Dimensional Analysis

Question 1

What is a physical quantity?

Answer 1

A physical quantity is a property of a material or system that can be quantified by measurement. A physical quantity can be expressed as the combination of a magnitude and a unit. For example, the physical quantity mass can be quantified as n kg where n is magnitude and kg is the unit.

Question 2

Name the two types of physical quantities.

Answer 2

Basic Quantities and Derived Quantities.

Question 3

_______ quantities make up _________ quantities.

Answer 3

base, derived.

Question 4

What is meant by, “All monomial derived physical quantities have the following power-law form.”

Answer 4

Q = σ A^aB^bC^c……. , where Q is the derived quantity and A, B, C, …. are base quantities. σ is a numerical constant and a,b,c, …… are real numbers.

Question 5

Any derived quantities involving special functions such as ln, sin, exp, …. should have their arguments ______________.

Answer 5

Dimensionless.

Question 6

Physical equations must be ______ ___________: both sides of an equation should have the _____ dimension.

Answer 6

dimensionally homogeneous, same.

Question 7

Give the 7 base quantities.

Answer 7

Property Symbol (SI) Dimension

Length m L

Mass kg M

Time s T

Temperature k θ

Current A I

Amount of Substance mol N

Luminous Intensity cd J

Angle rad σ

Question 8

Give the dimensions and SI unit of Force.

Answer 8

MLT^-2, Newton

Question 9

Give the dimensions and SI unit of Pressure.

Answer 9

Pascals, ML^-1T^-2.

Question 10

Give the dimensions and SI unit of Energy.

Answer 10

ML²T^-2, Joule.

Question 11

Give the dimensions and SI unit of Electric Charge.

Answer 11

IT, Q.

Question 12

Give the dimensions and SI unit of Electric Field.

Answer 12

MLT^-3I^-1, volt/meter.

Question 13

Give the dimensions and SI unit of Magnetic Field.

Answer 13

MT^-2I^-1, Tesla.

Question 14

Buckingham’s Π Theorem.

Answer 14

A general principle that you can extract the dependencies of a physical quantity in a systematic way.

Question 15

How does one extract the specific dependence of each physical quantity Q₀?

Answer 15

1. Identify a complete set of independent quantities {Q₁,Q₂,Q₃,….,Q_n}.

Note: The change of Q_i should not affect Q_j, such that i ≠ j.

Ex. {E_k, v, m} not an independent set, but {E_k, v}, {v, m}, or {E_k, m} in indep. sets.

2. Dimensional Consideration (Homogeneity)

[Q_i] = L^l_iM^m_it^t_i.

A. From the set of n independent quantities, identify two subsets of k dimensionally independent variables and (k-n) dimensionally dependent variables.

3. Two cases exists, k = n and k < n.

k = n, solve the equation using homogeneity.

k < n, one more step is needed from k = n.

A. For every dimensionally dependent quantity, cause this quantity to become dimensionless by dividing by the dimensionally independent quantities by using 2.

B. Multiply this dimensionless quantity by what was found in two to complete the equation.