Lami's theorem relates to the equilibrium of concurrent, coplanar forces acting on a rigid body or particle. It states that the magnitude of each force is proportional to the sine of the angle between the other two forces. The theorem can be derived using vector addition and the sine rule. It is useful for analyzing mechanical and structural systems in static equilibrium and for determining the magnitude of unknown forces, as demonstrated through an example of calculating the tension in two strings holding up a signboard.

1. shireen

2. Lami’s theorem relates the magnitudes of coplanar,
concurrent and non-collinear forces that maintain an
object in static equilibrium. The theorem is very useful
in analyzing most of the mechanical as well as
structural systems.

3. Lami’s Theorem Statement
Lami’s Theorem states, “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”.
Referring to the above diagram, consider three forces
A, B, C acting on a particle or rigid body making angles
α, β and γ with each other.
In the mathematical or equation form, it is expressed
as,

4. Lami’s Theorem Derivation
Now, let’s see how the theorem’s equation is derived.
Let FA, FB, and FC be the forces acting at a point. As per the
statement of the theorem, we take the sum of all forces acting at a
given point which will be zero.
i.e. FA + FB + FC = 0
The angles made by force vectors when a triangle is drawn are,

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

6. Hence, it is clearly seen that by applying sine rule
to complementary angles we arrive at the
required result for Lami’s theorem.
Now, we will see how Lami’s theorem is useful to
determine the magnitude of unknown forces for
the given system.

7. Lami’s Theorem Problems and
Solved Examples
Example 1: Consider an advertisement board hangs
with the help of two strings making an equal angle
with the ceiling. Calculate the tension in both the
strings in this case.

8. Solution: The free body diagram of the same
helps us to resolve the forces first. After resolving
the forces we will apply the required theorem to
get the value of tension in both the strings. Here,
the weight of the signboard is in a downward
direction, and the other force is the tension
generated by the signboard in both the strings. In
this case, the tension T in both the strings will be
the same as the angle made by them with the
signboard is equal.

9. Above figure represents the free body diagram of the
signboard. Applying the Lami’s Theorem we get,

10. Since sin (180 – θ) = sin θ and sin (2θ) = 2sinθ cosθ
So, we get, final tension force in the string T as,
i.e,T=mg/2 cosθ
The similar concept and equations can be applied
for a boy playing on a swing, and we arrive at the
same result.

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