed

v

, its mass changes to

m = m_{0} γ = \frac{m_{0}}{\sqrt{1 - v^{2} / c^{2}}}

Definition

Energy of an object moving at high speeds

The total energy of a particle moving with a speed

v

is given as

E = m c^{2}

= m_{0} c^{2} (1 - \frac{v^{2}}{c^{2}})^{- \frac{1}{2}}

= m_{0} c^{2} (1 + \frac{v^{2}}{2 c^{2}} + \frac{1}{2} . \frac{3}{4} \frac{v^{4}}{c^{4}} + . . .)

= m_{0} c^{2} + \frac{1}{2} m_{0} v^{2} + . . .

Definition

Kinetic energy of a particle from difference of mass energy and rest mass energy

The extra energy

(E - m_{0} c^{2})

is called the kinetic energy.
where

E = m c^{2} = m_{0} c^{2} + \frac{1}{2} m_{0} v^{2} + . . .

v << c

, the higher order terms in

E

can be neglected and hence the kinetic energy is

K = \frac{1}{2} m_{0} v^{2}

Definition

Momentum of a particle moving at high speeds

The momentum of a particle moving with a speed

v

is given as :

p = m v = \frac{m_{0} v}{\sqrt{1 - v^{2} / c^{2}}}

where

m_{0}

is the rest mass of the particle.

Definition

Relationship between energy and momentum of a particle having zero rest mass

In relativistic mechanics, we have
Momentum,

p = m v = \frac{m_{0} v}{\sqrt{1 - v^{2} / c^{2}}}

Energy,

E = m c^{2} = \frac{m_{0} c^{2}}{\sqrt{1 - v^{2} / c^{2}}}

so we have

E^{2} = m_{0}^{2} c^{4} + p^{2} c^{2}

For particles having zero rest mass like photons,

m_{0} = 0

and hence

E = p c

p = \frac{E}{c}

Definition

Define the critical size of a substance to sustain nuclear fission

The critical size must at least include enough fissionable material to reach critical mass. If the size of the reactor core is less than a certain minimum, fission neutrons escape through its surface and the chain reaction is not sustained.

Example

Calculate the energy released in a nuclear fission reaction

Example: In nuclear fission,

0.1

% mass is converted into energy. Find the energy released in the fission of

1 k g

mass in

k W h

.

Solution:

0.1

% mass is converted.

E = (\frac{1}{1000} \times 1 k g) (3 \times 10^{8})^{2} J

= 9 \times 10^{13} J

Let this energy is liberated per second then,
E per hour

= \frac{9 \times 10^{13}}{60 \times 60}

= 0.25 \times 10^{11} W h

= 2.5 \times 10^{7} k W h

Definition

Working principle of nuclear weapons

The splitting of atomic nuclei releases enormous energy. When a single free neutron strikes the nucleus of an atom of fissile (radioactive) material like uranium 235 or plutonium 239, it usually knocks two or three more neutrons free. Energy is released when those neutrons split off from the nucleus, and the newly released neutrons strike other uranium 235 (or plutonium 239) nuclei, splitting them in the same way, releasing more energy and more neutrons. This chain reaction spreads almost instantaneously. The atomic bomb (A-bomb) was a weapon of destruction that used the power released by the splitting of atomic nuclei.