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Total energy

Total energy E of a particle is

E = γ m c 2

where m is mass, c is the speed of light, γ = 1 1 u 2 c 2 , and u is the velocity of the mass relative to an observer.

Rest energy

Rest energy of an object is

E 0 = m c 2 .

This is the correct form of Einstein’s most famous equation, which for the first time showed that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. The implications of these first two equations regarding relativistic energy are so broad that they were not completely recognized for some years after Einstein published them in 1905, nor was the experimental proof that they are correct widely recognized at first. Einstein, it should be noted, did understand and describe the meanings and implications of his theory.

Calculating rest energy

Calculate the rest energy of a 1.00-g mass.

Strategy

One gram is a small mass—less than one-half the mass of a penny. We can multiply this mass, in SI units, by the speed of light squared to find the equivalent rest energy.

Solution

  1. Identify the knowns: m = 1.00 × 10 −3 kg ; c = 3.00 × 10 8 m/s.
  2. Identify the unknown: E 0 .
  3. Express the answer as an equation: E 0 = m c 2 .
  4. Do the calculation:
    E 0 = m c 2 = ( 1.00 × 10 −3 kg ) ( 3.00 × 10 8 m/s ) 2 = 9.00 × 10 13 kg · m 2 / s 2 .
  5. Convert units. Noting that 1 kg · m 2 /s 2 = 1 J, we see the rest energy is:
    E 0 = 9.00 × 10 13 J.

Significance

This is an enormous amount of energy for a 1.00-g mass. Rest energy is large because the speed of light c is a large number and c 2 is a very large number, so that m c 2 is huge for any macroscopic mass. The 9.00 × 10 13 J rest mass energy for 1.00 g is about twice the energy released by the Hiroshima atomic bomb and about 10,000 times the kinetic energy of a large aircraft carrier.

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Today, the practical applications of the conversion of mass into another form of energy , such as in nuclear weapons and nuclear power plants, are well known. But examples also existed when Einstein first proposed the correct form of relativistic energy, and he did describe some of them. Nuclear radiation had been discovered in the previous decade, and it had been a mystery as to where its energy originated. The explanation was that, in some nuclear processes, a small amount of mass is destroyed and energy is released and carried by nuclear radiation. But the amount of mass destroyed is so small that it is difficult to detect that any is missing. Although Einstein proposed this as the source of energy in the radioactive salts then being studied, it was many years before there was broad recognition that mass could be and, in fact, commonly is, converted to energy ( [link] ).

Photos of the Sun and of the Susquehanna Steam Electric Station are shown.
(a) The sun and (b) the Susquehanna Steam Electric Station both convert mass into energy—the sun via nuclear fusion, and the electric station via nuclear fission. (credit a: modification of work by NASA; credit b: modification of work by “ChNPP”/Wikimedia Commons)

Because of the relationship of rest energy to mass, we now consider mass to be a form of energy rather than something separate. There had not been even a hint of this prior to Einstein’s work. Energy-mass equivalence is now known to be the source of the sun’s energy, the energy of nuclear decay, and even one of the sources of energy keeping Earth’s interior hot.

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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