243 Work
A measure of energy transferred by a force acting through a displacement that is used to track energy handed from one system to another.
The line-integral definition. Work is the line integral of force along a path. This principle is used to compute the energy transferred by a known force.
The work of a force is
\[ W = \displaystyle\int\mathbf{F}\cdot d\mathbf{r} \]
where
- \(W\) is the work.
- \(\mathbf{F}\) is the force.
- \(d\mathbf{r}\) is a displacement along the path.
The parallel component. Only the component of force along the displacement contributes. Work is a scalar. This principle is used to obtain zero work when the force is perpendicular to the motion.
The infinitesimal work is
\[ dW = F_{\parallel}\,ds \]
where
- \(dW\) is the infinitesimal work.
- \(F_{\parallel}\) is the component of force along the path.
- \(ds\) is an infinitesimal path length.
The work-energy theorem. The work of the net force equals the change in kinetic energy. This principle is used to compute a speed change from a known work.
The work-energy theorem is
\[ W_{\mathrm{net}} = \Delta K \]
where
- \(W_{\mathrm{net}}\) is the work of the net force.
- \(\Delta K\) is the change in kinetic energy.
Path independence for conservative forces. For a conservative force the work depends only on the endpoints. Around a closed loop that work vanishes. This principle is used to define a potential energy.
Zero magnetic work. A magnetic force on a moving charge is perpendicular to the velocity and does no work. This principle is used to change direction without changing kinetic energy.
\(p\Delta V\) work. For a gas at pressure \(p\), the work of a volume change is \(-p\,\Delta V\) when \(p\) is constant. This principle is used to compute compression work.
The constant-pressure compression work is
\[ W = -p\,\Delta V \]
where
- \(W\) is the work done on the gas.
- \(p\) is the pressure.
- \(\Delta V\) is the change in volume.
Note: Everyday “work” is broader. In physics, work means energy transfer of this specific kind.
243.1 References
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — energy transfer; path dependence.
- Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — line integral of force.
- Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics. — parallel component; PdV work.
- Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — magnetic force does no work.
- Schwartz, M. Principles of Electrodynamics. Dover, 1972. — closed-loop work for conservative forces.