Question

torque: problem solving
a uniform yellow beam of length, l, is attached to a wire at one end. the other end is fixed by a hinge (the dot). a tension, t, is applied to the wire.
what is the torque due to the tension, t?
(note the coordinate system)
a. tl cosθ
b. tl sinθ
c. -tl cosθ
d. -tl sinθ
e. not enough information

Explanation:

Step1: Recall the formula for torque

The formula for torque is $\tau = rF\sin\theta$, where $r$ is the distance from the pivot point to the point of force application, $F$ is the force, and $\theta$ is the angle between the position vector $r$ and the force vector $F$. In this case, $r = l$ and $F = T$.

Step2: Determine the sign of the torque

The positive direction of torque is given as the clock - wise direction. The tension $T$ creates a counter - clockwise torque. So the torque should be negative.

Answer:

C. $-Tl\cos\theta$

(Note: There is a mistake in the initial formula application. The correct formula for torque when the angle between the beam (position vector) and the force is $\theta$ (if we consider the beam as the position vector) is $\tau=-Tl\cos\theta$. The angle used in the torque formula $\tau = rF\sin\phi$ where $\phi$ is the angle between the position vector and the force vector. If we assume the beam is along the x - axis (position vector) and the force makes an angle $\theta$ with the beam, then the angle between the position vector (beam) and the force is $(90^{\circ}-\theta)$ for the $\sin$ formula. But if we use the formula $\tau = rF_{\perp}$, where $F_{\perp}$ is the component of the force perpendicular to the position vector. The component of $T$ perpendicular to the beam (counter - clockwise, so negative) is $T\cos\theta$. So $\tau=-l\times T\cos\theta=-Tl\cos\theta$)