Equilibrant and Resultant Forces

In physics, when multiple forces act on an object, their combined effect can be represented by a single force called the resultant force. This resultant force is the vector sum of all the individual forces.

The equilibrant force is the single force that, when applied to an object along with other forces, results in a net force of zero. In other words, the equilibrant force is the force needed to achieve equilibrium, where there is no acceleration.

Relationship Between Equilibrant and Resultant

To achieve equilibrium, the equilibrant force must completely counteract the resultant force. This means the equilibrant must:

• Have the same magnitude as the resultant force.
• Act in the exact opposite direction to the resultant force.

When the equilibrant and resultant forces are added together as vectors, their sum is zero. This is why the equilibrant force is sometimes defined as the negative of the resultant force.

Let $\vec{R}$ be the resultant force and $\vec{E}$ be the equilibrant force. For equilibrium, the net force is zero:

\begin{equation*} \vec{R} + \vec{E} = \vec{0} \end{equation*}

This implies:

\begin{equation*} \vec{E} = -\vec{R} \end{equation*}

This vector equation shows that the magnitude of $\vec{E}$ is equal to the magnitude of $\vec{R}$ ($|\vec{E}| = |\vec{R}|$), and the direction of $\vec{E}$ is opposite to the direction of $\vec{R}$.

Analyzing the Options

• The first option states that the equilibrant is the same as the resultant in magnitude, but its direction is opposite to the resultant. This aligns perfectly with the definition and the vector relationship $\vec{E} = -\vec{R}$.
• The second option suggests the angle is perpendicular. If the equilibrant were perpendicular to the resultant, their vector sum would not be zero (unless both were zero).
• The third option says the equilibrant acts with the resultant. This would result in a net force of $2\vec{R}$ (if they are in the same direction), not zero.
• The fourth option states the direction is the same as the resultant. Similar to option 3, this would add to the resultant force, not balance it.

Therefore, the only statement that correctly describes the relationship between the equilibrant and the resultant force regarding magnitude and direction is that the equilibrant has the same magnitude but the opposite direction as the resultant.