Question 1.3: Mass is energy Here we will use the concept of significant f...
Mass is energy
Here we will use the concept of significant figures in one of the most famous equations in all of physics. Einstein’s theory of relativity predicts that mass and energy are equivalent. For instance, the energy E corresponding to the mass m of an electron is given by Einstein’s equation
E = mc² (equivalence of mass and energy),
where c is the speed of light. Calculate the value of E for an electron, using powers of ten and three significant figures.
SET UP We find the needed numbers inside the back cover or in Appendix E: m = 9.11 \times 10^{-31} kg and c = 3.00 \times 10^8 m/s.
SOLVE We use these numbers to calculate the value of E:
E = (9.11 \times 10^{-31} kg) (3.00 \times 10^8 m/s)^2
= (9.11)(3.00)^2 (10^{-31})(10^8)^2 (kg \cdot m^2)/s^2
= (82.0)(10^{[-31+(12\times 8)]})(kg \cdot m^2)/s^2
= 8.20 \times 10^{-14} kg \cdot m^2/s^2.
REFLECT Most pocket calculators can use scientific notation and do this addition of exponents automatically for you, but you should be able to do such calculations by hand when necessary. Incidentally, the value used for c in this example has three significant figures, even though two of them are zeros. To greater precision, c = 2.997925 \times 10^8 m/s.
Practice Problem: Using the values for m and c in Appendix E, calculate the energy of the electron to five significant figures. Answer: 8.1871 \times 10^{-14}.