Today lets deal with Moment of Inertia. Before that please check out the post Inertia posted in our website to get basic idea about inertia.
Moment of Inertia:
What is moment of inertia ??? Let us consider some examples and understand it.
In mechanics from so far we learnt, we understood that inertia is the tendency of the object to resist its motion. So in rotational mechanics, we define moment of inertia as an analogue to inertia in mechanics. It is the tendency of the object to resist the rotational motion.
In mechanics, inertia is calculated as the mass of the object, then how to calculate the moment of inertia in Rotational Mechanics. Lets follow this method given below
Consider an disc of mass m rotating about its axis with angular velocity ω, as shown below. At a distance r consider a small point mass dm moving with the velocity v.
So, the kinetic energy of the point object can be given as follow
dK = ½ dm v2
Σ dK = Σ ½ dm v2
But we know that v = r ω
Σ dK = Σ ½ dm r2ω2
Since w is a constant for the body’s rotation, we take that out as a common entity.
Σ dK =½ ( Σ dm r2 )ω2
Translational Kinetic Energy = ½ (inertia) (velocity)
Rotational Kinetic Energy = ½ (moment of inertia) (analogue of velocity)
Comparing above equation with Σ dK =½ ( Σ dm r2 )ω2, we get moment of inertia = Σ dm r2
= ∫ dm r2
So, for a point object, moment of inertia (I) is calculated = m r2
