The ratio of the dimensions of Planck’s constant and that of moment of inertia is the dimensions of: [PMT/NEET-2005]
a. time
b. frequency
c. angular momentum
d. velocity

Answer: b. frequency

Explanation:
To solve this, we need to find the dimensions of Planck’s constant and moment of inertia, then calculate their ratio.

Planck’s Constant (h)

From Planck’s equation $ E = hf $, where $ E $ is energy and $ f $ is frequency:

• Energy $ E $: $ [M\,L^{2}\,T^{-2}] $
• Frequency $ f $: $ [T^{-1}] $
• Therefore, $ h = \frac{E}{f} $ has dimensions: $ \frac{[M\,L^{2}\,T^{-2}]}{[T^{-1}]} = [M\,L^{2}\,T^{-1}] $

Moment of Inertia (I)

From $ I = mr^{2} $, where $ m $ is mass and $ r $ is distance:

• Mass $ m $: $ [M] $
• Distance squared $ r^{2} $: $ [L^{2}] $
• Therefore, $ I $ has dimensions: $ [M\,L^{2}] $

The Ratio

$ \frac{h}{I} = \frac{[M\,L^{2}\,T^{-1}]}{[M\,L^{2}]} = [T^{-1}] $

The dimension $ [T^{-1}] $ corresponds to frequency, since frequency is defined as the number of cycles per unit time, or $ \frac{1}{\text{time}} $.

Checking the other options:

• Time: $ [T] $ – doesn’t match
• Angular momentum: $ [M\,L^{2}\,T^{-1}] $ – doesn’t match
• Velocity: $ [L\,T^{-1}] $ – doesn’t match

Therefore, the correct answer is frequency.