Question

In the expression \( P = E l^2 m^{-5} G^{-2} \) where \( E, l, m \) and \( G \) represent Energy, Angular Momentum, Mass and Gravitational Constant, the dimensions of \( P \) are

• A. \( [M^1 L^2 T^{-2}] \)
• B. \( [M^0 L^2 T^{-2}] \)
• C. \( [M^0 L^0 T^0] \)
• D. \( [M^0 L^0 T^{-2}] \)

Hint:

Always substitute each quantity carefully and combine powers systematically to avoid mistakes.

Answer 1

The Correct Option is C

Step 1: Write dimensions of energy.
\[ [E] = [ML^2T^{-2}] \]

Step 2: Write dimensions of angular momentum.
\[ [l] = [ML^2T^{-1}] \]

Step 3: Write dimensions of mass.
\[ [m] = [M] \]

Step 4: Write dimensions of gravitational constant.
\[ [G] = [M^{-1}L^3T^{-2}] \]

Step 5: Substitute into given expression.
\[ P = E l^2 m^{-5} G^{-2} \]

Step 6: Simplify dimensions.
\[ [M L^2 T^{-2}] [M^2 L^4 T^{-2}] [M^{-5}] [M^2 L^{-6} T^4] \]
\[ = M^{1+2-5+2} L^{2+4-6} T^{-2-2+4} \]
\[ = M^0 L^0 T^0 \]

Step 7: Final Answer.
\[ \boxed{[M^0 L^0 T^0]} \]