Question

Which one of the following physical quantities has dimensions?

• A. Strain
• B. Poisson’s ratio
• C. Angle
• D. Gravitational constant
• E. Relative density

Hint:

Whenever a quantity is defined as a "ratio" of the same kind of physical quantities (like relative density or strain), it will be dimensionless.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Dimensionless quantities are usually ratios of similar physical quantities (e.g., length/length). Quantities that are not simple ratios typically have units and dimensions.

Step 3: Detailed Explanation:
- Strain: Ratio of change in dimension to original dimension (\(\Delta L / L\)). It is dimensionless.
- Poisson's ratio: Ratio of lateral strain to longitudinal strain. It is dimensionless.
- Angle: Ratio of arc length to radius (\(s / r\)). While it has units (radians), it is dimensionless (\([L^0M^0T^0]\)).
- Relative density: Ratio of density of a substance to density of water. It is dimensionless.
- Gravitational constant (G): From Newton's law \(F = G \frac{m_1 m_2}{r^2}\).
\[ G = \frac{F r^2}{m_1 m_2} \]
Dimensions of \(G = \frac{[MLT^{-2}][L^2]}{[M^2]} = [M^{-1}L^3T^{-2}]\).
It clearly has dimensions.

Step 4: Final Answer:
The Gravitational constant has dimensions.