Dimensional analysis class 11 physics

 

Dimensional analysis is a powerful tool in physics that helps you:

1. Check the correctness of equations

2. Convert units

3. Derive relationships between physical quantities

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⚙️ What is Dimensional Analysis?

It’s the method of expressing physical quantities in terms of basic dimensions:

• M → Mass

• L → Length

• T → Time

• (Others like A for current, K for temperature if needed)

🧠 Think of it like finding the DNA of any physical quantity.

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🧮 Common Examples

Quantity	Dimensional Formula
Velocity	$[LT^{-1}]$
Acceleration	$[LT^{-2}]$
Force (F = ma)	$[MLT^{-2}]$
Energy (E = F×d)	$[ML^2T^{-2}]$
Pressure (P = F/A)	$[ML^{-1}T^{-2}]$

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✅ Uses of Dimensional Analysis

1. Checking Equation Validity
   Example: Is $s = ut + \frac{1}{2}at^2$ dimensionally correct?
   LHS = $[L]$
   RHS = $[LT^{-1}][T] + [LT^{-2}][T^2] = [L] + [L]$ ✅ Valid.

2. Deriving a Formula (up to constant)
   Let’s say you don’t know the formula for time period of a pendulum.
   Assume: $T \propto l^a g^b$
   (where $l$ = length, $g$ = gravity)

   $$[T] = [L]^a [LT^{-2}]^b = L^{a + b} T^{-2b}$$

   Matching powers:

   • $a + b = 0$

   • $-2b = 1$ → $b = -1/2$, so $a = 1/2$

   Final relation: $T \propto \sqrt{\frac{l}{g}}$
   

3. Unit Conversions
   If your velocity is in km/h and you want m/s:

   $$1\ \text{km/h} = \frac{1000}{3600} = \frac{5}{18}\ \text{m/s}$$

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🚫 Limitations

• Can't find dimensionless constants (like 1/2 in $\frac{1}{2}mv^2$)

• Won’t work for equations with trig/log/exponential functions directly