Study Notes
What are Dimensions?
The dimensions of a physical quantity tell us the powers to which the fundamental quantities — mass (M), length (L), and time (T) — must be raised to represent that quantity.
In simple words, dimensions show how a physical quantity depends on M, L, and T.
Examples of Dimensions
1. Velocity
Velocity is defined as:
$$
\text{velocity} = \frac{\text{displacement}}{\text{time}}
$$
Displacement has dimension $L$ and time has dimension $T$.
So,
$$
\text{velocity} = \frac{L}{T} = [M^0 L^1 T^{-1}]
$$
Dimension of velocity:
- Mass = 0
- Length = 1
- Time = −1
2. Acceleration
Acceleration is defined as:
$$
\text{acceleration} = \frac{\text{velocity}}{\text{time}}
$$
Since velocity has dimension $LT^{-1}$:
$$
\text{acceleration} = \frac{LT^{-1}}{T} = [M^0 L^1 T^{-2}]
$$
Dimension of acceleration:
- Mass = 0
- Length = 1
- Time = −2
3. Force
Force is defined as:
$$
F = \text{mass} \times \text{acceleration}
$$
Mass has dimension $M$ and acceleration has dimension $LT^{-2}$:
$$
F = M \times LT^{-2} = [M^1 L^1 T^{-2}]
$$
Dimension of force:
- Mass = 1
- Length = 1
- Time = −2
Types of Variables and Constants
1. Dimensional Variable
These are quantities that:
- Have dimensions
- Have variable values
Examples:
- Acceleration
- Volume
- Force
2. Dimensional Constant
These are quantities that:
- Have dimensions
- Have fixed constant values
Examples:
- Gravitational constant (G)
- Planck’s constant
- Stefan’s constant
3. Non-Dimensional Variable
These are quantities that:
- Do not have dimensions
- Have variable values
Examples:
- Strain
- Angle
- Relative density
4. Non-Dimensional Constant
These are quantities that:
- Do not have dimensions
- Have constant values
Examples:
- $\pi$
- $e$
- 1
Important Non-Dimensional Quantities (for Exams)
You should remember these for entrance exams:
- 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)
- Dielectric constant
Dimensions and SI Units of Common Physical Quantities
| Physical Quantity | Dimensional Formula | SI Unit |
|---|---|---|
| Area $(A = L^2)$ | $[L^2]$ | m$^2$ |
| Volume $(V = L^3)$ | $[L^3]$ | m$^3$ |
| Density $\left(\rho = \frac{m}{V}\right)$ | $[M L^{-3}]$ | kg/m$^3$ |
| Speed $\left(v = \frac{d}{t}\right)$ | $[L T^{-1}]$ | m/s |
| Acceleration $\left(a = \frac{v}{t}\right)$ | $[L T^{-2}]$ | m/s$^2$ |
| Momentum $(p = mv)$ | $[M L T^{-1}]$ | kg·m/s |
| Force $(F = ma)$ | $[M L T^{-2}]$ | N |
| Impulse $(I = Ft)$ | $[M L T^{-1}]$ | N·s |
| Work $(W = Fd)$ | $[M L^2 T^{-2}]$ | J |
| Power $\left(P = \frac{W}{t}\right)$ | $[M L^2 T^{-3}]$ | W |
| Pressure $\left(P = \frac{F}{A}\right)$ | $[M L^{-1} T^{-2}]$ | N/m$^2$ |