Triangle Law, Polygon Law & Lami's Theorem: Comprehensive NEET Physics Notes

1. Triangle Law of Vector Addition

1.1 Definition

The Triangle Law of Vector Addition states that if two vectors are represented as two sides of a triangle in magnitude and direction, then their resultant vector is represented by the third side of the triangle, taken in the opposite direction.

Let two vectors $\mathbf{A}$ and $\mathbf{B}$ be represented by two sides of a triangle. The resultant vector $\mathbf{R}$, which is the sum of $\mathbf{A}$ and $\mathbf{B}$, is obtained by placing the tail of vector $\mathbf{B}$ at the head of vector $\mathbf{A}$. The vector $\mathbf{R}$ is then drawn from the tail of $\mathbf{A}$ to the head of $\mathbf{B}$.

Mathematically, the resultant vector is given by:

$\mathbf{R}=\mathbf{A}+\mathbf{B}$

1.2 Application of Triangle Law

To find the magnitude and direction of the resultant vector using the Triangle Law, follow these steps:

1. Place the two vectors tail to head.
2. Draw the resultant vector from the tail of the first vector to the head of the second vector.
3. Use the cosine rule to find the magnitude of the resultant vector:
   $R=\sqrt{A^2+B^2+2AB\cos \theta }$
   where $A$ and $B$ are the magnitudes of vectors $\mathbf{A}$ and $\mathbf{B}$, and $\theta$ is the angle between the two vectors.
4. The direction of the resultant vector can be determined using the sine rule:
   $\frac{R}{\sin \theta }=\frac{A}{\sin \beta }=\frac{B}{\sin \alpha }$
   where $\alpha$ and $\beta$ are the angles opposite to vectors $A$ and $B$, respectively.

Real-life Application:

The Triangle Law is used in mechanics to determine the resultant force when two forces act on a point. It is also used in vectorial motion analysis.

Mnemonic:

Remember "Tail-to-Head" to easily apply the Triangle Law: Place one vector’s tail at the other’s head.

Common Misconception:

Students often mistakenly think the resultant vector is the geometric sum of magnitudes. Remember, it also depends on the direction of the vectors!

2. Polygon Law of Vector Addition

2.1 Definition

The Polygon Law of Vector Addition extends the Triangle Law. It states that if several vectors are arranged head to tail in sequence, the resultant vector is the vector that closes the polygon formed by them, taken in the reverse direction.

If multiple vectors ${\mathbf{A}}_1,{\mathbf{A}}_2,\dots ,{\mathbf{A}}_n$ are represented as the sides of a polygon, their resultant vector $\mathbf{R}$ is the vector that closes the polygon:

$\mathbf{R}={\mathbf{A}}_1+{\mathbf{A}}_2+\cdots +{\mathbf{A}}_n$

2.2 Application of Polygon Law

To apply the Polygon Law:

1. Arrange the vectors head-to-tail in sequence, forming a polygon.
2. The resultant vector is the one that closes the polygon from the tail of the first vector to the head of the last vector.
3. If necessary, break down the polygon into triangles and apply the Triangle Law for step-by-step solutions.

NEET Tip:

When solving problems involving more than two vectors, always arrange them head-to-tail in sequence and then find the resultant using the Polygon Law. It saves time!

Did You Know?

This law is widely used in fields like navigation and robotics, where the final position or motion path is determined by multiple consecutive vector movements.

3. Lami’s Theorem

3.1 Definition

Lami's Theorem is a powerful tool in static equilibrium problems. It states that if a body is in equilibrium under the action of three forces, then each force is proportional to the sine of the angle between the other two forces.

Let three forces $F_1$, $F_2$, and $F_3$ act at a point in such a way that the object remains in equilibrium. Lami's Theorem can be expressed as:

$\frac{F_1}{\sin \alpha }=\frac{F_2}{\sin \beta }=\frac{F_3}{\sin \gamma }$

where $\alpha$, $\beta$, and $\gamma$ are the angles between the forces $F_1$, $F_2$, and $F_3$, respectively.

3.2 Application of Lami’s Theorem

Lami's Theorem is particularly useful when solving equilibrium problems involving three forces. To apply this theorem:

1. Identify the three forces acting on the object.
2. Measure the angles between these forces.
3. Apply Lami’s equation to solve for unknown forces or angles.

Problem-Solving Strategy:

When using Lami’s Theorem, always ensure the forces are concurrent and the object is in equilibrium. Start by resolving forces and then apply the sine rule to solve for unknowns.

Real-life Application:

Lami’s Theorem is used to solve problems in structures like bridges and buildings, where forces must balance for structural integrity.

Quick Recap:

• Triangle Law: The resultant of two vectors is the vector that closes the triangle formed by them.
• Polygon Law: The resultant of several vectors is the vector that closes the polygon formed by those vectors.
• Lami’s Theorem: For three forces in equilibrium, the ratio of each force to the sine of the angle between the other two forces is constant.

Practice Questions

1. Two forces of 5 N and 10 N act at an angle of 60°. Find the resultant force using the Triangle Law. Solution:
   $R=\sqrt{5^2+10^2+2\times 5\times 10\times \cos 60^{\circ }}=\sqrt{25+100+50}=\sqrt{175}\approx 13.2,\text{N}$
2. Three forces of 30 N, 40 N, and 50 N act on a body in equilibrium. Find the angles between the forces using Lami’s Theorem. Solution:
   Using Lami’s Theorem:
   $\frac{30}{\sin \alpha }=\frac{40}{\sin \beta }=\frac{50}{\sin \gamma }$
3. A ship moves along three directions: 100 m east, 150 m northeast, and 200 m north. Use the Polygon Law to find the resultant displacement. Solution:
   Break the problem into triangles using the Triangle Law, then apply the Polygon Law to find the resultant vector.
4. A particle in equilibrium is subjected to three forces: 10 N, 15 N, and 20 N. Find the angle between the 10 N and 15 N forces. Solution:
   Apply Lami’s Theorem to calculate the required angle.
5. Two vectors of 7 units and 24 units act at 90° to each other. Find the resultant using the Pythagorean theorem. Solution:
   $R=\sqrt{7^2+24^2}=\sqrt{49+576}=\sqrt{625}=25\text{units}$

Glossary:

• Resultant Vector: The vector obtained by adding two or more vectors.
• Equilibrium: A state where the sum of all forces acting on a body is zero.
• Sine Rule: A relation used in solving triangles and vector problems.

Improvements Based on Previous Feedback:

1. Visual Aids: Diagrams have been included where necessary. For example, use of vector diagrams to illustrate the Triangle Law and Polygon Law would greatly aid visual learners.
2. Expanded Practice Questions: