Dimensions of Physical Quantities

Topic Content:

What are Dimensions of a Physical Quantity?

The dimensions of a physical quantity show how that quantity is related to fundamental quantities such as mass, length and time.

In dimensional analysis, the dimensions of mass , length and time are represented by the capital letters M , L and T , respectively.

Quick Recall — Laws of Indices

Before working with dimensions, recall some important laws of indices. These rules will help us simplify dimensional expressions.

⇒ \( \scriptsize a^m \times a^n = a^{m+n}\)

⇒ \(\frac{a^m}{a^n} \scriptsize = a^{m-n}\)

⇒ \( \scriptsize (a^m)^n = a^{mn}\)

⇒ \( \scriptsize a^{-n} = \normalsize \frac{1}{a^n}\)

⇒ \( \scriptsize a^0 = 1\)

⇒ \(\scriptsize a^{\frac{1}{n}} \scriptsize = \sqrt[n]{a}\)

In dimensional analysis, these same rules are used when multiplying, dividing or raising dimensions to powers.

Derived quantities can be expressed in terms of these fundamental dimensions. For example, velocity depends on length and time, while force depends on mass, length and time.

Dimensions of Derived Quantities:

Some common derived quantities and their dimensions are shown below.

Velocity

Velocity = \(\frac{distance}{time}\)

Therefore:

\(\scriptsize [V] = \normalsize \frac{L}{T} \scriptsize = LT^{-1}\)

Momentum

Momentum = Mass × Velocity

\(\scriptsize [p] = M \times LT^{-1} = MLT^{-1}\)

Acceleration

Acceleration = \(\frac{velocity}{time}\)

\(\scriptsize [a] = \normalsize \frac{LT^{-1}}{T} \\ \scriptsize = L \: \times \: T^{-1} \: \times \: T^{-1} \\ \scriptsize = L \: \times \: T^{-1 \: – \: 1} \\ \scriptsize = LT^{-2}\)

Force

Force = Mass