Question

In Fig. 12-22, a vertical rod is hinged at its lower end and attached to a cable at its upper end. A horizontal force${\overrightarrow{\mathbf{F}}}_{\mathbf{a}}$is to be applied to the rod as shown. If the point at which then force is applied is moved up the rod, does the tension in the cable increase, decrease, or remain the same?

Short answer

The magnitude of the tension in the cable increases with the change in the position of the applied force to the rod.

Step-by-step solution

The given data

A figure for the rod–cable system is given.

Understanding the concept of equilibrium

We use the concept of static equilibrium, i.e., balancing of forces and torques.

Formulae:

The value of the net force at equilibrium,$F_{net=0}$ (i)

The value of the torque at equilibrium, ${\tau }_{net}=0$ (ii)

Calculation of the behavior of the tension

Initially, the rod is balanced when$F_a$is applied at the given position.

Thus,${\overrightarrow{F}}_{net}=0$and${\overrightarrow{\tau }}_{net}=0$are satisfied about the hinge that satisfies the equation of equilibrium considering equations (i) and (ii).

In this case, the torque is due to the applied force and the tension in the cable. The torque balancing equation says, $\overrightarrow{{\tau }_{force}}=\overrightarrow{{\tau }_{cable}}$

Now, when the position of the applied force is changed, the rod is still in equilibrium. Thus, $\overrightarrow{{\tau }_{net}}=0$ is still satisfied.

But now, though $F_a$is the same, its moment arm has increased.

Thus, the torque due to applied force has increased. So to balance this torque, the torque due to the tension in the cable should also increase. But for the cable, the moment arm cannot be altered; hence, the magnitude of the tension in the cable should increase.