Lami's Theorem Formula with Solved Examples for Students
Lami's Theorem is a fundamental result in statics relating the magnitudes of three coplanar, concurrent, and non-collinear forces holding a point in static equilibrium to the sines of the angles between the directions of those forces.
Mathematical Statement and Formula of Lami's Theorem for Concurrent Forces in Equilibrium
Lami's Theorem states: If three coplanar, concurrent, and non-collinear forces keep a particle in static equilibrium, then the magnitude of each force is proportional to the sine of the angle between the other two forces.
Let three forces, $\vec{F}_1$, $\vec{F}_2$, and $\vec{F}_3$, act at a point and the particle remains in equilibrium. Let $\alpha$, $\beta$, and $\gamma$ be the angles opposite to $\vec{F}_1$, $\vec{F}_2$, and $\vec{F}_3$ respectively, i.e., each angle is between the lines of action of the other two forces. Then:
$\displaystyle \frac{F_1}{\sin \alpha} = \frac{F_2}{\sin \beta} = \frac{F_3}{\sin \gamma}$
Geometric Interpretation Through Vector Diagram and Force Triangle
In the context of equilibrium, the vector sum of the three forces is zero:
$\vec{F}_1 + \vec{F}_2 + \vec{F}_3 = \vec{0}$
The three vectors can be represented as sides of a triangle taken in order, since they add to zero, thus forming a closed triangle—commonly referred to as the force triangle.
The angles between the directions of the forces at the point of concurrency correspond to the internal angles of the triangle between the force vectors. The length of each side is proportional to the magnitude of the corresponding force, and each side is opposite to the angle between the other two forces.
Stepwise Derivation of Lami’s Theorem Using the Law of Sines
Let $O$ be the point of concurrency. Let $\vec{F}_1$, $\vec{F}_2$, and $\vec{F}_3$ be three non-collinear, coplanar forces acting at $O$, and let the particle be at rest, i.e., in equilibrium. By the triangle law of vector addition, arrange the vectors head-to-tail such that $\vec{F}_1 + \vec{F}_2 + \vec{F}_3 = \vec{0}$ completes the triangle.
Denote the triangle by vertices $A$, $B$, and $C$, where sides opposite vertices $A$, $B$, and $C$ represent vectors $\vec{F}_1$, $\vec{F}_2$, and $\vec{F}_3$, respectively. The angles between the directions of the forces at $O$ are denoted by $\alpha$, $\beta$, and $\gamma$—where $\alpha$ is the angle between $\vec{F}_2$ and $\vec{F}_3$ at $O$, and so on for the other angles.
By the Law of Sines applied to triangle $ABC$:
$\displaystyle \frac{F_1}{\sin \alpha} = \frac{F_2}{\sin \beta} = \frac{F_3}{\sin \gamma}$
Each numerator represents the length (magnitude) of a force, and each denominator represents the sine of the angle between the other two forces. Thus, the derivation is established from vector addition and the Sine Law.
Application of Lami's Theorem to Evaluate Unknown Forces: Systematic Example
Example. A particle of mass $m$ is suspended from the ceiling by two identical strings, each making an angle $\theta$ with the ceiling, and is in equilibrium. Find the tension $T$ in each string.
The particle is acted upon by three concurrent, coplanar forces: the weight $mg$ vertically downward, and tensions $T$ along each string.
The angle between the two strings at the particle is $180^\circ - 2\theta$. The angle between the weight $mg$ and each string is $\theta$ (since each string makes an angle $\theta$ with the vertical).
Applying Lami’s Theorem:
$\displaystyle \frac{T}{\sin 90^\circ} = \frac{T}{\sin 90^\circ} = \frac{mg}{\sin (2\theta)}$
Since $\sin 90^\circ = 1$ and $\sin(2\theta) = 2\sin\theta \cos\theta$:
$\displaystyle T = \frac{mg}{2\sin\theta \cos\theta}$
Therefore, the tension in each string can be written as:
$\displaystyle T = \frac{mg}{2\cos\theta}$
Illustrative Numerical Problem Using Lami’s Theorem
Example. Given $m = 6\,\mathrm{kg}$, $g = 9.8\,\mathrm{m}/\mathrm{s}^2$, and $\theta = 45^\circ$, calculate the tension $T$ in each supporting string.
Substitute values in the formula derived:
$T = \dfrac{6 \times 9.8}{2 \cos 45^\circ}$
$\cos 45^\circ = \dfrac{1}{\sqrt{2}}$
$T = \dfrac{58.8}{2 \times \dfrac{1}{\sqrt{2}}}$
$T = \dfrac{58.8}{\dfrac{2}{\sqrt{2}}}$
$T = \dfrac{58.8 \sqrt{2}}{2}$
$T = 29.4 \sqrt{2}$
$T \approx 41.56\,\mathrm{N}$
This value represents the tension in each string required to keep the particle in static equilibrium.
Conditions and Limitations of Lami’s Theorem in Force System Resolution
For Lami's Theorem to be applicable, precisely three forces must act at a point, and these forces must be coplanar, concurrent, and non-collinear.
The theorem is not applicable if the number of forces is other than three, if the forces are not coplanar, or if the lines of action of the forces are collinear (forming a straight line instead of concurrent at a point).
Lami's theorem cannot be used when the angles between any two forces are $0^\circ$ or $180^\circ$, as the sine function becomes zero, leading to an undefined ratio.
The geometric basis of the theorem derives from the Law of Sines applied to triangles only, and thus, the theorem does not generalise to systems with more than three forces (i.e., polygons with more than three sides).
For detailed coverage of vector principles and mathematical reasoning underlying equilibrium conditions, refer to Mathematical Reasoning Explained.
FAQs on What Is Lami's Theorem? Understanding the Basics
1. What is Lami's Theorem?
Lami's Theorem is a rule in mechanics that states: If three coplanar, concurrent, and non-collinear forces are in equilibrium, each force is proportional to the sine of the angle between the other two forces.
Key points:
- Lami's Theorem is used for solving force equilibrium problems.
- The forces must act at a single point and in the same plane.
- It is commonly applied in physics and engineering, especially in statics problems involving three forces.
2. What is the formula for Lami's Theorem?
The mathematical expression of Lami's Theorem is:
- F1 / sin(α) = F2 / sin(β) = F3 / sin(γ)
- F1, F2, F3 are the magnitudes of the three forces.
- α, β, γ are the angles opposite to the respective forces.
3. Where is Lami's Theorem used?
Lami's Theorem is widely used in solving equilibrium problems involving three coplanar, concurrent forces.
Main applications include:
- Mechanical engineering for analyzing joint stability
- Physics problems on force analysis
- Determining unknown forces in supports or tension in cables
4. How do you apply Lami's Theorem?
To apply Lami's Theorem, ensure three forces act at a point in equilibrium and are coplanar.
Steps:
- Draw a clear force diagram showing all forces and angles.
- Identify the magnitudes of forces and the angles between them.
- Apply the formula: F1 / sin(α) = F2 / sin(β) = F3 / sin(γ)
- Solve for the unknown force(s).
5. What are the conditions of applicability of Lami's Theorem?
The following conditions must be satisfied for Lami's Theorem to be applicable:
- There must be exactly three forces acting.
- All forces should be coplanar (in the same plane).
- The forces must be concurrent (meet at a single point).
- The system should be in equilibrium (net force = 0).
6. State and prove Lami's Theorem.
Lami's Theorem states that if three coplanar, concurrent forces acting at a point keep a body in equilibrium, then each force is proportional to the sine of the angle between the other two forces.
Proof:
- Let three forces F1, F2, F3 act at point O.
- The angles between them are α, β, γ.
- Resolve forces into components and apply law of sines to the force triangle formed.
- It follows that F1 / sin(α) = F2 / sin(β) = F3 / sin(γ).
7. Give a real-life example where Lami's Theorem is used.
A real-life example of Lami's Theorem is in the analysis of forces in a suspended lamp held by two strings.
Applications:
- Calculating tensions in the cords holding a hanging sign.
- Determining forces in the legs of a tripod stand.
- Analyzing support forces in trusses and bridges.
8. What is the difference between Lami's Theorem and the Law of Sines?
Lami's Theorem relates to three forces in equilibrium, while the Law of Sines applies to any triangle.
- Lami's Theorem is for force equilibrium at a point and requires three forces.
- Law of Sines is a trigonometric relation for any triangle.
- Lami's Theorem is derived using the Law of Sines on the force triangle.
9. Can Lami's Theorem be used if the forces are not in equilibrium?
No, Lami's Theorem can only be used if the three coplanar forces are in equilibrium.
If the system is not in equilibrium, the theorem does not apply, and other laws of mechanics must be used to analyze the forces.
10. Why are only three forces considered in Lami's Theorem?
Lami's Theorem is restricted to three forces because the equilibrium of more than three coplanar concurrent forces requires vector addition methods or graphical analysis.
Key points:
- It is mathematically derived for three forces forming a closed triangle.
- More than three forces cannot form a single triangle for sine law application.
11. In which class and chapter is Lami's Theorem studied in CBSE?
Lami's Theorem is generally covered in Class 11 Physics in the chapter on Laws of Motion or Equilibrium of Forces as part of the CBSE syllabus under Statics.
