Understanding Equilibrium Conditions for Coplanar Concurrent Forces

The question asks for the necessary conditions for a system of coplanar concurrent forces to be in equilibrium. Let's break down what these terms mean and analyze the conditions required.

Key Concepts:

• Coplanar Forces: Forces that lie in the same plane.
• Concurrent Forces: Forces whose lines of action all intersect at a single common point.
• Equilibrium: A state where a body remains at rest or moves with constant velocity. For a force system, this means the net force and net moment acting on the body are both zero. Mathematically, this is represented as $\sum \vec{F} = 0$ and $\sum \vec{M} = 0$.

Analysis of Equilibrium for Concurrent Forces:

For a system of forces to be in equilibrium, two fundamental conditions must be met:

1. Translational Equilibrium: The vector sum of all forces must be zero. This means $\sum \vec{F} = 0$.
2. Rotational Equilibrium: The sum of the moments of all forces about any point must be zero. This means $\sum \vec{M} = 0$.

However, when forces are concurrent, they all pass through a single point (the point of concurrency). Let's consider the moment condition ($\sum \vec{M} = 0$) about this specific point of concurrency. The moment of a force about a point is given by $M = F \times d$, where $d$ is the perpendicular distance (moment arm) from the point to the line of action of the force. Since all forces are concurrent at a point P, the line of action of each force passes through P. Therefore, the perpendicular distance ($d$) for every force about point P is zero. This implies that the moment of each individual force about the point of concurrency is zero ($M_i = F_i \times 0 = 0$). Consequently, the sum of the moments about the point of concurrency is always zero ($\sum M_P = 0$), regardless of the magnitudes or directions of the forces, as long as they are concurrent.

Because the rotational equilibrium condition ($\sum M_P = 0$) is automatically satisfied for concurrent forces about their point of concurrency, the sole condition required to ensure equilibrium for a system of coplanar concurrent forces is the translational equilibrium condition: the vector sum of all forces must be zero ($\sum \vec{F} = 0$).

The vector equation $\sum \vec{F} = 0$ can be resolved into components along any set of axes. For convenience, we often choose two mutually perpendicular axes (like the horizontal x-axis and the vertical y-axis). If the sum of the force components along these two perpendicular axes are both zero, then the vector sum of the forces is zero.

• Sum of forces along the x-axis: $\sum F_x = 0$
• Sum of forces along the y-axis: $\sum F_y = 0$

If these two conditions hold, then $\sum \vec{F} = (\sum F_x)\hat{i} + (\sum F_y)\hat{j} = 0\hat{i} + 0\hat{j} = 0$. This ensures translational equilibrium, which is sufficient for concurrent forces.

Evaluating the Options:

Let's examine each option in the context of coplanar concurrent forces:

1. "The algebraic sum of the forces resolved in only one direction must be zero."

   This condition ($\sum F_x = 0$ or $\sum F_y = 0$) is insufficient. For example, if you only consider horizontal forces and their sum is zero, there might still be unbalanced vertical forces causing the body to move up or down.

2. "The algebraic sum of the forces acting on the body must be equal to the sum of the moments about any point."

   This statement is conceptually incorrect. It equates forces with moments, which have different units and represent different physical quantities (force causes linear acceleration, moment causes angular acceleration). For equilibrium, both the net force and the net moment must be zero independently.

3. "The algebraic sum of the moments of all forces about their point of concurrency must be zero."

   As explained earlier, this condition is always true for any set of concurrent forces, regardless of whether they are in equilibrium or not. It does not provide any constraint on the forces themselves, only on the reference point for calculating moments. Therefore, it is insufficient to guarantee equilibrium.

4. "The algebraic sum of the forces resolved in any two mutually perpendicular directions must be zero."

   This statement correctly represents the condition for translational equilibrium ($\sum F_x = 0$ and $\sum F_y = 0$). As demonstrated, for concurrent forces, translational equilibrium is the necessary and sufficient condition for the system to be in equilibrium. This option accurately captures this requirement.

Conclusion:

For a system of coplanar concurrent forces to be in equilibrium, the net force acting on the body must be zero. This is effectively stated by ensuring that the sum of the force components along any two mutually perpendicular directions is zero.