Dimensional Analysis

Question 1

What are Dimensional Quantities?

Answer 1

Dimensionless quantities are quantities expressed solely by a number and have no units.

Question 2

What are some applications of dimensional analysis?

Answer 2

• One application of dimensional analysis is in checking whether an equation is physically meaningful by ensuring that both sides of the equation have the same dimensions.
• When both sides of an equation have the same dimensions, the equation is said to have dimensional homogeneity or be dimensionally homogeneous.
• Another application of dimensional analysis is to predict the form of an equation.

Question 3

What are the base units of these dimensions?

• Mass
• Length
• Time
• Temperature
• Electric Current
• Luminosity

Answer 3

• Mass: M
• Length: L
• Time: T
• Temperature: θ
• Electric Current: I
• Luminosity: J

Question 4

What are the dimensions of the derived quantities?

• Area (A)
• Volume (V)
• Speed (v)
• Acceleration (a)

Answer 4

• Area (A): L²
• Volume (V): L³
• Speed (v): L T⁻¹
• Acceleration (a): L T⁻²

Question 5

What is the Law of dimensional analysis (principle of homogeneity)?

Answer 5

The equation is dimensionally correct if the dimensions on the left-hand side of the equation are equal to the dimensions on the right-hand side of the equation, if not, the equation is not dimensionally correct.

Question 6

What are dimensional variables?

Answer 6

Physical quantities that have dimensions but no fixed value.

Question 7

What are dimensionaless variables?

Answer 7

Physical quantities which have neither dimensions nor fixed value.

Question 8

What is a dimensional constant?

Answer 8

Physical quantities which appear as the constant of proportionality in a physical formula that have dimensions in basic quantities.

Example: Gravitational constant (G), Boltzmann constant (k), Planck’s constant, speed of light (c), etc.

Question 9

What is a dimensionaless constant?

Answer 9

Quantities which do not possess dimension but have fixed value.

Example: 1, 2, 3,𝜋, 𝑒, 𝑒𝑡𝑐.

Question 10

What are some limitations of dimensional analysis?

Answer 10

Limitations of dimensional analysis:

• Dimensional analysis only checks the units;
• Numeric factors have no units and can’t be tested
• Dimension analysis cannot be used to derive the exact form of a physical relation if the physical
  quantity depends upon more than three physical quantities (M, L & T).
• Dimensional analysis cannot be used to derive the relation involving trigonometrical and
  exponential functions.
• Dimensional analysis does not indicate whether a physical quantity is scalar or vector.