Work

Overview

Engineering reference data for Work in dynamics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Momentum

$p = mv$

Mass × velocity.

Work

$W = Fd\cos\theta$

Force × displacement × cos(angle).

Variables

Symbol	Description	Unit
$F$	Force	N
$m$	Mass	kg
$a$	Acceleration	m/s²
$v$	Velocity	m/s
$E_k$	Kinetic energy	J

Work Done by a Constant Force

Work is the product of a constant force acting on an object and the distance the object moves in the direction of that force. The work-energy theorem states that the net work done on an object equals its change in kinetic energy. $W_F = F s$ where $W_F$ is work (J, ft·lbf), $F$ is constant force (N, lbf), and $s$ is displacement in the direction of force (m, ft).

Example: A constant force of 20 N moves an object 30 m.
$W_F = (20\,\text{N})(30\,\text{m}) = 600\,\text{J}$

Work Done by a Spring Force

Springs exert a variable force described by Hooke's Law. The work done is proportional to the square of the displacement and equals the average force multiplied by the distance. $W_{\text{spring}} = \frac{1}{2} k s^2 = \frac{1}{2} F_{\text{spring\_max}} s$ where $k$ is the spring constant (N/m), $s$ is displacement from equilibrium (m), and $F_{\text{spring\_max}} = k s$ is the maximum spring force.

Example: A spring with constant $k = 1$ N/m is stretched 1 m.
$W_{\text{spring}} = \frac{1}{2} (1\,\text{N/m})(1\,\text{m})^2 = 0.5\,\text{J}$

Rotational Work

Work can also be done by a torque (moment) acting through an angular displacement. $W_M = T \theta$ where $W_M$ is rotational work (J, ft·lbf), $T$ is torque (N·m, ft·lbf), and $\theta$ is angular displacement (radians).

Example: A shaft applies a torque of 300 N·m over one revolution ($\theta = 2\pi$ rad).
$W_M = (300\,\text{N·m})(2\pi\,\text{rad}) \approx 1884\,\text{J}$

General Representations of Work

Work can be expressed as a path integral, accounting for force due to gravity or pressure. $W = \int \mathbf{F} \cdot d\mathbf{s} = \int m g \, dh = \int p \, dV$ where $m$ is mass (kg), $g$ is gravitational acceleration (m/s²), $h$ is height (m), $p$ is pressure (Pa), and $dV$ is the change in volume (m³).

References

• Original Source