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 momentum
D. time

Answer

To solve the problem of finding the ratio of the dimensions of Planck's constant (H) and the moment of inertia (I), we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Dimensions of Planck's Constant (H)**:
- Planck's constant is defined in the context of energy and frequency. The relationship is given by the equation:
\[ E = H \cdot f \]
where \(E\) is energy and \(f\) is frequency.
- Rearranging this gives:
\[ H = \frac{E}{f} \]
- The dimensions of energy (E) are:
\[ [E] = [M L^2 T^{-2}] \]
- The dimensions of frequency (f) are:
\[ [f] = [T^{-1}] \]
- Thus, the dimensions of Planck's constant can be calculated as:
\[ [H] = \frac{[E]}{[f]} = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}] \]

2. **Understand the Dimensions of Moment of Inertia (I)**:
- The moment of inertia is defined as:
\[ I = m r^2 \]
- The dimensions of mass (m) are:
\[ [m] = [M] \]
- The dimensions of radius (r) are:
\[ [r] = [L] \]
- Therefore, the dimensions of moment of inertia are:
\[ [I] = [M] \cdot [L^2] = [M L^2] \]

3. **Calculate the Ratio of the Dimensions**:
- Now we will find the ratio of the dimensions of Planck's constant to the dimensions of moment of inertia:
\[ \text{Ratio} = \frac{[H]}{[I]} = \frac{[M L^2 T^{-1}]}{[M L^2]} \]
- Simplifying this ratio:
\[ \text{Ratio} = \frac{[M L^2 T^{-1}]}{[M L^2]} = [T^{-1}] \]

4. **Interpret the Result**:
- The dimension \( [T^{-1}] \) corresponds to frequency. Therefore, the ratio of the dimensions of Planck's constant and the moment of inertia gives us the dimension of frequency.

### Final Answer:
The ratio of the dimensions of Planck's constant and that of the moment of inertia is the dimension of **frequency**.