Interpret the formula

The given expression is $P = E L^2 M^{-5} G^{-2}$ where:

• $E$ = energy
• $L$ = angular momentum
• $M$ = mass
• $G$ = gravitational constant

Write dimensions of each quantity

• Energy: $[E] = [M L^2 T^{-2}]$

• Angular momentum: $[L] = [M L^2 T^{-1}]$

• Mass: $[M] = [M]$

• Gravitational constant: From Newton's law, $F = \frac{G m_1 m_2}{r^2}$ so $[G] = \frac{[F][r^2]}{[M]^2} = \frac{(M L T^{-2})(L^2)}{M^2} = [M^{-1} L^3 T^{-2}]$

Compute dimensions of each factor in the formula

$[P] = [E][L]^2[M]^{-5}[G]^{-2}$

Substitute: $[P] = (M L^2 T^{-2})(M L^2 T^{-1})^2 (M)^{-5} (M^{-1}L^3T^{-2})^{-2}$

Simplify term by term

First, $(M L^2 T^{-1})^2 = M^2 L^4 T^{-2}$

and $(M^{-1}L^3T^{-2})^{-2} = M^2 L^{-6} T^4$

Therefore, $[P] = (M L^2 T^{-2})(M^2 L^4 T^{-2})(M^{-5})(M^2 L^{-6} T^4)$

Add powers of $M, L, T$

• Power of $M$: $1 + 2 - 5 + 2 = 0$

• Power of $L$: $2 + 4 + 0 - 6 = 0$

• Power of $T$: $-2 - 2 + 0 + 4 = 0$

Hence, $[P] = [M^0 L^0 T^0]$

Match with options

$[M^0 L^0 T^0]$ corresponds to Option D.

Comparison with stored correct answer

Stored correct answer = D

Our derived answer = D

So, they agree.