The Law of Moments:

The law of moments, also known as the principle of moments or Varignon's theorem, is a fundamental concept in mechanics. It states that when a number of coplanar forces acting on a particle are in equilibrium, the algebraic sum of their moments about any point in their plane is zero.

Understanding the Options:

To understand why option C is correct, let's briefly discuss the other options:

a) If the algebraic sum of the forces is zero, it means that the forces are balanced, but it doesn't necessarily imply that the forces are in equilibrium. So, option a) is incorrect.

b) The lines of action of the forces being at equal distances does not necessarily indicate equilibrium. It is possible for forces to be at equal distances but still result in a net moment. Therefore, option b) is incorrect.

c) The algebraic sum of the moments about any point in their plane being zero is a characteristic of equilibrium. This implies that the forces are balanced, and there is no net tendency for rotation. Therefore, option c) is correct.

d) The algebraic sum of the moments about any point being equal to the moment of their resultant force about the same point is not necessarily true. The resultant force may not pass through the same point, resulting in a different moment. So, option d) is incorrect.

Explanation:

The law of moments is derived from the principle of equilibrium in mechanics. When a particle is in equilibrium, the sum of the forces acting on it is zero, and the sum of the moments about any point is also zero.

Moment of a Force:

The moment of a force about a point is a measure of its tendency to cause rotation about that point. It is given by the product of the magnitude of the force and the perpendicular distance from the point to the line of action of the force.

Proof of the Law of Moments:

Consider a particle that is in equilibrium under the action of forces F1, F2, F3, ..., Fn. Let O be any point in the plane of the forces. The moment of force Fi about point O is given by Mi = Fi * di, where di is the perpendicular distance from point O to the line of action of force Fi.

According to the principle of equilibrium, the sum of the forces acting on the particle is zero:

ΣFi = F1 + F2 + F3 + ... + Fn = 0

Now, let's consider the sum of the moments about point O:

ΣMi = F1 * d1 + F2 * d2 + F3 * d3 + ... + Fn * dn

Since the particle is in equilibrium, the sum of the moments must be zero:

ΣMi = 0

Therefore,

F1 * d1 + F2 * d2 + F3 * d3 + ... + Fn * dn = 0

This equation implies that the algebraic sum of the moments about any point in the plane of the forces is zero.

Conclusion:

According to the law of moments, if a number of coplanar forces acting on a particle are in equilibrium, the algebraic sum of their moments about any point in their plane is zero. Therefore, option c) is the correct answer.