The Question is from neet-physics units-and-measurement priniciple-of-homoginity, and it was created by the author Mr. ANIL KUMAR.G

Question

A hypothetical equation is given as $\rho = a \sqrt{P} + bT^2$, where $\rho$ is density, $P$ is pressure, and $T$ is temperature.  The dimensions of $\frac{a}{b}$ are:

Mr. ANIL KUMAR.G · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [A]
## CONCEPT FRAMEWORK
	**Primary Concept**: Dimensional Analysis
	**Related Topics**: Physical Units, Fundamental Dimensions
	**Prerequisites**: Definition of Density, Pressure, and Temperature in terms of fundamental dimensions (M, L, T, \Theta)
## SOLUTION PATHWAY
**Given Information**:
	* Equation: $ho = a \sqrt{P} + bT^2$
	* $ho$ is density
	* $P$ is pressure
	* $T$ is temperature
**Method & Steps**:
	1. Determine the dimensions of each term.
		Reasoning: For the equation to be dimensionally consistent, each term must have the same dimensions as density.
		Formula/Rule:  Dimensions of density [$ho$] = $ML^{-3}$, Dimensions of pressure [$P$] = $ML^{-1}T^{-2}$, Dimensions of temperature [$T$] = $\Theta$
	2. Determine dimensions of 'a'.
		Working:  $[a\sqrt{P}]$ must have dimensions of density, so $[a\sqrt{ML^{-1}T^{-2}}] = [ML^{-3}]$, which gives $[a][M^{1/2}L^{-1/2}T^{-1}] = [ML^{-3}]$, and then $[a] = [M^{1/2}L^{-5/2}T^{1}]$
		Key Point: The square root operation halves the powers of the dimensions.
	3. Determine dimensions of 'b'.
		Working: $[bT^2]$ must have dimensions of density, so $[b\Theta^2] = [ML^{-3}]$, which gives $[b] = [ML^{-3}\Theta^{-2}]$
	4. Determine dimensions of $\frac{a}{b}$.
		Working:  $\frac{[a]}{[b]} = \frac{[M^{1/2}L^{-5/2}T^{1}]}{[ML^{-3}\Theta^{-2}]} = [M^{-1/2}L^{1/2}T^{1}\Theta^{2}]$
		Result:  The dimensions of $\frac{a}{b}$ are $[M^{-1/2}L^{1/2}T^{1}\Theta^{2}]$
		Verification: The calculated dimensions match option A.
 ## OPTION ANALYSIS
[A] ✓: This is the correct answer as derived in the solution.
[B] :x: This is incorrect. The powers of M, L and T are not correct, and also the sign of $\Theta$ is wrong.
[C] :x: This is incorrect. These are the dimensions of 'b' and not a/b.
[D] :x: This is incorrect. These are the dimensions of 'a' and not a/b.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Forgetting to consider the square root on the pressure term when finding the dimensions of 'a'. [Always remember to apply the square root operation to the dimensions as well.]
		2.[Error 2]: Mixing up the dimensions of density, pressure and temperature. [Always double-check the fundamental dimensions of physical quantities.]
	**Quick Tips**:
		1.[Helpful shortcut]: Always start by writing the dimensions of all the individual terms in the equation. This helps to keep track of the powers.
		2.[Memory aid]: Remember the fundamental dimensions: M for mass, L for length, T for time, $\Theta$ for temperature.

Mr. ANIL KUMAR.G · Answer

The correct Option is "[M$^{-1/2}$ L$^{1/2}$ T$^{1}$ \Theta$^{2}$]"

Mr. ANIL KUMAR.G · Answer

[M$^{1/2}$ L$^{-1/2}$ T$^{-1}$ \Theta$^{-2}$]

Mr. ANIL KUMAR.G · Answer

[M L$^{-3}$ \Theta$^{-2}$]

Mr. ANIL KUMAR.G · Answer

[M$^{1/2}$ L$^{-5/2}$ T$^{1}$]

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