Question

The ratio of the dimensions of Planck's constant and that of the moment of inertia is the dimension of:

• A. frequency
• B. velocity
• C. angular momentum
• D. time

Answer 1

The Correct Option is A

To determine the dimension of the ratio of Planck's constant to the moment of inertia, we first need to understand the dimensional formulas for each physical quantity involved:

1. Planck's Constant \( (h) \): The dimensional formula is expressed as [M^1L^2T^{-1}].
2. Moment of Inertia \( (I) \): The dimensional formula is expressed as [M^1L^2].

Next, we need to find the dimensional formula of the ratio \(\frac{h}{I}\):

\[ \text{Dimensions of } \frac{h}{I} = \frac{[M^1L^2T^{-1}]}{[M^1L^2]} = [T^{-1}] \]

The resulting dimensions are [T^{-1}], which correspond to the dimensional formula of frequency.

Let us now justify why this is the correct answer and why the other options are incorrect based on dimensions:

• Frequency: Dimensions are [T^{-1}]. This matches with the dimensions we derived.
• Velocity: Dimensions are [LT^{-1}]. This is incorrect.
• Angular Momentum: Dimensions are [M^1L^2T^{-1}]. This is incorrect.
• Time: Dimensions are [T^1]. This is incorrect.

Therefore, the correct answer is that the ratio of the dimensions of Planck's constant to the moment of inertia is the dimension of frequency.