The Question is from neet-physics units-and-measurement conversion-of-units-for-one-system-to-another, and it was created by the author NAGA SAI

Question

If E,m,J and G represent energy, mass, angular momentum and gravitational constant respectively, then the dimensional formula of $E{J^2}/{m^5}{G^2}$ is

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
	**Primary Concept**: Dimensional Analysis
	**Related Topics**: Units and Measurement, Fundamental Dimensions
	**Prerequisites**: Knowledge of basic physical quantities and their dimensions
## SOLUTION PATHWAY
**Given Information**:
	E = Energy, m = mass, J = angular momentum, G = Gravitational Constant
	Target: Find the dimensions of  $E{J^2}/{m^5}{G^2}$
**Method & Steps**:
	1.  Write the dimensional formulas for each quantity:
		Energy (E) =  $[ML^2T^{-2}]$
		Mass (m) = $[M]$
		Angular Momentum (J) = $[ML^2T^{-1}]$
		Gravitational Constant (G) = $[M^{-1}L^3T^{-2}]$
		Reasoning: These dimensions are standard and derived from the definitions of each quantity.
		Formula/Rule:  Dimensional formulas for standard physical quantities.
	2. Substitute the dimensional formulas into the given expression:
		$ \frac{E J^2}{m^5 G^2} = \frac{[ML^2T^{-2}] [ML^2T^{-1}]^2}{[M]^5 [M^{-1}L^3T^{-2}]^2} $ 
		Working: Substituting the dimensions and performing the operations.
		Key Point: Remember to square the dimensions when squaring the quantity.
	3. Simplify the expression:
		$ \frac{[ML^2T^{-2}] [M^2L^4T^{-2}]}{[M^5] [M^{-2}L^6T^{-4}]} =  \frac{[M^3L^6T^{-4}]}{[M^3L^6T^{-4}]} = [M^0L^0T^0]$
		Result:  The final result is dimensionless.
		Verification: The dimensions cancel out completely.
 ## OPTION ANALYSIS
[A] :x: Incorrect. The dimensions do not simplify to  $[MLT^{-2}]$
[B] :x: Incorrect. The dimensions do not simplify to  $[M^0L^0T^1]$
[C] :x: Incorrect. The dimensions do not simplify to  $[M^0L^2T^0]$
[D] ✓: Correct. The dimensions cancel out resulting in a dimensionless quantity
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Incorrectly remembering or deriving the dimensional formulas of quantities like angular momentum or the gravitational constant.
			[How to avoid]:  Memorize the standard dimensions or be able to derive them from the defining equations.
		2.[Error 2]: Making errors when simplifying the exponents of the dimensions.
			[How to avoid]: Carefully perform the operations on exponents.
	**Quick Tips**:
		1.[Helpful shortcut]: Remember to treat dimensions as algebraic quantities and simplify them accordingly.
		2.[Memory aid]: Write down the dimensions of each quantity separately before substituting into the expression to avoid errors.

NAGA SAI · Answer

$\left[ {{\rm{ML}}{{\rm{T}}^{ - 2}}} \right]$

NAGA SAI · Answer

$\left[ {{{\rm{M}}^0}{{\rm{L}}^0}{\rm{T}}} \right]$

NAGA SAI · Answer

$\left[ {{{\rm{M}}^0}{{\rm{L}}^2}{{\rm{T}}^0}} \right]$

NAGA SAI · Answer

The correct Option is "Dimensionless"

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