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).
- Dimensions of Mass (m): \( [M] \)
- Dimensions of Velocity (v): Velocity is defined as displacement (length) per unit time, so its dimensions are \( [L][T^{-1}] \).
- 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): \[ \text{Impulse} = F \cdot t \]
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Dimensions of Force (F): Force is mass times acceleration. Acceleration has dimensions of \( [L][T^{-2}] \): \[ [F] = [M][L][T^{-2}] = [M L T^{-2}] \]
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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.
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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: \[ [\theta] = [1] \quad (\text{dimensionless}) \]
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Dimensions of Time (t): In the dimensional analysis, time (t) has dimensions \( [T] \).
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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: \( [T^{-1}] \)