Study Notes
Dimension in Physics
Core Concept
The dimension of a physical quantity tells us:
To what powers the fundamental quantities (Mass \(M\), Length \(L\), Time \(T\)) must be raised to represent that quantity.
Mathematically written as:
\[ [M^a L^b T^c] \]
Where:
- \(a\) → power of mass
- \(b\) → power of length
- \(c\) → power of time
IOE exam mostly revolves around M, L, T system in Mechanics.
Building Dimensions from Definitions
Always start from the basic definition. Never memorize blindly.
1. Velocity
\[ \text{Velocity} = \frac{\text{Displacement}}{\text{Time}} \]
\[ = \frac{L}{T} \]
\[ \boxed{[M^0 L^1 T^{-1}]} \]
Interpretation:
- No mass dependence
- Linear in length
- Inverse in time
2. Acceleration
\[ \text{Acceleration} = \frac{\text{Velocity}}{\text{Time}} \]
\[ = \frac{LT^{-1}}{T} \]
\[ \boxed{[M^0 L^1 T^{-2}]} \]
3. Force
\[ F = ma \]
\[ = M \times LT^{-2} \]
\[ \boxed{[M^1 L^1 T^{-2}]} \]
This is one of the most important dimensions in Mechanics.
Types of Variables and Constants
1. Dimensional Variable
- Has dimension
- Value changes
Examples:
- Acceleration
- Force
- Volume
2. Dimensional Constant
- Has dimension
- Value fixed
Examples:
- Gravitational constant \(G\)
- Planck’s constant
- Stefan’s constant
MCQ Trap:
Don’t confuse gravitational constant with acceleration due to gravity \(g\).
3. Non-Dimensional Variable
- No dimension
- Value changes
Examples:
- Strain
- Angle
- Relative density
4. Non-Dimensional Constant
- No dimension
- Fixed value
Examples:
- \( \pi \)
- \( e \)
- 1
Important Non-Dimensional Quantities
These are favorite IOE MCQ targets:
- Relative density
- Angle and solid angle
- Strain
- Poisson’s ratio
- Refractive index
- Mechanical equivalent of heat
- Emissivity
- Magnetic susceptibility
- Electric susceptibility
- Relative permittivity
- Relative permeability
- Coefficient of friction
- Loudness (Decibel is unit of intensity level, but loudness itself is dimensionless ratio)
- Dielectric constant
Exam Trick
If quantity is a ratio of similar quantities, it is usually dimensionless.
Example:
\[
\text{Strain} = \frac{\Delta L}{L}
\]
\[ \Rightarrow \frac{L}{L} = 1 \]
Important Dimensional Formulas (High Priority Table)
| Physical Quantity | Formula | Dimension | SI Unit |
|---|---|---|---|
| Area | \(L^2\) | \([L^2]\) | m\(^2\) |
| Volume | \(L^3\) | \([L^3]\) | m\(^3\) |
| Density | \(\rho = \frac{m}{V}\) | \([ML^{-3}]\) | kg/m\(^3\) |
| Speed | \(v = \frac{d}{t}\) | \([LT^{-1}]\) | m/s |
| Acceleration | \(a = \frac{v}{t}\) | \([LT^{-2}]\) | m/s\(^2\) |
| Momentum | \(p = mv\) | \([MLT^{-1}]\) | kg·m/s |
| Force | \(F = ma\) | \([MLT^{-2}]\) | N |
| Impulse | \(I = Ft\) | \([MLT^{-1}]\) | Ns |
| Work | \(W = Fd\) | \([ML^2T^{-2}]\) | J |
| Power | \(P = \frac{W}{t}\) | \([ML^2T^{-3}]\) | W |
| Pressure | \(P = \frac{F}{A}\) | \([ML^{-1}T^{-2}]\) | N/m\(^2\) |
Powerful Observation for IOE
Notice patterns:
- Work and Energy → same dimension
- Impulse and Momentum → same dimension
- Pressure = Force / Area → reduces one length power
This pattern recognition helps eliminate wrong options instantly.
Common MCQ Traps
- Confusing impulse with force
- Thinking angle has dimension (it does not)
- Mixing up work and power dimensions
- Forgetting negative powers (especially \(T^{-2}\))
Rapid Recognition Technique
If question asks dimension of:
- Energy → Think \(F \times d\)
- Pressure → Think \(F/A\)
- Momentum → Think \(m \times v\)
- Power → Think energy per second
Never panic. Reduce to M, L, T.
Practice MCQs
1. Which of the following is dimensionless?
A. Strain
B. Pressure
C. Momentum
D. Work
Answer
A. Strain
2. Dimension of impulse is same as:
A. Force
B. Momentum
C. Energy
D. Pressure
Answer
B. Momentum
3. Dimension of pressure is:
A. \(MLT^{-2}\)
B. \(ML^{-1}T^{-2}\)
C. \(ML^2T^{-2}\)
D. \(MLT^{-1}\)
Answer
B. \(ML^{-1}T^{-2}\)
Summary
- Dimension expresses a quantity in terms of powers of \(M, L, T\).
- Velocity → \([LT^{-1}]\)
- Acceleration → \([LT^{-2}]\)
- Force → \([MLT^{-2}]\)
- Dimensional vs Non-dimensional classification is very important.
- Most IOE questions rely on reduction to basic definitions.
- Pattern recognition of dimensions saves time in MCQs.
Master dimensions properly — this is one of the highest scoring and lowest effort areas in Mechanics.