Answer to Question #173580 in Mechanics | Relativity for Owolabi Ibukun

Question

Explain the importance of these concept (moment of inertia) in dealing with problems concerning rotating body

Answer 1

MOMENT OF INERTIA ($I$)?

$unit=kgm^2$

The moment of inertia of a given rigid body about any axis is defined as

the sum of the product of the masses of each and every particle of given body

and the square of its distance from the given axis.

It actuallly derive from the newton's first law of motion ....or law of inertia.

If system of discrete masses;

$I=m_1r_1+m_2r_2+....+....+m_nr_n \\ \therefore I=\displaystyle\sum_{i=1}^nm_ir_i \\$

If rigid body of continuous matter (mass);

$$I=\intop r^2.dm$$

$\bigstar$ In translation motion mass is the measure of inertia and in Rotational motion moment of inertia is measure of inertia.

APPLICATION IN PROBLEMS RELATED TO ROTATING BODY:

1) In Rotational motion moment of inertia is measure of inertia.

Greater the moment of Inertia,larger is the torque required to produce a given angular acceleration in it.

2)this implies that

$\boxed{\alpha\propto{1\over I}}$

$\\ \therefore\boxed{Torque=I \alpha}$ here, $I=\\$ Moment of inertia

$\alpha=$ Angular acceleration

this can be thought of as newton's second law for rotation.The moment of inertia is rotational mass and the torque is rotational force.

$\bigstar$ So, this can be used to to calculate

1)TORQUE of an rotating body and is usefull in solving problems related to rotating body.

It is also used to calculate :

2) ANGULAR MOMEMNTUM

$\boxed{L=I\omega}$ $\omega=$ angular velocity

3) KINETIC ENERGY

$\boxed{K={1\over2}I\omega^2}$