20.6 Relativistic energy  (Page 2/12)

Calculating rest energy: rest energy is very large

Calculate the rest energy of a 1.00-g mass.

Strategy

One gram is a small mass—less than 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. $m=1\text{.00}\times {\text{10}}^{-3}\text{kg}$ ; $c=3\text{.}\text{00}\times {\text{10}}^8\text{m/s}$
2. Identify the unknown. $E_0$
3. Choose the appropriate equation. $E_0={\mathrm{mc}}^2$
4. Plug the knowns into the equation.
   $\begin{array}{lll}{E}_0& =& {\mathrm{mc}}^2=(1.00\times {\text{10}}^{-3}\text{kg})(3.00\times {\text{10}}^8\text{m/s}{)}^2\\ & =& 9.00\times {\text{10}}^{\text{13}}\text{kg}\cdot {\text{m}}^2{\text{/s}}^2\end{array}$
5. Convert units.

   Noting that $1\text{kg}\cdot {\text{m}}^2{\text{/s}}^2=\mathrm{1\; J}$ , we see the rest mass energy is

   $E_0=9\text{.}\text{00}\times {\text{10}}^{\text{13}}\text{J}.$

Discussion

This is an enormous amount of energy for a 1.00-g mass. We do not notice this energy, because it is generally not available. Rest energy is large because the speed of light $c$ is a large number and $c^2$ is a very large number, so that ${\mathrm{mc}}^2$ is huge for any macroscopic mass. The $9\text{.}\text{00}\times {\text{10}}^{\text{13}}\text{J}$ 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. If a way can be found to convert rest mass energy into some other form (and all forms of energy can be converted into one another), then huge amounts of energy can be obtained from the destruction of mass.