What is Lami’s theorem?

Short Answer:

Lami’s Theorem states that if three forces acting simultaneously on a body keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces. This theorem is used when three concurrent and non-parallel forces act on a body in equilibrium. It helps to find the magnitude or direction of unknown forces easily. Lami’s theorem is an important concept in Engineering Mechanics and is often applied to problems involving supports, strings, pulleys, and structures in equilibrium.

Detailed Explanation :

Lami’s Theorem

Lami’s Theorem is a fundamental principle in Engineering Mechanics that provides a mathematical relationship between three concurrent forces acting on a body in equilibrium. It is particularly useful when only three forces act on a single point, and the body remains stationary or in equilibrium under their combined effect.

According to Lami’s Theorem:
If three concurrent and non-parallel forces acting on a body keep it in equilibrium, then each force is proportional to the sine of the angle between the other two forces.

Mathematically, it is expressed as:

Where,

  • F₁, F₂, F₃ = Magnitudes of the three forces,
  • α, β, γ = Angles between the other two forces respectively.

In simple terms, each force is related to the sine of the angle opposite to it. The theorem is mainly used to find unknown magnitudes or directions when a body is under the action of three forces that are in equilibrium.

Explanation of the Theorem

Consider a body at point O acted upon by three forces F₁, F₂, and F₃. These forces are concurrent (i.e., they meet at the same point) and act in different directions such that the body remains in equilibrium — meaning, the net effect of all forces is zero.

According to Lami’s theorem:

Here,

  • α is the angle between the forces F₂ and F₃,
  • β is the angle between F₃ and F₁,
  • γ is the angle between F₁ and F₂.

This relationship is derived from the triangle law of forces, which states that when three forces are in equilibrium, they can be represented by the three sides of a closed triangle taken in order.

Derivation of Lami’s Theorem

Let three forces F₁, F₂, and F₃ act at a point O, keeping the body in equilibrium. By the triangle law of forces, these three forces can be represented by the sides of a triangle taken in order.

Draw a triangle ABC, such that the sides AB, BC, and CA represent the magnitudes and directions of the forces F₁, F₂, and F₃ respectively.

In this triangle,

  • The side opposite to F₁ is the angle α,
  • The side opposite to F₂ is the angle β, and
  • The side opposite to F₃ is the angle γ.

Using the Law of Sines in triangle ABC, we have:

This is the mathematical expression of Lami’s Theorem.

Hence, the theorem is derived from the Triangle Law of Forces, and it provides an easy and direct way to calculate unknown forces or angles in systems with three concurrent forces in equilibrium.

Conditions for Applying Lami’s Theorem

  1. The body must be in equilibrium (no motion).
  2. Only three forces must act on the body.
  3. The forces must be concurrent (they meet at a common point).
  4. The forces must be non-parallel.

If these conditions are not satisfied, Lami’s theorem cannot be applied.

Applications of Lami’s Theorem

Lami’s theorem is widely used in Engineering Mechanics and Structural Analysis because it provides a simple way to solve equilibrium problems involving three forces. Some common applications are:

  1. Finding unknown tensions:
    It is used to find the tension in strings, cables, or supports when a body is hanging or supported by two or more ropes.
  2. Support reactions:
    Used to find the reactions at supports such as pins, rollers, or hinges when subjected to forces at an angle.
  3. Force analysis in machines:
    Helps in calculating the forces acting on different machine components like linkages or trusses.
  4. Static equilibrium problems:
    Useful for solving equilibrium conditions in systems where only three forces are acting at a joint.
  5. Verification of experimental results:
    It is also verified using a Lami’s theorem apparatus in mechanics laboratory experiments.

Advantages of Lami’s Theorem

  • It provides a direct relationship between forces and angles.
  • Simplifies the solution of three-force equilibrium problems.
  • Reduces complex vector calculations to simple trigonometric relations.
  • Helps determine both magnitude and direction of unknown forces accurately.

Limitations of Lami’s Theorem

  • It is applicable only when exactly three forces act on a body.
  • It cannot be used if the forces are non-concurrent or parallel.
  • The forces must be coplanar, i.e., lie in the same plane.
  • The body must be in equilibrium (net force = 0).

Practical Example

Suppose a small ring is suspended by two strings making angles of 30° and 60° with the horizontal and carrying a weight of 100 N vertically downward.
By applying Lami’s theorem, we can easily find the tensions in both strings without constructing a parallelogram or using vector resolution.

Conclusion

In conclusion, Lami’s Theorem is a simple yet powerful principle in Engineering Mechanics used to analyze systems where three concurrent forces act on a body in equilibrium. It relates each force to the sine of the angle between the other two forces. The theorem helps engineers solve problems related to supports, cables, and structures with ease. However, it is applicable only for three coplanar concurrent forces in equilibrium. Understanding and applying Lami’s theorem is essential for solving practical problems in statics and structural mechanics.