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Step-by-Step Solution
Step 1: Write down the given expression
The quantity given is
$f = \sqrt{\frac{h c^5}{G}}$.
Step 2: Note the dimensions of each constant
Planck’s constant ($h$): $[h] = M^1 L^2 T^{-1}$
Speed of light ($c$): $[c] = L^1 T^{-1}$
Gravitational constant ($G$): $[G] = M^{-1} L^3 T^{-2}$
Step 3: Substitute dimensions into the given expression
In the expression $f = \sqrt{\frac{h c^5}{G}}$, we substitute the dimensional formulas:
$[f] = \sqrt{\frac{[h] \cdot [c]^5}{[G]}}$.
Thus,
$[f] = \sqrt{\frac{(M^1 L^2 T^{-1}) \times (L^1 T^{-1})^5}{M^{-1} L^3 T^{-2}}}.$
Step 4: Simplify the dimensions inside the square root
First handle $[c]^5 = (L^1 T^{-1})^5 = L^5 T^{-5}$:
Now the numerator becomes:
$ (M^1 L^2 T^{-1}) \times (L^5 T^{-5}) = M^1 L^{2+5} T^{-1 -5} = M^1 L^7 T^{-6}.$
Hence,
$[f] = \sqrt{\frac{M^1 L^7 T^{-6}}{M^{-1} L^3 T^{-2}}}.$
When dividing, we subtract exponents:
$M^1 / M^{-1} = M^{1 - (-1)} = M^2,$
$L^7 / L^3 = L^{7 - 3} = L^4,$
$T^{-6} / T^{-2} = T^{-6 - (-2)} = T^{-4}.$
Therefore inside the square root, we get
$M^2 L^4 T^{-4}.$
Step 5: Apply the square root
Taking the square root:
$[f] = \sqrt{M^2 L^4 T^{-4}} = M^1 L^2 T^{-2}.$
Step 6: Recognize the physical dimension
The dimension $M^1 L^2 T^{-2}$ is the dimension of Energy (the same as that of work, e.g., joule in SI).
Hence, the correct answer is Energy.
