The Question is from neet-physics , and it was created by the author VENKATESWARARAO

Question

Dimensional analysis is used to check the validity of an equation relating the speed of sound (v) in a gas to its pressure (P) and density ($\rho$). Which limitation prevents dimensional analysis from distinguishing between $v = \sqrt{P/\rho}$ and $v = \sqrt{14P/\rho}$?

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [Option A]
## CONCEPT FRAMEWORK
	**Primary Concept**: Dimensional Analysis
	**Related Topics**: Units, Physical Quantities
	**Prerequisites**: Understanding of fundamental dimensions (length, mass, time), ability to represent physical quantities with dimensions.
## SOLUTION PATHWAY
**Given Information**:
	* Relationship between speed of sound (v), pressure (P), and density ($ho$)
	* Two equations: $v = \sqrt{P/ho}$ and $v = \sqrt{14P/ho}$
**Method & Steps**:
	1. **Determine the dimensions of each variable**
		Reasoning: We need to express the physical quantities in terms of fundamental dimensions (M, L, T).
		Formula/Rule: [P] = ML$^{-1}$T$^{-2}$ (Pressure = Force/Area), [$ho$] = ML$^{-3}$ (Density = Mass/Volume), [v] = LT$^{-1}$ (Speed = Distance/Time)
	2. **Analyze the dimensions of both equations**
		Working: For both equations, consider the right-hand side:  $\sqrt{P/ho}$ has dimensions $\sqrt{(ML^{-1}T^{-2})/(ML^{-3})} = \sqrt{L^2T^{-2}} = LT^{-1}$, which are the dimensions of speed (v). Similarly, $\sqrt{14P/ho}$ also has dimensions $\sqrt{(ML^{-1}T^{-2})/(ML^{-3})} = \sqrt{L^2T^{-2}} = LT^{-1}$.
		Key Point: The number 14 is dimensionless.
	3. **Identify the limitation of dimensional analysis**
		Result: Dimensional analysis shows that both equations are dimensionally consistent, i.e. each side has the same dimensions. However, it cannot distinguish between the equations due to the dimensionless constant (1 in first equation and 14 in the second equation).
		Verification: Dimensional analysis does not involve numerical values, but only the dimensions of physical quantities.
 ## OPTION ANALYSIS
[A] ✓: Dimensional analysis only checks for dimensional consistency and cannot determine the values of dimensionless constants. It can't distinguish between $\sqrt{P/ho}$ and $\sqrt{14P/ho}$ because '1' and '14' are dimensionless numbers.
[B] :x: Dimensional analysis can handle fractional exponents, as shown in the analysis of the square root.
[C] :x: Dimensional analysis does not assume a linear relationship between variables. It only checks that the dimensions on both sides of the equation match, regardless of the mathematical form of the equation.
[D] :x: Dimensional analysis can handle equations with more than three variables. The limitation is not the number of variables but rather the inability to fix dimensionless constants.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Confusing dimensional consistency with the exact form of the equation. Dimensional analysis only confirms that an equation is *possible*, not necessarily *correct*.
		2.[Error 2]: Thinking that dimensional analysis can find dimensionless constants. It cannot; it only deals with dimensions.
	**Quick Tips**:
		1.[Helpful shortcut]: Always write down the dimensions of the variables before attempting the analysis.
		2.[Memory aid]: Remember that dimensionless constants are pure numbers without units and thus don't affect the dimensional analysis.

VENKATESWARARAO · Answer

The correct Option is "It cannot determine dimensionless constants."

VENKATESWARARAO · Answer

It cannot handle fractional exponents.

VENKATESWARARAO · Answer

It assumes a linear relationship between variables.

VENKATESWARARAO · Answer

It fails when more than three variables are involved.

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