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: \([T^{-1}]\)
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