Answer
We can rank the situations in order of the magnitude of the torque applied to the handle, from smallest to largest:
$e \lt a = b = d \lt c$
Work Step by Step
The magnitude of the torque can be expressed as $\tau = r~F~sin~\theta$, where $r$ is the displacement between the rotation axis and the point where the force is applied, $F$ is the force, and $\theta$ is the angle between the force vector and the displacement vector $r$.
We can find the magnitude of the torque for each situation:
(a) $\tau = r~ F~sin~\theta$
$\tau = (0.50~m)(20~N)~sin~90^{\circ}$
$\tau = 10~N\cdot m$
(b) $\tau = r~ F~sin~\theta$
$\tau = (0.25~m)(40~N)~sin~90^{\circ}$
$\tau = 10~N\cdot m$
(c) $\tau = r~ F~sin~\theta$
$\tau = (0.25~m)(80~N)~sin~60^{\circ}$
$\tau = 17.3~N\cdot m$
(d) $\tau = r~ F~sin~\theta$
$\tau = (0.25~m)(80~N)~sin~30^{\circ}$
$\tau = 10~N\cdot m$
(e) $\tau = r~ F~sin~\theta$
$\tau = (0.50~m)(40~N)~sin~0^{\circ}$
$\tau = 0$
We can rank the situations in order of the magnitude of the torque applied to the handle, from smallest to largest:
$e \lt a = b = d \lt c$
