237 Mass
A measure of an object’s resistance to changes in its motion that is used to tell how stubbornly it keeps its current velocity.
Inertial mass. Inertial mass quantifies resistance to acceleration under a net force. This principle is used to write Newton’s second law as \(\mathbf{F}=m\mathbf{a}\).
Newton’s second law is
\[ \mathbf{a} = \dfrac{\mathbf{F}_{\mathrm{net}}}{m} \]
where
- \(\mathbf{a}\) is the acceleration.
- \(\mathbf{F}_{\mathrm{net}}\) is the net force.
- \(m\) is the inertial mass.
Gravitational mass. Gravitational mass sets the strength of mutual attraction between bodies. This principle is used to write Newton’s law of gravitation.
Newton’s law of gravitation is
\[ F = G\dfrac{m_{1}m_{2}}{r^{2}} \]
where
- \(F\) is the gravitational force.
- \(G\) is the gravitational constant.
- \(m_{1}\) and \(m_{2}\) are the gravitational masses.
- \(r\) is the separation.
The equivalence principle. The equivalence principle identifies inertial mass with gravitational mass. This principle is used to treat a gravitational field as locally equivalent to an acceleration.
\(E_{0}=mc^{2}\). Rest energy is mass times \(c^{2}\). This principle is used to convert a mass defect into energy.
The mass-energy relation is
\[ E_{0} = mc^{2} \]
where
- \(E_{0}\) is the rest energy.
- \(m\) is the rest mass.
- \(c\) is the speed of light.
Lorentz invariance of rest mass. Rest mass is a Lorentz scalar: its value is the same in every inertial frame. A Lorentz scalar is a number unchanged by a change of inertial frame. This principle is used to treat \(m\) as an intrinsic property of a particle.
Massless photons. Photons have exactly zero rest mass and always move at the speed of light. This principle is used to distinguish massive particles from massless radiation.
Note: In everyday language mass is often mixed up with weight. Weight is the gravitational pull on an object. Mass is a property of the object itself. Rest mass is not conserved in relativity: rest mass and kinetic energy can convert. In general relativity, mass is a source of spacetime curvature.
237.1 References
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — inertia; intrinsic scalar.
- Carroll, S. M. Spacetime and Geometry. Cambridge University Press, 2019. — invariance; curvature source.
- Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory: The Theoretical Minimum. Basic Books, 2017. — rest energy.
- Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — equivalence of inertial and gravitational mass.
- Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics. — center of mass; massless photons.