Question

Find the dimension of the following
I. Impulse
ii. Angular vel

Answer 1

To find the dimensions of different physical quantities, we can express them in terms of the fundamental quantities: mass (M), length (L), Time (T), and in some cases, electric current (I), temperature (Θ), amount of substance (N), and luminous intensity (J).

### I. Impulse
Impulse is defined as the change in momentum. Momentum (p) is given by the product of mass (m) and velocity (v).

1. **Dimensions of Mass (m)**: \( [M] \)
2. **Dimensions of Velocity (v)**: Velocity is defined as displacement (length) per unit time, so its dimensions are \( [L][T^{-1}] \).
3. **Dimensions of Momentum (p)**:
\[
[p] = [m][v] = [M][L][T^{-1}] = [M L T^{-1}]
\]

Impulse (I) can also be expressed as force (F) multiplied by time (t):
\[
\text{Impulse} = F \cdot t
\]

4. **Dimensions of Force (F)**: Force is mass times acceleration. Acceleration has dimensions of \( [L][T^{-2}] \):
\[
[F] = [M][L][T^{-2}] = [M L T^{-2}]
\]

5. Therefore, the dimensions of impulse can be expressed as:
\[
\text{Impulse} = [F][t] = [M L T^{-2}][T] = [M L T^{-1}]
\]

Thus, the dimension of impulse is:
\[
\text{Impulse} = [M L T^{-1}]
\]

### II. Angular Velocity
Angular velocity is defined as the rate of change of angular displacement with respect to time.

1. **Dimensions of Angular Displacement**: Angular displacement (θ) is measured in radians. Radians are dimensionless, but we will express angular velocity as the change in angular displacement over time:
\[
[\theta] = [1] \quad (\text{dimensionless})
\]

2. **Dimensions of Time (t)**: In the dimensional analysis, time (t) has dimensions \( [T] \).

3. **Dimensions of Angular Velocity (ω)**:
\[
\omega = \frac{d\theta}{dt}
\]
Therefore, the dimension of angular velocity is:
\[
[\omega] = \frac{[\theta]}{[T]} = \frac{[1]}{[T]} = [T^{-1}]
\]

Thus, the dimension of angular velocity is:
\[
\text{Angular Velocity} = [T^{-1}]
\]

### Summary
- **Impulse**: \( [M L T^{-1}] \)
- **Angular Velocity**: \( [T^{-1}] \)