Animated Solution for Mathematics - Vector Algebra: 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

Imagine you are standing on a vast, frictionless plane, holding a rigid rod. You have two forces, $A$ and $B$, pushing down on this rod in the exact same direction. This is the classic scenario of 'like parallel forces.'

In the world of Statics, we are obsessed with simplification. We don't want to deal with two forces if we can replace them with one 'super-force' that does the exact same job. This is the concept of the Resultant.

Since $A$ and $B$ are parallel and pointing in the same direction, their combined effect is simply their sum:

This resultant $R$ acts at a specific point, maintaining the balance of the rod. But now, we introduce a twist: a couple of moment $H$.

A couple is one of the most beautiful concepts in physics. It is a pure rotational agent consisting of two equal and opposite forces separated by a distance.

Because the forces are equal and opposite, their vector sum is zero. They don't push the rod left or right; they don't push it up or down. They only make it spin.

When we introduce this couple $H$ into our system, we are adding a pure turning effect without adding any translational force. The net force of our system remains $R=A+B$.

Finding the Equilibrium

The system now has a net force $R$ and a net moment $H$. To keep the system equivalent to the original state, we must find a way to represent this new configuration with a single resultant.

If we have a force $R$ and we want to account for an additional moment $H$, we can achieve this by simply shifting the line of action of $R$. By moving the point of application of the force, we change the moment it creates.

If we shift our resultant $R$ by a distance $x$, we create a new moment about the original position equal to $R\cdot x$. For the system to remain equivalent to the original state, this new moment must exactly balance the applied couple $H$.

Therefore, we set up our fundamental equation:

We know that $R=A+B$. Substituting this into our balance equation, we get:

Our goal is to find the displacement $x$. With a simple algebraic step, we isolate $x$:

This result is profound. It tells us that the displacement of the resultant is directly proportional to the magnitude of the couple and inversely proportional to the sum of the forces.

This is a clean, elegant relationship that governs how systems of forces behave in equilibrium. Whenever you see a problem involving couples and resultants, remember this: a couple is just a 'moment' waiting to be balanced by a shift in the resultant's position.