Unit and Measurement - Dimensions and Dimensional Analysis (Applied Physics-1)

Flashcards covering the core concepts of dimensions, dimensional formulas, dimensional equations, unit systems, and the method and limitations of dimensional analysis as presented in the notes.

What is dimension?

The powers to which the fundamental quantities are raised to represent a physical quantity; represented by square brackets around the quantity.

What is a dimensional formula?

The expression showing which base quantities and their powers represent the dimensions of a physical quantity, e.g., [A] = [M^a L^b T^c].

What is a dimensional equation?

An equation obtained by equating a physical quantity with its dimensional formula, e.g., [A] = [M^a L^b T^c].

List the seven base quantities and their symbols.

Length [L], Mass [M], Time [T], Electric current [A], Thermodynamic temperature [K], Luminous intensity [cd], Amount of substance [mol].

What is a dimensionless quantity?

A quantity with no dimensions; its value is a pure number. Examples include plane angle, solid angle, and ratios of similar quantities.

What are dimensionless constants?

Constants that have no dimensions, e.g., refractive index, relative density.

What are dimensional constants?

Constants that carry dimensions, e.g., gravitational constant G, electric permittivity ε0, coefficient of elasticity.

What is the dimensional formula for area?

[L^2]

What is the dimensional formula for volume?

[L^3]

What is the dimensional formula for mass density ρ?

[M L^-3]

What is the dimensional formula for frequency?

[T^-1]

What is the dimensional formula for velocity?

[L T^-1]

What is the dimensional formula for acceleration?

[L T^-2]

What is the dimensional formula for force?

[M L T^-2]

What is the dimensional formula for impulse?

[M L T^-1]

What is the dimensional formula for work and energy?

[M L^2 T^-2]

What are the SI base units for mass, length, and time?

Mass: kilogram (kg), Length: metre (m), Time: second (s). CGS equivalents are gram (g), centimetre (cm), second (s).

How many dynes equal one newton?

1 N = 10^5 dynes.

What is the relation between joules and ergs?

1 J = 10^7 erg.

What is the principle of homogeneity of dimensions?

An equation is dimensionally correct if the dimensions of all terms on both sides are the same; only like quantities can be added or subtracted.

Is the equation x = x0 + v0 t + (1/2) a t^2 dimensionally correct?

Yes; each term has dimension of length [L].

What is the dimensional analysis result for centripetal force when F depends on m, v, and r?

F ∝ m v^2 / r.

What is the dimensional analysis result for a simple pendulum’s period when it depends on mass m, length l, and gravity g?

T ∝ sqrt(l/g) (mass cancels out; constant factor is dimensionless).

What is the dimensionally derived relation between escape velocity v, gravity g, and planet radius R?

v ∝ sqrt(g R) (up to a dimensionless constant).

Name some limitations of dimensional analysis.

Cannot determine dimensionless constants; cannot handle trigonometric or exponential dependencies; difficult to guess factors; not valid when there are more than three independent variables or when an equation has more than one term; may fail for multiple quantities with the same dimensions.