Understanding Lami's Theorem

Lami's theorem is a fundamental principle in statics, which is a branch of mechanics dealing with bodies at rest under the action of forces. The theorem provides a relationship between the magnitudes of three coplanar, concurrent forces acting on a body and the sines of the angles between the forces, when the body is in equilibrium.

For Lami's theorem to be applicable, the following conditions must be met:

• There must be exactly three forces acting on the body.
• These three forces must be coplanar (acting in the same plane).
• These three forces must be concurrent (their lines of action must meet at a single point).
• The body must be in a state of equilibrium, meaning the net force and net torque acting on it are zero.

If these conditions are satisfied, Lami's theorem states that each force is directly proportional to the sine of the angle between the other two forces.

Mathematically, if three forces $F_1$, $F_2$, and $F_3$ are acting at a point and are in equilibrium, and $\alpha$ is the angle between $F_2$ and $F_3$, $\beta$ is the angle between $F_1$ and $F_3$, and $\gamma$ is the angle between $F_1$ and $F_2$, then Lami's theorem can be written as:

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

This equation shows that the ratio of each force to the sine of the angle opposite to it (the angle between the other two forces) is constant.

Analyzing the Statements

Let's examine the given options in light of the correct statement of Lami's theorem:

1. If three forces acting on a body are said to be in equilibrium, then each force is directly dependent on sine of the angle between the other two forces.

   This statement aligns perfectly with Lami's theorem. It specifies three forces and the condition of equilibrium, stating the direct relationship between each force and the sine of the angle between the other two forces. This is the core principle of Lami's theorem.

2. If three forces are non-coplanar, then each force is directly dependent on sine of the angle between the other two forces.

   This statement is incorrect because Lami's theorem is applicable only for coplanar forces. If the forces are non-coplanar, a different approach (like vector addition in three dimensions) is required for equilibrium analysis.

3. If three forces are not in equilibrium, then each force is inversely proportional to sine of the angle between the other two forces.

   This statement is incorrect on two counts. First, Lami's theorem applies only when the forces are in equilibrium. Second, it states a direct proportionality, not an inverse proportionality, between the force and the sine of the angle.

4. If two forces are in equilibrium, then each force is directly proportional to sine of the angle between them.

   This statement is incorrect because Lami's theorem specifically deals with three forces. Equilibrium involving two forces simply implies that the two forces must be equal in magnitude, opposite in direction, and collinear. The concept of angles between 'other two forces' doesn't apply here.

Conclusion

Based on the analysis of Lami's theorem and the given options, the statement that accurately describes Lami's theorem is the one that mentions three forces in equilibrium and the direct dependence of each force on the sine of the angle between the other two forces.