Moment of Inertia: An Introduction
Consider a wheel and a uniform disc with the same masses, rotating about the same axis. If we try to stop them, it is a bit more difficult to stop the wheel than the disc. This is due to the force required to stop any rotating object is directly proportional to the product of the square of the distance from the axis of rotation to the particles and mass of the object. This is the moment of inertia.
What is the Moment of Inertia?
Moment of inertia is a quantity expressed by the body which is resisting angular acceleration. In simple words, Moment of Inertia can be explained as the amount of torque that a body needs for specific angular acceleration in the rotational axis. The moment of inertia is specified with the chosen axis of rotation. The moment of inertia mostly depends on the distribution of mass around the axis of rotation. The moment of inertia varies depending upon which axis of rotation we choose.
Moment of Inertia is expressed by the symbol ‘I’. And formula of moment of inertia is given by \[I = \sum {m{r^2}} \]
where I is the moment of inertia,
m is mass and
r is the distance from the axis of rotation.
What is the Unit of Moment of Inertia?
We have seen that the formula of the moment of inertia is \[I = m{r^2}\].
From this, the unit of mass is Kg and the unit of r, i.e., distance is m. Therefore, the moment of the Inertia unit is Kg.m2. This is the SI unit of the moment of inertia. C.G.S. unit is g.cm2.
The moment of inertia of an area is given by \[I = A{d^2}\]. Therefore, the unit of moment of inertia of an area is m4. This is the SI unit.
Moment of inertia of a mass is given by \[I = m{r^2}\]. Therefore, the units of the moment of inertia of mass are Kg.m2 and g.cm2.
Interesting Facts
As the speed of an object increases, rotational inertia also increases.
When air resistance is there, it causes rotational inertia to decrease.
The more the mass of an object is, the more the inertia will be.
Solved Problems
What will be the moment of inertia when you are spinning a 4kg solid ball of radius 0.4m?
Solution:
Given,
m= 4kg
r=0.4m
We know that inertia for a solid sphere is \[I = \dfrac{2}{5}m{r^2}\].
Therefore, \[I = \dfrac{2}{5}4 \times {0.4^2} = 0.256Kg{m^2}\].
A solid cylinder of mass 10Kg is rolling down on a road. The cylinder has a radius of 0.1 m. Its acceleration is 4.rad/s2. Find the inertia and torque.
Solution:
Given,
m= 10Kg
r= 0.1m
a= 4 rad/s2
We know that inertia for a solid cylinder is \[I = \dfrac{1}{2}m{r^2}\].
Therefore, \[I = \dfrac{1}{2}10 \times {0.1^2} = 0.05Kg{m^2}\].
Torque= I x a = 0.05*4=0.2 N.m
Summary
Moment of inertia is a quantity expressed by the body which is resisting angular acceleration. In simple words, the Moment of Inertia can be explained as the amount of torque that a body needs for specific angular acceleration in the rotational axis. The units of the moment of inertia are Kg.m2 and g.cm2.
FAQs on Moment of Inertia Units
1. On what factors, the moment of inertia depends on?
The moment of inertia depends on the density of the material - The moment of inertia is directly proportional to density.
Shape and size of the body - Due to change in the shape and size and size of the object, mass distribution around the axis of rotation changes. This affects the moment of inertia.
Axis of rotation, distribution of mass relative to the axis affects the moment of inertia.
Moment of inertia depends on the distribution of mass.
Moment of inertia does not depend on the angular velocity of the body.
2. What is the importance of inertia in our daily life?
Inertia has many important uses: design of safety devices for vehicles, including seat belts and many others, that provide external force to stop a motion of a body in event of a sudden change in the environment. In space travel, when a device escapes Earth’s Gravity, it continues on its given trajectory till it encounters another gravitational field or object. We can send space probes to great distances without any additional fuel required other than the fuel that is needed to “escape” Earth.
3. What will happen if there is no inertia?
In linear motion, the mass of the body is responsible for its inertia. Therefore, if inertia wouldn’t have been there, all bodies would have had no mass. For example, if there had been no inertia, the body could have started agitated and then stopped at random. A force of 1000N wouldn't have changed the state of motion or made the body travel the greatest distances. In absence of inertia, electrons in the nucleus would have ceased to revolve, putting matter's existence in doubt. At first, it appears as if the body would have started moving and come to rest even under vacuum and other conditions and without the application of any force.
