Question

Find the dimension of the following
I. Impulse
ii. Angular velocity
iii. moment of force
IV. Frequency

Answer 1

To find the dimensions of physical quantities, we use the basic dimensions of mass \([M]\), length \([L]\), and time \([T]\). Let's analyze each quantity.

### I. Impulse
Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It can be expressed as:

\[
\text{Impulse} = \text{Force} \times \text{Time}
\]

The dimension of force is given by:

\[
\text{Force} = \text{mass} \times \text{acceleration} = [M][LT^{-2}] = [MLT^{-2}]
\]

Thus, the dimension of impulse is:

\[
[\text{Impulse}] = [MLT^{-2}] \times [T] = [MLT^{-1}]
\]

### II. Angular Velocity
Angular velocity is defined as the rate of change of angular displacement with respect to time. Its dimension can be expressed as:

\[
\text{Angular Velocity} = \frac{\text{Angle}}{\text{Time}}
\]

The dimension of angle (radians) is dimensionless, so we have:

\[
[\text{Angular Velocity}] = [T^{-1}]
\]

### III. Moment of Force
The moment of force (or torque) is defined as the product of the force and the perpendicular distance from the axis of rotation:

\[
\text{Moment of Force} = \text{Force} \times \text{Distance}
\]

As we have established earlier, the dimension of force is \([MLT^{-2}]\), and the dimension of distance is \([L]\). Therefore, the dimension of the moment of force is:

\[
[\text{Moment of Force}] = [MLT^{-2}] \times [L] = [ML^2T^{-2}]
\]

### IV. Frequency
Frequency is defined as the number of occurrences of a repeating event per unit time. The dimension can be expressed as:

\[
\text{Frequency} = \frac{1}{\text{Time}} = [T^{-1}]
\]

### Summary of Dimensions
- Impulse: \([MLT^{-1}]\)
- Angular Velocity: \([T^{-1}]\)
- Moment of Force: \([ML^2T^{-2}]\)
- Frequency: \([T^{-1}]\)