A quantity is given by $f = \sqrt (h c^{5} / G)$ , where $c$ is speed of light, $G$ is universal gravitational constant and $h$ is the Planck’s constant. Dimension of $f$ is that of;

Area

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Energy

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Volume

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Momentum

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Solution

The correct option is B
Energy

Step 1: Given data
$f = \sqrt (h c^{5} / G)$
$c =$ speed of light,
$G =$ universal gravitational constant
$h =$ Planck’s constant
Step 2: Find dimension of $f$
A quantity is given by
$f = \sqrt (h c^{5} / G)$
The dimension of Planck's constant is given by $h = [M L^{2} T^{- 1}]$ .
The dimension of speed of light is $[L^{1} T^{- 1}]$ .
The dimension of universal gravitational constant is given by $[M^{- 1} L^{3} T^{- 2}]$ .
Put all the dimensions in quantity
$f = \sqrt{\frac{[M L^{2} T^{- 1}] {[L^{1} T^{- 1}]}^{5}}{M^{- 1} L^{3} T^{- 2}}} f = \sqrt{\frac{[M L^{7} T^{- 6}]}{M^{- 1} L^{3} T^{- 2}}} f = \sqrt{M^{2} L^{4} T^{- 4}} f = M L^{2} T^{- 2}$
The dimensional formula for $f$ is same as that of Energy.
Hence, option B is correct.

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