Question

a uniform yellow beam of length, i is attached to a wire at one end. the other end is fixed by a hinge (blue dot). a tension, t, is applied to the wire. what is the torque due to the tension, t? (hint: the coordinate system) a. tl cos β b. -tl sin θ c. -tl cos (θ + β) d. -tl sin (θ - β) e. -tl sin (θ - β)

Explanation:

Step1: Recall the formula for torque

The formula for torque \(\tau = rF\sin\theta\), where \(r\) is the distance from the pivot, \(F\) is the force, and \(\theta\) is the angle between the position vector \(r\) and the force vector \(F\). In this case, \(r = l\), \(F=T\). The angle between the wire (where the force \(T\) acts) and the perpendicular to the bar (for torque calculation) needs to be considered. Using the right - hand rule and angle relations. The angle between the force vector \(T\) and the lever arm (perpendicular to the bar) is \((\theta-\beta)\). Also, considering the direction (negative sign for the direction of rotation).

Step2: Calculate the torque

The torque \(\tau=-Tl\sin(\theta - \beta)\) (the negative sign is because of the direction of rotation which is opposite to the positive direction of the coordinate system assumed for torque).

Answer:

E. \(-Tl\sin(\theta - \beta)\)