The dimension of angular velocity is:
Try it first: pick an option, then read the explanation.
Correct answer: B. T⁻¹
Asked in SSC JE 30/10/2020 (Evening); SSC JE 2014 (Morning)
Detailed Explanation
Angular velocity ω is the rate at which an angle changes with time.
\[ \omega = \frac{d\theta}{dt} \]
An angle in radians is arc length ÷ radius, i.e. length ÷ length, so it has no dimensions. Hence \([\omega] = [\text{T}^{-1}]\), unit rad/s.
Why “T⁻¹” Is Correct
Since the angle is dimensionless, the only dimension left is 1/time: [T⁻¹], option (b).
| Quantity | Formula | Dimensions |
|---|---|---|
| Angular velocity | dθ/dt | T⁻¹ |
| Angular acceleration | dω/dt | T⁻² |
| Linear velocity | v = ωr | LT⁻¹ |
| Mass moment of inertia | Σmr² | ML² |
| Torque / moment | F × r | ML²T⁻² |
Why the Other Options Are Wrong
- A. L²T: L²T has length squared, which angular velocity does not have.
- C. T²: T² is not the dimension of any rate.
- D. LT³: LT³ includes length, which angular velocity does not have.
Key Point
Angular velocity: [T⁻¹]; angular acceleration: [T⁻²].
Key Concepts for Students
- ω = 2πN/60 for N rpm.
- Frequency also has dimension T⁻¹.
- Radian is a dimensionless (supplementary) unit.
Practice more: this is question 8 in Moment MCQs – Part 1. Try the whole set and check every answer instantly.