Torsional moment is a twisting force that is applied to a beam, causing it to rotate around its longitudinal axis. It is important to understand where this torsional moment is applied in order to analyze the behavior of the beam under the applied force.

The correct answer is option 'A', which states that the torsional moment is applied at the centroid of the beam. Let's understand why this is the case:

The centroid:
The centroid of a beam is the geometric center, which is also the center of mass, of the beam's cross-sectional area. It is the point where the beam's mass is evenly distributed in all directions. In simple terms, the centroid is the point around which the beam will tend to rotate under the influence of a torsional moment.

Reasoning:
When a torsional moment is applied to a beam, it creates a twisting force that tends to rotate the beam around its longitudinal axis. This twisting force is directly proportional to the distance from the centroid of the beam to the point where the torsional moment is applied.

If the torsional moment is applied at the centroid of the beam, the distance from the centroid to the point of application is zero. As a result, the twisting force will be zero, and the beam will not experience any torsional deformation.

On the other hand, if the torsional moment is applied at any other point away from the centroid, the distance between the centroid and the point of application will be non-zero. This will result in a twisting force that causes the beam to deform and rotate.

Conclusion:
In conclusion, the torsional moment is applied at the centroid of the beam because it is the point around which the beam tends to rotate without experiencing any torsional deformation. Applying the torsional moment at any other point will result in torsional deformation and rotation of the beam.