AP Physics C Mechanics Question of the Day

1. The value of the $\alpha$ versus $t$ graph at $t=t_f$
2. The slope of the $\alpha$ versus $t$ graph at $t=t_f$
3. The area under the $\omega$ versus $t$ graph from $t=0$ to $t=t_f$
4. The area under the $\alpha$ versus $t$ graph from $t=0$ to $t=t_f$ (correct answer)

Explanation: By definition, $\alpha = \frac{d\omega}{dt}$. Rearranging gives $d\omega = \alpha dt$. To find the total change in angular velocity, $\Delta\omega$, we integrate this expression over the time interval: $\Delta\omega = \int_{0}^{t_f} \alpha(t) dt$. The definite integral of a function represents the area under the curve of that function. Thus, the change in angular velocity is the area under the $\alpha$ versus $t$ graph.