The dimension of omega ($\omega$), which represents angular velocity, is [T-1].
Understanding Angular Velocity
Angular velocity ($\omega$) is a fundamental physical quantity in rotational motion. It describes the rate at which an object rotates or revolves relative to another point, i.e., the rate of change of its angular displacement. It is typically measured in radians per second (rad/s) in the International System of Units (SI).
The formula for angular velocity is given by:
$\omega = \frac{d\theta}{dt}$
Where:
- $d\theta$ represents the change in angular displacement.
- $dt$ represents the change in time.
Dimensional Analysis of Angular Velocity
To determine the dimension of angular velocity, we analyze the dimensions of its components:
- Angular displacement ($\theta$): Although measured in radians, an angle is considered a dimensionless quantity in dimensional analysis. This is because it is defined as the ratio of arc length to radius (length/length), making its dimension [L$^0$].
- Time ($t$): The dimension of time is [T].
Therefore, the dimension of angular velocity is:
$\text{Dimension of } \omega = \frac{\text{Dimension of angular displacement}}{\text{Dimension of time}} = \frac{[\text{L}^0]}{[\text{T}]} = [\text{T}^{-1}]$
This indicates that angular velocity is solely dependent on time and has no fundamental dimensions of mass or length.
Common Physical Quantities and Their Dimensions
For a clearer understanding, here's a table summarizing angular velocity and some related physical quantities with their respective dimensional formulas:
| Physical Quantity | Formula | Dimension |
|---|---|---|
| Angular velocity ($\omega$) | $\omega = d\theta/dt$ | [T-1] |
| Acceleration (a) | $a = dv/dt$ | [L1T-2] |
| Momentum (p) | $p = mv$ | [M1L1T-1] |
| Angular momentum (L) | $L = rp = mvr$ | [M1L2T-1] |
This table clearly illustrates how different physical quantities are composed of fundamental dimensions of mass (M), length (L), and time (T).
