Dimensional Analysis in Physics

Study Notes

Units and Dimensions

Multiples and submultiples of metric units are based on various powers of ten.
The word "dimension" denotes the physical nature of a quantity.
Dimensions of Length, Mass, and Time are L, M, and T, respectively.

Derived Quantities

Area (A) = L²
Volume (V) = L³
Speed (v) = L/T
Acceleration (a) = L/T²
Force (F) = M L T⁻²
Work (W) = F.d = M L² T⁻²
Frequency (f) = T⁻¹
Momentum (p) = M L T⁻¹

MKS and CGS Systems

MKS system: m², m³, m/s, m/s², Newton ( Kg.m/s²), Joule (Kg.m²/s²)
CGS system: cm², cm³, cm/s, cm/s², dyne (gm.cm/s²), erg (gm.cm²/s²)
1 Newton = 10⁵ dyne, 1 Joule = 10⁷ erg

Dimensional Analysis Theory

The dimensional analysis is used to check the formula, since the dimension of the Left-hand side and the Right-hand side of the formula must be the same.
The dimensions of the Left-hand side (L.H.S.) and the Right-hand side (R.H.S.) are written in brackets [ ].

Verification of Physical Laws

Example 1: x = ½ at², where x is the distance, a is the acceleration, and t is the time.
Example 2: v = v₀ + a t, where v is the velocity, v₀ is the initial velocity, a is the acceleration, and t is the time.
Example 3: Poiseuille's law, Q = (π r⁴ P) / 8 η L, where Q is the volumetric flow rate, P is the pressure, r is the radius of the tube, η is the viscosity, and L is the length of the tube.

Derivation of Some Relations

Example: When a particle suspended in a thread vibrates under the acceleration due to earth, the periodic time T (the time of one period) and the length L of the thread are related to each other.
The relationship is T ∝ L⁵⁰ g⁻⁵⁰, which is correct only if the dimensions of both sides are the same.

Dimensions of the Constants of Proportionate

Example: The general law of gravity states that there is a gravitational force between two bodies of mass m and M and of distance r between the two centers.
The constant G has dimensions of M⁻¹ L³ T⁻².

Density

Density is defined as the amount of mass contained in a unit volume, and is a property of any substance.

Description

This quiz covers the concept of dimensional analysis in physics, including the use of metric units and their multiples and submultiples, and the representation of physical quantities using dimensions such as Length, Mass, and Time.