Understanding Forces in Equilibrium and Lami's Theorem

The question describes a specific principle related to a body that is not moving or accelerating. This state is known as equilibrium. Specifically, it talks about a body being in equilibrium under the influence of three forces that all act at the same point. These are called concurrent forces.

What is Lami's Theorem?

Lami's theorem is a fundamental principle in statics, applicable when three concurrent forces are in equilibrium. It provides a relationship between the magnitudes of these forces and the sines of the angles opposite to them.

The theorem states that if three concurrent forces are acting on a body and the body is in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

Let's consider three concurrent forces $\mathbf{F}_1$, $\mathbf{F}_2$, and $\mathbf{F}_3$ acting at a point and keeping the body in equilibrium. Let the angle between $\mathbf{F}_2$ and $\mathbf{F}_3$ be $\alpha$, the angle between $\mathbf{F}_1$ and $\mathbf{F}_3$ be $\beta$, and the angle between $\mathbf{F}_1$ and $\mathbf{F}_2$ be $\gamma$. According to Lami's theorem:

$$ \frac{F_1}{\sin \alpha} = \frac{F_2}{\sin \beta} = \frac{F_3}{\sin \gamma} $$

Here, $F_1$, $F_2$, and $F_3$ are the magnitudes of the forces.

Analyzing the Options

Let's look at the given options in the context of the question:

• Parallelogram law of forces: This law is used to find the resultant of two forces acting at a point. It states that if two forces acting at a point are represented in magnitude and direction by the adjacent sides of a parallelogram, their resultant is represented by the diagonal passing through that point. This doesn't describe the equilibrium condition for three forces.
• Varignon's theorem: Also known as the principle of moments, this theorem states that the moment of a resultant force about a point is equal to the algebraic sum of the moments of its components about the same point. This is related to moments, not directly to the relationship between concurrent forces and angles in equilibrium as described.
• Transmissibility of forces: This principle states that a force acting on a rigid body can be considered to act anywhere along its line of action without changing the external effects (like translation and rotation) on the body. This principle deals with the point of application, not the relationship between multiple forces in equilibrium.
• Lami's theorem: As discussed above, Lami's theorem precisely matches the statement given in the question: "If a body is in equilibrium, under the action of three concurrent forces, each force is proportional to the sine of the angle between the other two forces".

Therefore, the theorem that states this principle is Lami's theorem.

Revision Table: Key Principles in Statics

Additional Information on Concurrent Forces and Equilibrium

For a body to be in equilibrium under the action of concurrent forces, two conditions must be met:

• The vector sum of all forces acting on the body must be zero ($\Sigma \mathbf{F} = 0$).
• The sum of moments of all forces about any point must be zero ($\Sigma \mathbf{M} = 0$). For concurrent forces, if the first condition ($\Sigma \mathbf{F} = 0$) is met, the second condition ($\Sigma \mathbf{M} = 0$) is automatically satisfied if we take the moment about the point of concurrency.

Lami's theorem is essentially a special case of the equilibrium condition ($\Sigma \mathbf{F} = 0$) when only three concurrent forces are involved. It can be derived using the sine rule from a triangle formed by representing the three forces vectorially (if $\mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 = 0$, they form a closed triangle).

Lami's theorem is very useful for solving problems involving three forces in equilibrium, such as a weight suspended by two ropes, forces in truss members at a joint, etc.