$$A$$ and $$B$$ are two like parallel forces. A couple of moment $$H$$ lies in the plane of $$A$$ and $$B$$ and is contained with them. The resultant of $$A$$ and $$B$$ after combining is displaced through a distance

Solution & Explanation

1. Understand the Initial System of Parallel Forces

We are given two like parallel forces, $$A$$ and $$B$$. "Like" parallel forces act in the exact same direction along parallel lines of action.

The resultant ($$R$$) of two like parallel forces is simply the algebraic sum of their magnitudes, and it acts in the same direction:

$$R = A + B$$

Initially, this resultant force $$R$$ passes through a specific point (let's call it $$C$$) on the line segment connecting the points of application of forces $$A$$ and $$B$$, satisfying the principle of moments: $$A \cdot AC = B \cdot BC$$.

2. Introduce the Couple of Moment $$H$$

A coplanar couple of moment $$H$$ is introduced into the plane containing forces $$A$$ and $$B$$.

In rigid-body mechanics, a couple consists of two equal and opposite forces whose net translational force vector is exactly zero. Therefore, adding a couple to a system of forces does not alter the net total force magnitude or direction of the system. The new overall resultant force ($$R'$$) must still be:

$$R' = A + B$$

3. Calculate the Displacement of the Resultant Line of Action

While the couple doesn't change the magnitude of the total resultant force, its rotational moment shifts the position where the net force effectively acts.

To find the perpendicular distance ($$d$$) by which the line of action of the resultant shifts to balance out the extra torque introduced by the couple, we use the foundational relationship between a force, a displacement, and a couple's moment:

$$\text{Moment of the Couple} = \text{Resultant Force} \times \text{Displacement}$$

$$H = R \cdot d$$

Substitute the value of the combined resultant force ($$R = A + B$$) into the expression:

$$H = (A + B) \cdot d$$

Isolating the displacement variable ($$d$$):

$$d = \frac{H}{A + B}$$

Concept Check: Introducing a pure rotational moment $$H$$ to a system with an existing net translational force $(A+B)$ creates a structural torque imbalance. To restore equivalence without changing the net force magnitude, the entire translational force vector must slide sideways by a distance inversely proportional to the system's total linear strength.

Correct Option Key: Option D ($$\frac{H}{A+B}$$)