Question

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the figure shows an overhead view of a large door that is free to pivot on a hinge along its left edge. a force f is applied to the door in four different ways. the magnitude of the force is the same in each case. in which case is the torque on the door the largest?
a
b
hinge
hinge
f
f
c
d
hinge
hinge
f
f
b
d
a
c

Explanation:

Step1: Recall the formula for torque

Torque $\tau = rF\sin\theta$, where $r$ is the distance from the pivot (hinge) to the point of force application, $F$ is the force magnitude, and $\theta$ is the angle between the position vector $r$ (from pivot to force - application point) and the force vector $F$.

Step2: Analyze each case

• Case A:

Let the length of the door be $L$. Here, $r = L$ (distance from hinge to the end of the door) and $\theta = 90^{\circ}$, so $\sin\theta=1$. Then $\tau_A=LF\times1 = LF$.

• Case B:

$r = L$, but $\theta<90^{\circ}$ (since the force is not perpendicular to the door). Using $\tau = rF\sin\theta$, and since $\sin\theta<1$ (for $\theta\in(0,90^{\circ})$), $\tau_B=LF\sin\theta

• Case C:

$r < L$ (force is applied at a point closer to the hinge than the end of the door) and $\theta = 90^{\circ}$. So $\tau_C=rF\times1

• Case D:

The position vector $r$ (from hinge to the end of the door) and the force vector $F$ are parallel ($\theta = 180^{\circ}$ or $\theta = 0^{\circ}$). Then $\sin\theta = 0$, and $\tau_D=rF\sin\theta=0$.

Answer:

A.