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Which of the following is a dimensional ...

Which of the following is a dimensional constant?

A

Refractive index

B

Poisson's ratio

C

Relative density

D

Gravitational constant

Text Solution

AI Generated Solution

The correct Answer is:
To determine which of the given options is a dimensional constant, we will analyze each option step by step. ### Step 1: Analyze the Refractive Index - The refractive index (μ) is defined as the ratio of the speed of light in vacuum to the speed of light in a medium. - Mathematically, it is expressed as: \[ \mu = \frac{c}{v} \] where \(c\) is the speed of light in vacuum and \(v\) is the speed of light in the medium. - Since it is a ratio of two speeds, it is dimensionless (it has no units). - **Conclusion**: The refractive index is NOT a dimensional constant. ### Step 2: Analyze Poisson's Ratio - Poisson's ratio (σ) is defined as the ratio of lateral strain to longitudinal strain. - Mathematically, it is expressed as: \[ \sigma = \frac{\text{lateral strain}}{\text{longitudinal strain}} \] - Both lateral strain and longitudinal strain are dimensionless quantities (they are ratios of lengths). - **Conclusion**: Poisson's ratio is NOT a dimensional constant. ### Step 3: Analyze Relative Density - Relative density (specific gravity) is defined as the ratio of the density of a substance to the density of water. - Mathematically, it is expressed as: \[ \text{Relative Density} = \frac{\text{Density of substance}}{\text{Density of water}} \] - Since it is a ratio of two densities, it is also dimensionless. - **Conclusion**: Relative density is NOT a dimensional constant. ### Step 4: Analyze Gravitational Constant - The gravitational constant (G) is used in the formula for gravitational force: \[ F = \frac{G m_1 m_2}{r^2} \] - Rearranging gives: \[ G = \frac{F r^2}{m_1 m_2} \] - The dimensions of force (F) are \( [M L T^{-2}] \), and the dimensions of distance squared (\(r^2\)) are \( [L^2] \). The dimensions of mass (\(m\)) are \( [M] \). - Thus, the dimensions of G can be calculated as: \[ [G] = \frac{[M L T^{-2}] [L^2]}{[M]^2} = [M^{-1} L^3 T^{-2}] \] - **Conclusion**: The gravitational constant has dimensions and is considered a dimensional constant. ### Final Conclusion Among the options analyzed, the only dimensional constant is the **gravitational constant (G)**. ---

To determine which of the given options is a dimensional constant, we will analyze each option step by step. ### Step 1: Analyze the Refractive Index - The refractive index (μ) is defined as the ratio of the speed of light in vacuum to the speed of light in a medium. - Mathematically, it is expressed as: \[ \mu = \frac{c}{v} \] ...
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