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. Lami's Theorem is applied in a static analysis of structural and mechanical systems. 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,

\[\frac{A}{sin\alpha }\]=\[\frac{B}{sin\beta }\]=\[\frac{C}{sin\gamma}\]

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,

 \[\frac{A}{sin(180-\alpha)}\]=\[\frac{B}{sin(180-\beta)}\]=\[\frac{C}{sin(180-\gamma)}\]

So, by Lami’s theorem, we have,

\[\frac{A}{sin\alpha }\]=\[\frac{B}{sin\beta }\]=\[\frac{C}{sin\gamma}\]

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 weighs 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,

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

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,

\[\frac{T}{sin\theta}\]=\[\frac{mg}{2sin\theta cos\theta }\]

That is, \[T=\frac{mg}{2cos\theta }\]

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 be remembered 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 with each other). Also, this mathematically says 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 are not supposed 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.

Lami’s Theorem

Lami’s Theorem 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’. Lami’s theorem describes that if three forces acting at a single point are in equilibrium, then each of the forces is directly proportional to the sine of the angle between the remaining two forces. This theorem gives the conditions of equilibrium for three forces acting at a point. Lami’s theorem also defines that if three forces stand in at a position in symmetry, each force is comparable to the sine of the angle between the other two forces. The angle between the force vectors is taken when all the three vectors will emerge from the particle.

Derivation of Lami’s Theorem

Let P, Q, R be the three concurrent forces in equilibrium as shown in figure

Since the forces are vectors, it can be moved to form a triangle as shown in figure

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Applying the sine rule,

\[\frac{p}{(sin(180-\alpha ))}\]

=\[\frac{Q}{(sin(180-\beta ))}\]

=\[\frac{R}{(sin(180-\theta ))}\]

\[\frac{P}{(sin\alpha )}\]

=\[\frac{Q}{(sin\beta )}\]

=\[\frac{R}{(sin\theta )}\].    Hence Proved

Example of Lami's Theorem

Example: Let 45 degrees be the angle made by the strings with the signboard having a mass of 6 kg, then what is the value of the tension T in both the strings?

Solution: Given, m = 6 kg, g = 9.8 m/s², 𝜃 = 45 degrees

Using the derived formula from example 1, we get,

T = \[\frac{mg}{2cos\theta }\]

i.e. T = 6 x 9.8 / 2cos45 = 41.6 N

Therefore, the tension in both the strings to hold the signboard exactly horizontal is 41.6 N.

Limitations of Lami’s Theorem

There are various limitations of Lami’s Theorem:

• In Lami’s Theorem, there should be three forces only.

• The three forces should be coplanar, that means they should be in a single plane.

• The three forces should be concurrent, that means their line of action meeting at a point.

• The three forces of Lami’s Theorem should be non-linear, which means their line of action does not overlap each other. It means no angles between those three forces should be equal to 180 in degree 

• The three forces of Lami’s Theorem should be radially inward or outward & opposite. It means the 3 angles between those 3 forces should not be greater than 180 degrees.

• The three forces of Lami’s Theorem must be in equilibrium.

Applications of Lami’s Theorem

Lami’s theorem has been obtained from the Sine Rule for triangles. By representing the forces as lines as in a free-body diagram and translating them in such a way that one head touches the tail of another, then it will be noticed that when there are three forces, if they are supposed to cancel each other, they resultantly form a triangle. If they are not supposed to cancel each other, they form an open curve. The Sine Rule is only applicable for triangles only and hence Lami's Theorem is only applicable to three forces, but not for the 'n' number of forces.