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

Options

A. frequency
B. velocity
C. angular momention
D. time

Explanation

To solve the problem, we need to find the ratio of the dimensions of Planck's constant (h) and the moment of inertia (I). Let's break down the steps:

### Step 1: Identify the dimensions of Planck's constant (h)
Planck's constant relates energy (E) and frequency (ν) through the equation:
\[ E = hν \]
From this, we can express Planck's constant as:
\[ h = \frac{E}{ν} \]
The dimension of energy (E) is given by:
\[ [E] = [M][L^2][T^{-2}] \]
And the dimension of frequency (ν) is:
\[ [ν] = [T^{-1}] \]
Thus, the dimensions of Planck's constant can be calculated as follows:
\[ [h] = \frac{[E]}{[ν]} = \frac{[M][L^2][T^{-2}]}{[T^{-1}]} = [M][L^2][T^{-2}] \cdot [T] = [M][L^2][T^{-1}] \]

### Step 2: Identify the dimensions of moment of inertia (I)
The moment of inertia (I) is defined as:
\[ I = m r^2 \]
Where:
- \( m \) is mass with dimension \([M]\)
- \( r \) is distance with dimension \([L]\)

Thus, the dimensions of moment of inertia are:
\[ [I] = [M][L^2] \]

### Step 3: Find the ratio of the dimensions of Planck's constant to the moment of inertia
Now we can find the ratio:
\[ \frac{[h]}{[I]} = \frac{[M][L^2][T^{-1}]}{[M][L^2]} \]

### Step 4: Simplify the ratio
When we simplify the ratio:
\[ \frac{[h]}{[I]} = \frac{[M][L^2][T^{-1}]}{[M][L^2]} = [T^{-1}] \]

### Step 5: Interpret the result
The dimension \([T^{-1}]\) corresponds to frequency, as frequency is defined as the reciprocal of time.

### Final Answer
Thus, the ratio of the dimensions of Planck's constant and that of the moment of inertia is the dimension of frequency.