Question

What is the dimensional formula of moment of inertia?

Answer 1

Dimensional formula of a quantity is defined as the expression that is used for the unit of a physical quantity. It shows which of the quantities are involved in making any fundamental quantity. A dimensional formula uses Mass (M), Length (L) and Time (T) as its fundamental quantities.

Step-By-Step Explanation:
Step I:
Moment of inertia is the quantity that explains how much amount of torque is needed by a body for angular acceleration about a rotational axis. It is also known as rotational inertia. It’s formula is written as the sum of the product of mass and square of distance from the axis of rotation. It can be written as
$I = m{r^2}$---(i)
Where m is the sum of the product of mass
And r is the distance from the axis of rotation or radius of gyration
And I is the moment of inertia

Step II:
Dimensional formula of mass is ${M^1}{L^0}{T^0}$
Dimensional formula of radius of gyration ${M^0}{L^1}{T^0}$

Step III:
Substitute the values of mass and radius in equation (i),
$I = [{M^1}{L^0}{T^0}] \times {[{M^0}{L^1}{T^0}]^2}$
$I = {M^{1 + 0}}{L^{0 + 2}}{T^{0 + 0}}$
$I = [{M^1}{L^2}{T^0}]$

Step IV:
Therefore the dimensional formula of moment of inertia is written as $[{M^1}{L^2}{T^0}]$

Note: It is to be noted that the dimensional formulas or equations of any physical quantity are of great importance. In order to check whether a physical relation is correct or not, dimensional formulas are used. But the value of the dimensionless constants can not be determined. If there is any physical quantity that depends on more than three factors, then dimensional formulas or equations can not prove the correctness of any equation.