Consider a rotating disc. The kinetic energy of the disc can be represented as
$K.E.=\frac{1}{2}\sum _{}^{}{m}_{i}{v}_{I}^{2}.................\left(1\right)\phantom{\rule{0ex}{0ex}}$
But in this case each mass does not have same linear velocity hence it is easier to realize using angular velocity. Angular velocity, is equal for all the masses. We also know,
Substituting the value ofin (1) we get,
From the above equation, we get,
$\sum _{}^{}{m}_{i}{r}_{i}^{2}$is the moment of inertia of the system.
Unit of moment of inertia is
$kg{m}^{2}$There are two theorems to calculate moment of inertia about any arbitrary axis:
o Parallel axis theorem and
o Perpendicular axis theorem
According to the parallel axis theorem, the moment of inertia, of a body about any axis is equal to the moment of inertia, which is a parallel axis through the centre of gravity of the body plus
$M{d}^{2}$, where M is the mass of the body and d is the distance between the two axes.
According to perpendicular axes theorem if the moment of inertia is I about z-axis and the two axes, say x-axis and y-axis are mutually perpendicular to the original axis, then we can state:
${I}_{z}={I}_{x}+{I}_{y}$We can represent all moment of inertia as:
$I=M{K}^{2}$Here, K is known as radius of gyration about the considered axis.
Inertia is of three types:
o Inertia of rest,
o Inertia of motion and
o Inertia of direction.
• The tendency of a body to remain in its state of rest is called inertia of rest.
• For example: When we shake a tree, the fruits or dry leaves fall down from the tree
• The tendency of a body to remain in its state of uniform motion in a straight line is called inertia of motion.
• For example:
o An athlete runs for a certain distance before taking a jump so that his inertia of motion may help him to take a longer jump.
o If a horse which is running fast stops suddenly, the rider is thrown forward if he is not firmly seated
• The inability of a body to change its direction of motion by itself is called inertia of direction.
• For example: The force exerted on the steering wheel of a car changes its direction of motion.
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