Dimensionless Quantity Identification

A dimensionless quantity is one that has no physical dimensions, meaning it does not depend on fundamental units like length, mass, or time. These quantities are often ratios of similar physical quantities.

Analyzing Physical Quantities

Let's examine the dimensions of each option:

• Velocity: This is the rate of change of displacement with respect to time. Its dimensions are Length per Time ($LT^{-1}$). For example, meters per second (m/s).
• Angle: An angle, particularly when measured in radians, is defined as the ratio of arc length to the radius of the circle. Since both arc length and radius have dimensions of length, the ratio $\frac{\text{Length}}{\text{Length}}$ cancels out, resulting in dimensions of $L^0$, meaning it is dimensionless.
• Mass: Mass is a fundamental physical quantity and has its own dimension, typically denoted by $M$.
• Area: Area is a measure of two-dimensional space. Its dimensions are Length squared ($L^2$). For example, square meters (m$^2$).

Conclusion

Based on the dimensional analysis, only Angle is a dimensionless quantity.