Lami's Theorem

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What is Lami's Theorem?

Lami's Theorem is related to the magnitudes of concurrent, coplanar, and non-collinear forces that maintain an object in static equilibrium. The Theorem is so useful to analyze most of the mechanical and structural systems as well. The proportionality constant is similar for all the given three forces. The Lami's Theorem is applied in a static analysis of structural and mechanical systems. The Lami's Theorem is named after Bernard Lamy.

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Statement of Lami's Theorem

Lami 's Theorem states, "When 3 forces related to the vector magnitude acting at the point of equilibrium, each force of the system is always proportional to the sine of the angle that lies between the other two forces." By the diagram given above, let us consider the three forces as A, B, C acting on a rigid body/particle making angles α, β, and γ with each other.

It is expressed in the mathematical or equation form as,

\[\frac{A}{sinα}\] = \[\frac{B}{sinβ}\] = \[\frac{C}{sinγ}\]


Lami's Theorem Derivation

Now, let us see how to state and prove Lami's Theorem or the Lami's theorem derivation.

Let FA, FB, and FC are the forces acting at a common point. As per Lami's theorem statement, we take the sum of all the three forces acting which will be zero at a given point.

That is, FA + FB + FC = 0

The angles made by the force vectors when a triangle is drawn given by,

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We write the angles in terms of complementary angles and use the law of triangle of vector addition. Then, by applying the sine rule, we will get,

\[\frac{A}{sin(180-α)}\] = \[\frac{B}{sin(180 - β)}\] = \[\frac{C}{sin(180 - γ)}\]

So, by Lami’s theorem, we have,

\[\frac{A}{sinα}\] = \[\frac{B}{sinβ}\] = \[\frac{C}{sinγ}\]

Hence, by applying the sine rule to complementary angles, we clearly see that we reach the required result for Lami's Theorem.

Now, let us see how Lami's Theorem is useful to determine the magnitude of unknown forces for the given system.


Example of Lami's Theorem

Now, let us understand Lami's theorem problems and solved examples.

Problem

Consider an advertisement board that hangs using two strings, making an equal angle with the ceiling. Calculate the tension in this case in both the strings.

Solution

A similar free-body diagram helps us to resolve the forces first. After resolving the forces, we'll apply the Theorem that we require to get the value of tension in both the strings. Here, the signboard weight towards the downward direction and another force is the tension generated by the signboard in both strings. Here, in this case, the tension 'T' in both the strings will be similar because the angle made by both strings with the signboard is equal.

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The above image represents the free body diagram of the signboard. Applying the Lami's Theorem we get,

\[\frac{T}{sin(180 - θ)}\] = \[\frac{mg}{sin(2θ)}\]

Since, the value of sin (180 – θ) = sin θ, and sin (2θ) = 2sinθ cosθ

Therefore, we get the final tension force in the string T as given below,

\[\frac{T}{sinθ}\] = \[\frac{mg}{2SinθCosθ}\]

That is, T = \[\frac{mg}{2Cosθ}\]

The same concept, along with the equations, can apply for a boy who is playing on a swing, and we reach the same result.


Limitations of Lami's Theorem

The limitations of the Lami's Theorem are pointed below, which must need to remember before application.

  • There should exist only three forces 

  • The three forces are to be coplanar (i.e., should be in a single plane)

  • And, the three forces should remain concurrent (their line of action meeting at a point)

  • Those forces should also be non-linear (their line of action should not overlap on each other). Also, this mathematically tells that no angles between those three forces should be equal to 180 in degrees.

  • Radially, those three forces should be inward or outward and opposite. Mathematically, the three angles between those three forces should not be higher than 180 in degrees

  • And most important is, those three forces must be in the equilibrium point


Applicability of Lami's Theorem

This theorem has been obtained from the Sine Rule for triangles. If we represent the forces as lines as in a free-body diagram and translate them in such a way that one head touches the tail of another, we will notice that when there are three forces, if they are supposed to cancel each other, they resultantly form a triangle.

If they do not suppose to cancel each other, they form an open curve. The Sine Rule is only applicable for triangles and not for all polygons. Therefore, Lami's Theorem is only applicable to three forces, but not for 'n' number of forces.

FAQ (Frequently Asked Questions)

1. Give the applications of Lami's Theorem?

Ans. A few of Lami's theorem applications can be given as follows.

  • To find the length of sides of a right triangle (a triangle which is of one right-angled corner)

  • It's been incorporated into the log calculators and the operation of about every electronic device on the planet

  • Besides Engineering Calculations, it can also be used in the math for Oceanography, Geology, Meteorology, Aerospace, or anywhere either the Trigonometry or log calculations are incorporated

  • One of the uses in Oceanography is, to determine the sound speed in the water, while sometimes it is also used to calculate the range of a sound source in water

2. Solve the given problem using Lami's Theorem?

Ans. 

Problem Statement

A baby is playing in a swing, hanging with the help of two identical chains, which is at rest. Identify the forces acting on the baby by applying the Lami's Theorem and find the tension acting on the chain?

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Solution

The baby and the chains are both modeled as a particle hung by two identical strings as represented in the figure. Here, three forces are acting on the baby.

  1. Downward gravitational force along the negative 'y' direction (mg)

  2. Tension (T) exists along the two strings

These three forces are coplanar and concurrent as well, as shown in the below-given figure.

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By using Lami’s theorem,

T / Sin (180 - θ) = mg / Sin (2θ)

Since, sin (180 – θ) = sin θ, and sin (2θ) = 2sinθ cosθ

Therefore, we get the final tension force in the string T as given below,

T / Sin  θ =  mg / 2Sin θ cos θ

So, by this, we get the tension on each string as,

T = mg / 2 cos θ