Question

what is dimensional analysis????

method based numerical or
example....

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Answer 1

Answer:

Dimensional Anaylysis

• analysis using the fact that physical quantities added to or equated with each other must be expressed in terms of the same fundamental quantities (such as mass, length, or time) for inferences to be made about the relations between them.

example

1. Determine the relevant quantities from

physics considerations. This may mean

identifying the equation(s) that determine

how the system behaves (e.g., F = ma).

The wave speed v should depend on the

ambient density ρ0 (cf. mass of particle)

and the ambient pressure p0 (cf. restoring

force on particle).

2. Determine the fundamental units ([M],

[L], [T]) of each quantity. You can use an

equation that contains the quantity (e.g.,

F = ma) if you know the units of

everything else in the equation.

v ∼ [L][T] −1

ρ0 ∼ mass/volume ∼ [M][L] −3

p0 ∼ force/area ∼ ([M][L][T] −2 )/[L] 2 ∼ [M][L] −1 [T] −2

3. Postulate an equation relating the

quantities, with unknown exponents

(a, b, c, . . . ), which may be fractions.

v = C (ρ0) a (p0) b

, with C a dimensionless

constant.

4. Substitute units from 2 and combine

exponents.

[L][T] −1 ∼ [M] a [L] −3a [M] b [L] −b [T] −2b =⇒ [L][T] −1

∼[M]a+b [L] −3a−b [T] −2b

5. Equate exponents of [M], [L], [T] on

left and right sides of 4 and solve the

resulting equations simultaneously.

[M]: 0 = a + b

[L]: 1 = −3a − b

[T]: −1 = −2b

=⇒ b = 1/2 =⇒ a = −1/2

=⇒ answer: v = C

q

p0/ρ0

6. Always check your results by plugging

back into the equations.

0 = −1/2 + 1/2

√

1 = −3 × (−1/2) − 1/2 = 3/2 − 1/2

√

−1 = −2 × 1/2

√

If you have identified the most relevant

quantities, the (undetermined!)

dimensionless coefficient will typically be

of order unity (e.g., between 1/3 and 3).

C ≈ 1.2 for air!

Answer 2

Dimensional Analysis:-

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed.

Method based numerical or example.....

Dimensional analysis can only tell the dimensional quantities that constitute the formula and not the constant values.

A and 1/2

Example....

1. Simple Example:- The time periods of a harmonic oscillator.

2.A more Complex Example:- The energy of a vibrating conducting of wire.

3. A third Example :- Demand Versus capacity for a desk that is rotating.

What is the Dimensional Analysis of kinetic energy:-

kinetic energy has a dimensional formula of,

${m}^{2} \: \: \: {l}^{2} \: \: \: \: {t}^{ - 2}$

where,

• M = mass of object

• L = length of object

• T = time taken

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