Graphical Analysis of Rotational Work

From constant to variable torque

For a constant torque $\tau$, the work done over an angular displacement $\Delta\theta$ is:

On a graph of torque versus angular position, this product equals the area of the rectangle formed between the horizontal torque line and the $\theta$-axis. The base of the rectangle is $\Delta\theta$, and the height is $\tau$.

When torque varies with angular position, the same principle applies: work equals the area under the torque versus angular position curve. This generalizes the constant-torque formula to any torque function.

Dimensional check: The horizontal axis has units of radians (dimensionless) and the vertical axis has units of $\text{N} \cdot \text{m}$. The product $(\text{N} \cdot \text{m}) \cdot (\text{rad}) =\text{N} \cdot \text{m}= \text{J}$, confirming that area gives work.

Sign conventions: Regions where $\tau > 0$ (above the axis) contribute positive work—energy enters the system. Regions where $\tau < 0$ (below the axis) contribute negative work—energy leaves the system. The total work is the algebraic sum of all signed areas.

Decomposing piecewise-linear graphs

Break the region under the curve into rectangles, triangles, and trapezoids.

Rectangle: When torque is constant over an interval,

Triangle: When torque changes linearly from zero (or to zero),