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Mathematics for Physics

235 Force

An interaction that is used to change the motion of an object.

Force as a push or pull. A force is a push or a pull exerted by an identifiable agent. This principle is used to name the source of every force in a free-body diagram.

Superposition. Independent forces on one body add as vectors. This principle is used to replace a many-force problem by one net force.

The superposition of forces is

\[ \mathbf{F}_{\mathrm{net}} = \sum_{i}\mathbf{F}_{i} \]

where

  • \(\mathbf{F}_{\mathrm{net}}\) is the net force.
  • \(\mathbf{F}_{i}\) is an individual force.

Newton’s second law. The net force equals mass times acceleration. This principle is used to compute how a known force changes the motion.

Newton’s second law is

\[ \mathbf{F}_{\mathrm{net}} = m\mathbf{a} \]

where

  • \(\mathbf{F}_{\mathrm{net}}\) is the net force.
  • \(m\) is the mass.
  • \(\mathbf{a}\) is the acceleration.

Newton’s third law. Forces occur in equal and opposite pairs between two bodies. This principle is used to cancel internal forces in a system.

Newton’s third law is

\[ \mathbf{F}_{12} = -\mathbf{F}_{21} \]

where

  • \(\mathbf{F}_{12}\) is the force on body 1 by body 2.
  • \(\mathbf{F}_{21}\) is the force on body 2 by body 1.

Force from a potential. A conservative force is minus the derivative of potential energy with position. This principle is used to obtain \(\mathbf{F}\) from \(U\).

The conservative-force relation is

\[ \mathbf{F} = -\dfrac{dU}{d\mathbf{r}} \]

where

  • \(\mathbf{F}\) is the force.
  • \(U\) is the potential energy.
  • \(\mathbf{r}\) is the position.

Note: A push or a pull is the everyday name for a force. In general relativity, gravity is described as the curvature of spacetime rather than as a force in the Newtonian sense.

235.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — push or pull; agent; superposition; instantaneous action.
  2. Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics, Vol. I. — material origin; action-reaction pairs.
  3. Logan, J. D. A First Course in Differential Equations. Springer, 2015. — acceleration law.
  4. Jaffe, R. L., & Taylor, W. The Physics of Energy. Cambridge University Press, 2018. — energy transformation.
  5. Susskind, L., & Friedman, A. Quantum Mechanics: The Theoretical Minimum. Basic Books, 2014. — force from potential.
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press, 2019. — gravity as curvature.