Question

If a body is in rotational equilibrium, under the action of three forces, the lines of action of the three forces must be either concurrent or parallel.

Answer 1

Recall the condition for a rotating body to be in equilibrium. Check whether this equilibrium condition for rotational motion is satisfied by the three forces which are concurrent or parallel or by the forces which are not parallel or non-concurrent.

Complete step by step answer:
-The condition of equilibrium for the rotational motion is that the sum of the torques on the object due to all forces acting on the object must be zero.If three forces are concurrent, then all the three forces must be passing through a common point.Suppose there are three forces ${F_1}$, ${F_2}$ and ${F_3}$.
-Let the two forces ${F_1}$ and ${F_2}$ pass through a point O and the force ${F_3}$ does not pass through the point O. Hence, the torque due these two forces ${F_1}$ and ${F_2}$ is zero. But the torque due to the force ${F_3}$ is not zero.In such a condition, the net torque on the rotating body is not zero. Hence, the body will not be in equilibrium.
For the net torque on the body to be zero, the torque due to the force ${F_3}$ must be equal to zero which is only possible if the force ${F_3}$ passes through the point O.Hence, the line of action of the three forces must be concurrent, for the rotating body to be in equilibrium.
-Now let us assume that the two forces ${F_1}$ and ${F_2}$ are parallel and are in the same direction and the third force ${F_3}$ is not parallel to the forces ${F_1}$ and ${F_2}$.For the rotating body to be in equilibrium, the torque due to the forces ${F_1}$ and ${F_2}$ must be equal to the torque due to the third force ${F_3}$. This will happen only if the force ${F_3}$ is parallel and opposite to the forces ${F_1}$ and ${F_2}$.

Hence, the line of action of the three forces must be parallel, for the rotating body to be in equilibrium.

Note: The students may get confused about how the torque is zero when the force passes through a common point. The formula of torque involves a term of sine of angle between the force vector and the position vector. When the forces pass through a common point, this angle is zero for all the forces and the sine of angle zero is also zero. Hence, the torque also becomes zero.