Dimensional Analysis in Physics

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.