Torque-Time Graphs

When examining torque vs time graphs, the area under the curve in a given interval of time represents the total angular impulse delivered over that duration of time.

For a constant torque, the graph forms a horizontal line. The area bounded by this line is a rectangle with height equal to the average torque $\tau_{\text{avg}}$ and width equal to the time interval $\Delta t$.

For negative torques, the area is negative and therefore contributes a negative impulse to the total.

Non-Constant Torques

When the torque varies continuously, such as during a collision where torque ramps up and down, the graph often forms shapes like triangles or trapezoids. To calculate the total impulse, decompose the area under the curve into simple geometric regions like rectangles and triangles. Calculate the area of each section individually.

Summing these individual geometric areas yields the total impulse. Units of newton-meters ($\mathrm{N}\cdot \text{m}$) for torque and seconds ($\mathrm{s}$) for time result in impulse units of newton-meter-seconds ($\mathrm{N} \text{m}\cdot \mathrm{s}$).