Write dimensions of the given constants

We need an expression having the dimensions of time $[T]$.

• Universal gravitational constant: $[G] = \frac{[F][r^2]}{[m_1][m_2]} = \frac{(MLT^{-2})L^2}{M^2} = M^{-1}L^3T^{-2}$

• Planck constant: $[h] = [\text{energy}]\cdot [\text{time}] = (ML^2T^{-2})T = ML^2T^{-1}$

• Speed of light: $[c] = LT^{-1}$

Assume the required time is proportional to

$t \propto G^a h^b c^d$

So, $[T] = [G]^a[h]^b[c]^d$

Substituting dimensions: $T^1 = (M^{-1}L^3T^{-2})^a(ML^2T^{-1})^b(LT^{-1})^d$

Therefore, $M^{-a+b}L^{3a+2b+d}T^{-2a-b-d} = T^1$

Compare powers of $M, L, T$

Since there is no $M$ or $L$ on the left side:

• For mass: $-a+b=0 \implies b=a$

• For length: $3a+2b+d=0$ Using $b=a$: $3a+2a+d=0 \implies 5a+d=0 \implies d=-5a$

• For time: $-2a-b-d=1$ Using $b=a$ and $d=-5a$: $-2a-a-(-5a)=1$ $2a=1 \implies a=\frac12$

Hence, $b=\frac12, \qquad d=-\frac52$

Form the expression

$t \propto G^{1/2}h^{1/2}c^{-5/2}$

$t \propto \sqrt{\frac{Gh}{c^5}}$

Match with options

This corresponds to: $\boxed{\sqrt{\frac{Gh}{c^5}}}$

So the correct option is C.

Verification with stored answer

Stored correct answer: C

Our derived answer: C

Hence, they agree.