What are Dimensionless Quantities in Physics and Engineering?

In physics and engineering, a dimensionless quantity is a quantity that has no units or dimensions associated with it. It is a quantity that can be expressed as a pure number, without any reference to length, time, mass, or other physical quantities.

Examples of dimensionless quantities include:

1. Pi (π), which is the ratio of a circle's circumference to its diameter, and is a pure mathematical constant with no units.
2. The speed of light (c), which is a fundamental constant of nature that describes the maximum speed at which information or matter can travel, and has no units.
3. Planck's constant (h), which is a fundamental constant of quantum mechanics that describes the relationship between energy and frequency, and has no units.
4. The fine-structure constant (α), which is a fundamental constant of quantum electrodynamics that describes the strength of the electromagnetic interaction between charged particles, and has no units.
5. The Euler-Mascheroni constant (γ), which is a mathematical constant that appears in many areas of mathematics and physics, and has no units.

Dimensionless quantities are often used in scientific formulas and equations because they can be easily compared and combined without worrying about units. For example, the equation for the period of an oscillator (T) can be written as:

T = 2π √(m / k)

where m is the mass of the oscillator, k is the spring constant, and π is a dimensionless constant. This equation shows that the period of the oscillator depends only on the mass and the spring constant, and not on any other physical quantities such as length or time.