Lami's Theorem - Free MCQ Practice Test with solutions, GATE Civil Engineering

Concept:
Lami's Theorem:- It states that when three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

So according to the theorem.


Let,
Force in member AB = F_AB
Force in member BC = F_BC
from fig α = 90° + 30°  = 120°
β = 90° + 60° = 150°
γ = 90°
From Lami's theorem,

Hence the correct answer is an option (1) i.e 100√3 N.

Detailed Explanation:
Lami's Theorem states that, "If three coplanar forces acting at a point be in equillibrium, then each force is proportional to the sine of the angle betweenthe other two".
Let us consider three forces A, B, and C acting on a particle or a rigid body making angles α, β, and γ with each other, then

Additional Information
Lambert's Law:

• It is related to Illumination.
• Lambert's cosine law: It states that illumination, E at any point on a surface is directly proportional to the cosine of the angle between the normal at that point and the line of flux.
• Law of Inverse Squares: The illumination of a surface is inversely proportional to the square of the distance between the surface and the light source

E = I/d²
Where E = Illuminance
I = Luminous Intensity
D = Distance between the surface and the source

Lami's theorem:
It states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces.
Consider three forces F_A, F_B, and F_C acting on a particle or rigid body making angles α, β, and γ with each other.

Then from Lami's theorem,

Calculation:
Given:
W = 730 N, And the pulley is smooth.
First string ABCD is split into two parts and consider the joints B and C separately.
Let, T₁ = Tension in String AB, T₂ = Tension in String BC, T₃ = Tension in String CD, T₄ = Tension in String DE

Since at joint B and C, three forces are acting on both points. But at B all three forces are unknown and at point C only two forces are unknown. So Apply Lami's theorem first at joint C:
∵ Pulley is smooth, T₃ = T₄ = 730 N
 Apply Lami's theorem first at joint C:

⇒ T₂ = 249.674 N And W₂ = 685.98 N
Now, the value of T₂ is known at point B.
Apply Lami's theorem first at joint B:

LAMI’S THEOREM:

• If a system of Three forces is in equilibrium, then, each force of the system is proportional to sine of the angle between the other two forces (and constant of proportionality is the same for all the forces). Thus, we have,

The formula for Lami's Theorem:

Calculation:
Given:

A = P₁, B = P₂ = 300N, C = P₃, and
α = 90° + 30° = 120°, β = 90° + 60° = 150° and  γ = 360° - 120° - 150° = 90°
on salving 

P₃ = 600N
only option 1 is correct.

Lami's theorem:
It states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces. Consider three forces F_A, F_B, F_C acting on a particle or rigid body making angles α, β and γ with each other.

Therefore, 
Calculation:
Given:

We have angle BAC = 180° - (30° + 60°)
angle BAC = 90°
By Lami’s Theorem, 
T_AC = 400 N

Lami's theorem:
It states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces.
Consider three forces F_A, F_B, F_C acting on a particle or rigid body making angles α, β and γ with each other.

Then from Lami's theorem,

Lami's theorem:
It states that if three forces acting at a point are in equilibrium, each force is proportional to the sine of the angle between the other two forces.
Consider three forces F_A, F_B, F_C acting on a particle or rigid body making angles α, β and γ with each other.

Therefore,

Important Points
Triangle Law of Forces: It states that if two forces acting simultaneously on a particle, be represented in magnitude and direction by the two sides of a triangle, taken in order; their results may be represented in magnitude and direction by the third side of the triangle, taken in the opposite order.”

Law of the parallelogram of force: 
It states that if two forces, acting at the point of a body, be represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram contained in between the two adjacent forces.