Study Notes
Use of Dimensional Formula
Dimensional analysis is not theory for memorization — it is a powerful tool in IOE Mechanics.
There are three major applications:
- Conversion of units
- Checking consistency of equations
- Deriving formula
Master these — they directly appear in MCQs.
1. Conversion of One System of Units to Another
General Formula
If:
- \(M_1, L_1, T_1\) → fundamental units in first system
- \(M_2, L_2, T_2\) → fundamental units in second system
- Dimensional formula of quantity = \(M^a L^b T^c\)
Then,
\[ \boxed{ n_2 = n_1 \left(\frac{M_1}{M_2}\right)^a \left(\frac{L_1}{L_2}\right)^b \left(\frac{T_1}{T_2}\right)^c } \]
Where:
- \(n_1\) = numerical value in first system
- \(n_2\) = numerical value in second system
Example: Convert 1 Joule into erg
Step 1: Identify dimensional formula of energy.
Energy (Work) = Force × Distance
\[ [MLT^{-2}] \times [L] = [ML^2T^{-2}] \]
So,
\[ \text{Energy} = [ML^2T^{-2}] \]
Step 2: Identify systems
- Joule → SI (MKS)
- Erg → CGS
So,
\[ n_2 = 1 \left(\frac{1\,kg}{1\,g}\right)^1 \left(\frac{1\,m}{1\,cm}\right)^2 \left(\frac{1\,s}{1\,s}\right)^{-2} \]
Convert:
\[ 1\,kg = 1000\,g = 10^3 g \]
\[ 1\,m = 100\,cm = 10^2 cm \]
Substitute:
\[ n_2 = (10^3)^1 (10^2)^2 \]
\[ = 10^3 \times 10^4 = 10^7 \]
\[ \boxed{1 \text{ Joule} = 10^7 \text{ erg}} \]
IOE Shortcut Insight
If length power is squared, its conversion factor also gets squared.
Students often forget to raise conversion factors to correct powers.
2. Checking Consistency of Equation
Principle of Homogeneity
In any valid physical equation:
\[ \textbf{Dimensions of LHS = Dimensions of RHS} \]
If not → equation is definitely wrong.
But:
If dimensionally correct → may still be wrong physically.
Important IOE logic.
Example: Check \(v^2 = u^2 + 2as\)
Dimensions:
\[ [v] = [LT^{-1}] \]
\[ [a] = [LT^{-2}] \]
\[ [s] = [L] \]
Now check each term.
LHS
\[ [v^2] = [LT^{-1}]^2 = [L^2T^{-2}] \]
RHS
\[ [u^2] = [LT^{-1}]^2 = [L^2T^{-2}] \]
\[ [2as] = [1] [LT^{-2}] [L] = [L^2T^{-2}] \]
Both terms same.
Therefore:
\[ \boxed{\text{Equation is dimensionally consistent}} \]
IOE Trap
Dimensionally consistent equations are not necessarily correct.
Example:
\[
v = u + at^2
\]
This is dimensionally inconsistent.
But:
\[
v = u + at
\]
is dimensionally correct.
Always check time power carefully.
3. Deriving an Equation Using Dimensional Analysis
This is powerful in MCQs.
Example: Viscous Force on a Sphere
Given:
Viscous force \(F\) depends on:
- Coefficient of viscosity \( \eta \)
- Radius \(r\)
- Velocity \(v\)
So assume:
\[ F \propto \eta^a r^b v^c \]
\[ F = k \eta^a r^b v^c \]
Where \(k\) is dimensionless constant.
Step 1: Write Dimensions
\[ [F] = [MLT^{-2}] \]
\[ [\eta] = [ML^{-1}T^{-1}] \]
\[ [r] = [L] \]
\[ [v] = [LT^{-1}] \]
Step 2: Apply Homogeneity
\[ [MLT^{-2}] = [ML^{-1}T^{-1}]^a [L]^b [LT^{-1}]^c \]
\[ = [M^a L^{-a+b+c} T^{-a-c}] \]
Step 3: Compare Powers
Mass:
\[
a = 1
\]
Time:
\[
-a - c = -2
\]
Substitute \(a=1\):
\[ -1 - c = -2 \]
\[ c = 1 \]
Length:
\[
-a + b + c = 1
\]
\[ -1 + b + 1 = 1 \]
\[ b = 1 \]
So:
\[ a = b = c = 1 \]
\[ F = k \eta r v \]
Given proportional constant \(k = 6\pi\)
\[ \boxed{F = 6\pi \eta r v} \]
This is Stokes' Law.
IOE Strategy Tips
- Always compare powers of M, L, T separately.
- Solve algebra carefully.
- If exponent becomes negative unexpectedly, recheck signs.
- Dimensional method cannot give numerical constant (like \(6\pi\)) — it must be given.
Practice MCQs
1. 1 Joule equals:
A. \(10^5\) erg
B. \(10^6\) erg
C. \(10^7\) erg
D. \(10^8\) erg
Answer
C. \(10^7\) erg
2. Which statement is correct?
A. Dimensionally correct equations are always correct
B. Dimensionally incorrect equations may be correct
C. Dimensionally inconsistent equations are always wrong
D. All of the above
Answer
C. Dimensionally inconsistent equations are always wrong
3. Using dimensional analysis, force proportional to \( \eta r v \) gives exponents:
A. 1,1,1
B. 1,2,1
C. 2,1,1
D. 1,1,2
Answer
A. 1,1,1
Final Summary
Dimensional formula helps in:
- Unit conversion between systems
- Checking equation validity
- Deriving relations between physical quantities
Golden Rule:
\[ \text{LHS dimensions} = \text{RHS dimensions} \]
Master dimensional analysis — it is one of the safest and most scoring parts in IOE Mechanics.