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. What is the SI unit for moment of inertia?
The SI unit for moment of inertia is the kilogram metre squared (kg·m²). This unit represents an object's resistance to being rotated around a specific axis. In the CGS system, the corresponding unit is the gram centimetre squared (g·cm²).
2. How is the unit for moment of inertia derived from its formula?
The unit is derived directly from the basic formula for a point mass, I = mr². In this formula, 'm' stands for mass and 'r' is the perpendicular distance from the axis of rotation. The unit is therefore a product of the SI unit for mass (kg) and the square of the SI unit for distance (m), which results in kg·m².
3. What is the dimensional formula for moment of inertia?
The dimensional formula for moment of inertia is [M¹L²T⁰]. This shows that the quantity depends on mass to the first power (M¹), length to the second power (L²), and does not depend on time (T⁰).
4. What key factors determine an object's moment of inertia?
An object's moment of inertia is determined by a few key factors:
- The total mass of the object.
- The shape and size of the object, which dictates how its mass is distributed.
- The specific axis of rotation chosen. The same object will have a different moment of inertia if rotated around a different axis.
5. How is moment of inertia different from an object's mass?
While both concepts measure resistance to changes in motion, they apply to different situations. Mass measures linear inertia (resistance to changes in straight-line motion) and is a constant property of an object. Moment of inertia measures rotational inertia (resistance to changes in spinning motion) and is not constant; it depends on how the object's mass is distributed relative to its axis of rotation.
6. Can you explain moment of inertia with a real-world example?
A great example is a spinning ice skater. When the skater pulls their arms in close to their body, they decrease their moment of inertia by bringing their mass closer to the axis of rotation. This causes their spinning speed to increase. To slow down, they extend their arms, increasing their moment of inertia and making it harder to rotate quickly.
7. Why does mass farther from the axis of rotation have a greater effect on moment of inertia?
This is because the distance from the axis is squared in the moment of inertia formula (I = Σmr²). The r² term means that a particle of mass located twice as far from the axis will contribute four times as much to the total moment of inertia. This gives mass at a greater distance significantly more influence on the object's resistance to rotation.
8. Does an object's spinning speed affect its moment of inertia?
No, the moment of inertia is an inherent physical property based on an object's mass and shape for a given axis. It does not depend on the angular velocity (how fast it's spinning) or angular acceleration. Instead, it is the property that determines how much torque is needed to change that angular velocity.
