241 Power
A measure of how quickly energy is transferred or transformed that is used to compare fast and slow energy changes.
Power as \(dW/dt\). Power is the time rate of work. This principle is used to convert a work interval into an instantaneous rate.
The definition of power is
\[ P = \dfrac{dW}{dt} \]
where
- \(P\) is the power.
- \(W\) is the work.
- \(t\) is time.
\(\mathbf{F}\cdot\mathbf{v}\). Instantaneous mechanical power is the scalar product of force and velocity. This principle is used to compute the power delivered by a known force.
The mechanical power is
\[ P = \mathbf{F}\cdot\mathbf{v} \]
where
- \(P\) is the power.
- \(\mathbf{F}\) is the force.
- \(\mathbf{v}\) is the velocity.
The watt. The SI unit of power is the watt, one joule per second. This principle is used to report laboratory and engineering rates.
Rotational power. For rotation the power is torque times angular velocity. This principle is used to compute the power of a shaft or a motor.
The rotational power is
\[ P = \boldsymbol{\tau}\cdot\boldsymbol{\omega} \]
where
- \(P\) is the power.
- \(\boldsymbol{\tau}\) is the torque.
- \(\boldsymbol{\omega}\) is the angular velocity.
Electrical power \(IV\). In a circuit the electrical power is the product of current and voltage drop. This principle is used to compute heating in a resistor.
The electrical power is
\[ P = IV \]
where
- \(P\) is the power.
- \(I\) is the current.
- \(V\) is the voltage drop.
Note: A large energy change done slowly can have small power. A small energy change done quickly can have large power. Ideal capacitors and inductors average zero power over an AC cycle.
241.1 References
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — force times velocity; AC averages.
- Shankar, R. Fundamentals of Physics. Yale University Press. — watt; relativistic four-force.
- Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — electrical power VI.
- Jaffe, R. L., & Taylor, W. The Physics of Energy. Cambridge University Press, 2018. — energy rate; fluid flow scaling.
- Feynman, R. P., Leighton, R. B., & Sands, M. The Feynman Lectures on Physics. — rotational power.