Question

What is the dimensional formula for the radius of gyration?

Answer 1

Step 1: Definition

1. The radius of gyration is defined as the imaginary distance from the centroid at which the area of cross-section is imagined to be focused at a point in order to obtain the same moment of inertia.
2. The radius of gyration is denoted by K.
3. Formula: $\mathrm{K}=\sqrt{\frac{\mathrm{I}}{\mathrm{m}}}$. Here, I is the moment of inertia, m is the mass.

Step 2: Dimensional formula for the radius of gyration

We know

Dimensional formula of Mass $=\left[{\mathrm{M}}^1{\mathrm{L}}^0{\mathrm{T}}^0\right]$

Dimensional formula of Moment of Inertia $=\left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^0\right]$

Putting these values in the formula of the radius of gyration, we get

Dimensional formula of the radius of gyration $=\sqrt{\frac{\left[{\mathrm{M}}^1{\mathrm{L}}^2{\mathrm{T}}^0\right]}{\left[{\mathrm{M}}^1{\mathrm{L}}^0{\mathrm{T}}^0\right]}}$

$=\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^0\right]$

Therefore, the dimensional formula of the radius of gyration is $\left[{\mathrm{M}}^0{\mathrm{L}}^1{\mathrm{T}}^0\right]$.