Rest Mass Energy

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Rest Mass Energy

By: Shiv Tailor

Rest mass energy is the energy an object has when is neither moving nor is it in a potential field. The famous equation

[math]\displaystyle{ E=mc^2 }[/math]

demonstrates the mass energy equivalence where

E is the internal energy in joules

m is the mass in kilograms

c is the speed of light in a vacuum (approximately [math]\displaystyle{ 3.00 \times 10^{8} {\rm \ m/s} }[/math])

This relationship was shown by Albert Einstein in 1905.

Albert Einstein in 1947

A Mathematical Model

This is a very simple equation but it can be rewritten in many ways.

[math]\displaystyle{ E/(c^2)=m }[/math]

This is especially important because it says that all the energy, regardless of form, can be equated to mass

A Computational Model

The best way to visualize this mass-energy equivalence to think about a pan on a stove. As the pan heats up one would see that the pan gets hotter, and one could infer that the internal energy of the pan goes up. This change in energy can be equated to mass. This is shown in the examples.

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Simple

What is the rest mass energy of an object that weighs 7 kg and is going 40 mi/h?

Since it asks for rest mass energy, we ignore the movement.

[math]\displaystyle{ E = mc^2 }[/math]

[math]\displaystyle{ E = (7 {\rm \ kg})*(3e8 {\rm \ m/s})^2 }[/math]

[math]\displaystyle{ E = 6.3e17 {\rm \ J} }[/math]

Middling

 What is the mass of an object that has a rest mass energy of 1e16 J
and is traveling through a medium where the speed of light is 2e8 m/s?

The rest mass energy always uses the speed of light in a vacuum (c) which is ~3e8 m/s.

[math]\displaystyle{ E = mc^2 }[/math]

[math]\displaystyle{ m = \frac{E}{c^2} }[/math]

[math]\displaystyle{ m = \frac{1e16 {\rm \ J}}{3e8 {\rm \ m/s}^2} }[/math]

[math]\displaystyle{ m = 0.11 {\rm \ kg} }[/math]

Difficult

A pot with mass 0.5 kg is filled with 1.2 kg of water. After some time 10,000 kJ of heat have been added to the pot
and 40,000 kJ of heat have been added to the water. Ignoring evaporation what is the mass of the pot and the water?

Initial rest mass energy:

[math]\displaystyle{ E_i = m_i c^2 }[/math]

[math]\displaystyle{ E_i = (1.5 {\rm \ kg})*(3e8 {\rm \ m/s})^2 }[/math]

[math]\displaystyle{ E_i= 1.35e17 {\rm \ J} }[/math]

Final rest mass energy:

[math]\displaystyle{ E_f = (10,000 {\rm \ kJ})*(1000 {\rm \ J/kJ}) + (40,000 {\rm \ kJ})*(1000 {\rm \ J/kJ}) + 1.35e17 {\rm \ J} }[/math]

[math]\displaystyle{ E_f = 1.3500000005e17 {\rm \ J} }[/math]

[math]\displaystyle{ m_f = \frac{E_f}{c^2} }[/math]

[math]\displaystyle{ m_f = \frac{1.3500000005e17 {\rm \ J}}{3e8 {\rm \ m/s}^2} }[/math]

[math]\displaystyle{ m_f = 1.5000000006 {\rm \ kg} }[/math]

**note: The mass change is almost 0. Why?**

History

Mass-energy equivalence was proposed by Albert Einstein in 1905 in his paper Does the Inertia of a Body Depend upon its Energy Content

See also

See the energy section of this wiki.

Further reading

https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence

References

http://einsteinpapers.press.princeton.edu/vol2-trans/186