Moment of inertia

Moment of inertia

Edited By Vishal kumar | Updated on Jul 02, 2025 05:44 PM IST

The moment of inertia of a rigid body about a given axis of rotation is the sum of the products of the masses of the various particles and squares of their perpendicular distance from the axis of rotation.

In this article, we will cover the concept of moment of inertia. This topic falls under the broader category of rotational motion, which is a crucial chapter in Class 11 physics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), National Eligibility Entrance Test (NEET), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE and more. Generally, this concept is used to solve various questions in exams.

This Story also Contains
  1. Definition of Moment of Inertia
  2. Radius of Gyration (K)
  3. Solved Examples Based On Moment Of Inertia
  4. Summary
Moment of inertia
Moment of inertia

Let's read this entire article to gain an in-depth understanding of the moment of inertia.

Definition of Moment of Inertia

Moment of inertia (I) of a body is a measure of its ability to resist change in its rotational state of motion. It play the same role in rotatory motion as is played by mass in translatory motion.

Formula

$\text { Moment of inertia of a particle } I=m r^2$

Where: m is the mass of the particle and r is the perpendicular distance of the particle from the rotational axis.

Moment of Inertia For System of Particle

$\begin{aligned}
& I=m_1 r_1^2+m_2 r_2^2+\ldots \ldots \ldots m_n r_n^2 \\
= & \sum_{i=1}^n m_i r_i^2
\end{aligned}$

(This is Applied when masses are placed discreetly)

Moment of Inertia For Continuous Body

$I=\int r^2 d m$

Where: r is the perpendicular distance of a particle of mass dm of a rigid body from the axis of rotation

- Dimension $=\left[M L^2\right]$
- S.I. unit $=k g-m^2$

  • It depends on mass, distribution of mass and on the position of the axis of rotation.
  • It does not depend on angular velocity, angular acceleration, torque, angular momentum and rotational kinetic energy.
  • It is a tensor quantity.

Radius of Gyration (K)

Radius of Gyration of a body about an axis is the effective distance from the axis where the whole mass can be assumed to be concentrated so that the moment of inertia remains the same.

  • Formula- $K=\sqrt{\frac{I}{M}}$
    or, $I=M K^2$

  • It does not depend on the mass of the body

  • It depends on the shape and size of the body, distribution of mass of the body w.r.t. the axis of rotation etc.

  • Dimension- $M^o L^1 T^o$

  • S.I. unit: Meter.

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Moment of Inertia of Two Point Masses About Their Centre of Mass

Let two masses m1 and m2 at a distance r and from their centre of mass they are at a distance r1 and r2 respectively.

Then,

$
\begin{aligned}
& r_1+r_2=r_1 \\
& m_1 r_1=m_2 r_2
\end{aligned}
$

From equation (1) and (2)
$
r_1=\frac{m_2}{m_1+m_2} * r
$

And,
$
r_2=\frac{m_1}{m_1+m_2} * r
$
$
\begin{aligned}
& I=I_1+I_2=m_1 r_1^2+m_2 r_2^2 \\
& I=\frac{m_1 m_2}{m_1+m_2} * r^2 \\
& I_1=m_2 I, \text { and } I_2=m_1 I
\end{aligned}
$

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Solved Examples Based On Moment Of Inertia

Example 1: Four point masses (each of mass m) are arranged in the x-y plane. The moment inertia of masses about the y-axis is :

1) $6 m a^2$
2) $4 m a^2$
3) $2 m a^2$
4) $m a^2$

Solution:

Moment of inertia for the system of particle

$\begin{aligned}
& I=m_1 r_1^2+m_2 r_2^2+\ldots \ldots \ldots m_n r_n^2 \\
& =\sum_{i=1}^n m_i r_i^2
\end{aligned}$

So, Moment of inertia of masses= $m(0)^2+m(2 a)^2+m(a)^2+m(a)^2=6 m a^2$

Example 2: A massless equilateral triangle $E F G$ of side ' $a$ ' (As shown in the figure) has three particles of mass $m$ situated at its vertices. The moment of inertia of the system about the line $E X$ perpendicular to $E G$ in the plane of $E F G$ is $\frac{N}{20} m a^2$ where $N$ is an integer. The value of $N$ is:


1) 5

2) 10

3) 20

4) 25

Solution:

The moment of inertia of the system about the line passing through EX is
$
I=I_E+I_F+I_G
$

As we know for point mass $m$ at separation $\mathrm{r}$ from the axis of rotation is
$
I=m r^2
$

Point masses separation from the axis of rotation at positions $\mathrm{E}, \mathrm{F} \& \mathrm{G}$ are 0 , $a / 2$, a respectively
$
I=m(0)^2+m(a / 2)^2+m(a)^2=\frac{5 m a^2}{4}=\frac{25 m a^2}{20}
$
comparing with, $\frac{N}{20} m a^2$

N=25

Hence, the answer is option (4).

Example 3: A wheel of mass 10kg has a moment of inertia of 90 kg-m2 about its own axis, the radius of the gyration will be:

1) 1

2) 2

3) 3

4) 4

Solution

$\begin{aligned}
& I=m K^2 \\
& 90=10\left(K^2\right) \\
& K=3
\end{aligned}$

Hence, the answer is the option (3).

Example 4: Let the moment of inertia of a hollow cylinder of length 30 cm (inner radius 10 cm and outer radius 20 cm ), about its axis be I. The radius (in cm) of a thin cylinder of the same mass such that its moment of inertia about its axis is also I, is :

1) 16

2) 18

3) 14

4) 12

Solution

Radius of gyration (K)

The radius of Gyration of a body about an axis is the effective distance from the axis where the whole mass can be assumed to be concentrated so that the moment of inertia remains the same.

wherein

$\begin{aligned}
& I=M K^2 \\
& K=\sqrt{\frac{I}{M}}
\end{aligned}$

$
I=\frac{m\left(20^2+10^2\right)}{2}
$
now let the moment of inertia of a thin cylinder
$
\begin{aligned}
& I=m k^2 \\
& I=m k^2=m \frac{20^2+10^2}{2} \\
& k=\sqrt{\frac{400+100}{2}}=5 \sqrt{10} \mathrm{~cm} \\
& k \approx 16 \mathrm{~cm}
\end{aligned}
$

Hence, the answer is option (1).

Summary

The moment of inertia for a rigid body is a physical quantity that combines mass and shape in Newton's equations of motion, momentum, and kinetic energy.
The moment of inertia is applied in both linear and angular moments, although it manifests itself in planar and spatial movement in rather different ways. One scalar quantity defines the moment of inertia in planar motion.

Frequently Asked Questions (FAQs)

1. What is moment of inertia, and why is it important in rotational motion?
Moment of inertia is a measure of an object's resistance to rotational acceleration, similar to how mass resists linear acceleration. It's important in rotational motion because it determines how easily an object can be rotated. Objects with larger moments of inertia require more torque to achieve the same angular acceleration as objects with smaller moments of inertia.
2. How is moment of inertia different from mass?
While mass is a measure of an object's resistance to linear acceleration, moment of inertia is a measure of an object's resistance to rotational acceleration. Mass is a scalar quantity that depends only on the amount of matter, while moment of inertia is a tensor quantity that depends on both the mass and its distribution relative to the axis of rotation.
3. Why does a figure skater spin faster when they pull their arms close to their body?
When a figure skater pulls their arms close to their body, they decrease their moment of inertia. Due to the conservation of angular momentum, the decrease in moment of inertia results in an increase in angular velocity, making them spin faster. This demonstrates the relationship between moment of inertia and angular velocity in a constant angular momentum system.
4. Why do planets with rings, like Saturn, have such high moments of inertia?
Planets with rings, like Saturn, have high moments of inertia because a significant portion of their mass is distributed far from their axis of rotation. The rings, being located at a large distance from the planet's center, contribute disproportionately to the total moment of inertia. This high moment of inertia makes these planets more resistant to changes in their rotational motion.
5. Why do ice skaters start a spin with their arms outstretched and end with their arms close to their body?
Ice skaters use the principle of conservation of angular momentum to increase their spinning speed. They start with arms outstretched, which increases their moment of inertia. As they pull their arms in close to their body, their moment of inertia decreases. Since angular momentum (L = Iω) must be conserved, the decrease in moment of inertia (I) results in an increase in angular velocity (ω), making them spin faster.
6. How does the distribution of mass affect an object's moment of inertia?
The distribution of mass greatly affects an object's moment of inertia. Mass located farther from the axis of rotation contributes more to the moment of inertia than mass closer to the axis. This is why a hollow cylinder has a larger moment of inertia than a solid cylinder of the same mass and radius when rotating about their central axes.
7. Can an object have different moments of inertia?
Yes, an object can have different moments of inertia depending on the axis about which it's rotating. For example, a rod has a different moment of inertia when rotating about its center compared to when rotating about one of its ends. The moment of inertia is always specific to a particular axis of rotation.
8. How does the shape of an object affect its moment of inertia?
The shape of an object significantly affects its moment of inertia. Objects with mass distributed farther from the axis of rotation have larger moments of inertia. For example, a hollow sphere has a larger moment of inertia than a solid sphere of the same mass and radius when rotating about their centers. This is why the shape of objects is crucial in designing things like flywheels or gymnastic equipment.
9. How does changing the axis of rotation affect an object's moment of inertia?
Changing the axis of rotation can significantly affect an object's moment of inertia. Generally, the moment of inertia increases as the axis of rotation moves farther from the object's center of mass. This is because more mass is distributed farther from the axis, contributing more to the moment of inertia. The parallel axis theorem quantifies this relationship.
10. What is the parallel axis theorem, and how is it used?
The parallel axis theorem states that the moment of inertia of an object about any axis parallel to an axis passing through its center of mass is equal to its moment of inertia about the center of mass axis plus the product of its mass and the square of the perpendicular distance between the two axes. This theorem is used to calculate the moment of inertia about any axis parallel to a known axis, simplifying many rotational motion problems.
11. How does the moment of inertia affect the period of a physical pendulum?
The moment of inertia significantly affects the period of a physical pendulum. The period (T) of a physical pendulum is given by T = 2π√(I/mgd), where I is the moment of inertia about the pivot point, m is the mass, g is the acceleration due to gravity, and d is the distance from the pivot to the center of mass. A larger moment of inertia results in a longer period, meaning the pendulum swings more slowly. This relationship is crucial in the design of pendulum clocks and in understanding the behavior of various oscillating systems.
12. How does the moment of inertia of a molecule affect its rotational spectrum in spectroscopy?
The moment of inertia of a molecule directly affects its rotational spectrum in spectroscopy. The energy levels of a rotating molecule are inversely proportional to its moment of inertia. Molecules with smaller moments of inertia have larger spacings between rotational energy levels, resulting in higher frequency spectral lines. Conversely, molecules with larger moments of inertia have smaller energy level spacings and lower frequency spectral lines. This relationship allows scientists to determine molecular structures and compositions through rotational spectroscopy.
13. What is the relationship between torque, moment of inertia, and angular acceleration?
The relationship between torque (τ), moment of inertia (I), and angular acceleration (α) is given by the rotational form of Newton's Second Law: τ = Iα. This equation shows that for a given torque, an object with a larger moment of inertia will experience less angular acceleration, while an object with a smaller moment of inertia will experience more angular acceleration.
14. How does the concept of moment of inertia apply to everyday objects like doors?
The concept of moment of inertia applies to doors in determining how easily they can be opened or closed. A door's moment of inertia about its hinges affects how much force is needed to start it moving. Doors with more mass or with the mass distributed farther from the hinges (like wider doors) have larger moments of inertia and require more force to open or close. This is why large, heavy doors often feel more difficult to move than lighter ones.
15. What is the relationship between moment of inertia and angular momentum?
The relationship between moment of inertia (I) and angular momentum (L) is given by the equation L = Iω, where ω is the angular velocity. This equation shows that for a given angular velocity, an object with a larger moment of inertia will have more angular momentum. This relationship is crucial in understanding phenomena like the conservation of angular momentum in rotating systems.
16. What is the significance of the perpendicular axis theorem in calculating moments of inertia?
The perpendicular axis theorem states that for a planar object, the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about two perpendicular axes in the plane that intersect at the point where the perpendicular axis passes through the plane. This theorem simplifies calculations of moments of inertia for complex shapes by allowing them to be broken down into simpler components. It's particularly useful for understanding the rotational behavior of flat objects like discs or irregular shapes.
17. Why is the moment of inertia important in understanding the precession of gyroscopes and the Earth?
The moment of inertia is crucial in understanding precession because it affects how an object responds to external torques. In gyroscopes and the Earth, the large moment of inertia about the spin axis resists changes to the spin orientation. When an external torque is applied perpendicular to the spin axis, instead of toppling over, the object precesses - its axis of rotation slowly traces out a cone. The rate of precession is inversely proportional to the moment of inertia and the spin rate. This explains why the Earth's axis precesses slowly, completing a cycle every 26,000 years, a phenomenon known as the precession of the equinoxes.
18. How does the moment of inertia of a solid sphere compare to that of a hollow sphere of the same mass and radius?
A hollow sphere has a larger moment of inertia than a solid sphere of the same mass and radius when rotating about their centers. This is because more of the mass in a hollow sphere is distributed farther from the axis of rotation. Specifically, the moment of inertia of a solid sphere is (2/5)MR², while for a thin hollow sphere it's (2/3)MR², where M is the mass and R is the radius.
19. What is the significance of the radius of gyration in relation to moment of inertia?
The radius of gyration (k) is a characteristic length that relates an object's moment of inertia to its mass. It's defined by the equation I = Mk², where I is the moment of inertia and M is the mass. The radius of gyration represents the distance from the axis of rotation at which all the mass of the object could be concentrated to give the same moment of inertia. It's useful for comparing the rotational inertia of different objects.
20. How does the moment of inertia of a rod change when rotated about different axes?
The moment of inertia of a rod changes significantly depending on the axis of rotation. For a rod of length L and mass M:
21. Why is the moment of inertia important in the design of flywheels?
The moment of inertia is crucial in flywheel design because flywheels are used to store rotational energy. A higher moment of inertia allows a flywheel to store more energy at a given angular velocity. Engineers often design flywheels with most of their mass concentrated at the rim to maximize the moment of inertia, allowing for efficient energy storage and smooth power delivery in various applications, from engines to power plants.
22. How does the moment of inertia of a system change when objects are added or removed?
When objects are added to or removed from a rotating system, the total moment of inertia changes. Adding mass increases the moment of inertia, especially if it's added far from the axis of rotation. Removing mass decreases the moment of inertia. The exact change depends on the mass and position of the added or removed objects relative to the axis of rotation. This principle is used in various applications, from adjustable exercise equipment to variable-inertia flywheels in engines.
23. How does the moment of inertia of a solid cylinder compare to that of a hollow cylinder of the same mass and radius?
A hollow cylinder has a larger moment of inertia than a solid cylinder of the same mass and radius when rotating about their central axes. For a solid cylinder, I = (1/2)MR², while for a thin hollow cylinder, I = MR². This difference occurs because the mass in the hollow cylinder is distributed farther from the axis of rotation, contributing more to the moment of inertia.
24. What is the importance of moment of inertia in the design of gyroscopes?
Moment of inertia is crucial in gyroscope design because it affects the gyroscope's stability and resistance to changes in orientation. A larger moment of inertia makes the gyroscope more resistant to external torques, maintaining its orientation more effectively. This is why gyroscopes often have most of their mass concentrated at the rim of a wheel, maximizing the moment of inertia about the spin axis and enhancing their stability and precision in applications like navigation systems and attitude control in spacecraft.
25. How does the moment of inertia of a planet affect its rotational stability?
The moment of inertia of a planet plays a significant role in its rotational stability. Planets with larger moments of inertia are more resistant to changes in their rotational motion. This resistance helps maintain a stable rotation rate and axis orientation over long periods. For example, Earth's large moment of inertia contributes to the stability of its 24-hour day and the orientation of its rotational axis, which is crucial for maintaining stable climate patterns.
26. Why is the concept of moment of inertia important in the design of rotating machinery?
The concept of moment of inertia is crucial in designing rotating machinery because it affects the machine's performance, energy efficiency, and stability. Machines with larger moments of inertia require more energy to start and stop but maintain more consistent speeds under varying loads. Engineers must balance these factors when designing components like flywheels, turbines, and centrifuges to optimize performance, reduce vibrations, and ensure smooth operation.
27. How does the moment of inertia of a rotating object affect its kinetic energy?
The moment of inertia directly affects the rotational kinetic energy of an object. The rotational kinetic energy is given by the equation KE = (1/2)Iω², where I is the moment of inertia and ω is the angular velocity. This means that for a given angular velocity, an object with a larger moment of inertia will have more rotational kinetic energy. This relationship is important in understanding energy storage in rotating systems and the behavior of objects in rotational motion.
28. Why do long objects like poles or javelins rotate end-over-end when thrown?
Long objects like poles or javelins tend to rotate end-over-end when thrown because this rotation minimizes their moment of inertia about the center of mass. The moment of inertia is smallest when the object rotates perpendicular to its long axis, allowing for faster rotation. This orientation also provides the least air resistance. As a result, even if not initially thrown in this orientation, air resistance and torque will tend to align the object perpendicular to its trajectory, leading to the characteristic end-over-end rotation.
29. How does changing the mass distribution of a rotating object affect its angular acceleration?
Changing the mass distribution of a rotating object affects its moment of inertia, which in turn affects its angular acceleration. According to τ = Iα, where τ is torque, I is moment of inertia, and α is angular acceleration, for a given torque, increasing the moment of inertia (by moving mass farther from the axis of rotation) decreases the angular acceleration. Conversely, decreasing the moment of inertia (by concentrating mass closer to the axis) increases the angular acceleration. This principle is used in various applications, from adjustable exercise equipment to variable-inertia flywheels in engines.
30. What role does moment of inertia play in the stability of spinning tops?
Moment of inertia plays a crucial role in the stability of spinning tops. A larger moment of inertia about the spin axis makes the top more resistant to changes in its orientation, increasing its stability. This is why many tops are designed with most of their mass concentrated around the rim. The high moment of inertia results in a high angular momentum, which resists the torque due to gravity that would otherwise cause the top to fall. This resistance to change in orientation is what keeps the top upright and spinning for an extended period.
31. How does the concept of moment of inertia apply to the rotation of galaxies?
The concept of moment of inertia is crucial in understanding the rotation of galaxies. Galaxies, being vast collections of stars, gas, and dust, have enormous moments of inertia. This large moment of inertia explains why galaxies rotate relatively slowly despite the immense gravitational forces involved. It also affects the distribution of matter within the galaxy, as objects with higher angular momentum (related to moment of inertia) tend to orbit farther from the galactic center. Understanding galactic moments of inertia has been key in identifying discrepancies that led to the theory of dark matter.
32. What is the significance of the moment of inertia tensor in describing the rotation of non-symmetrical objects?
The moment of inertia tensor is crucial for describing the rotation of non-symmetrical objects because it accounts for the object's resistance to rotation about any arbitrary axis. Unlike symmetrical objects that can be described by a single scalar value, non-symmetrical objects require a 3x3 matrix (the inertia tensor) to fully characterize their rotational behavior. This tensor allows for the calculation of the object's moment of inertia about any axis and helps predict complex rotational motions, including wobbling and precession, which are common in real-world objects like satellites or asteroids.
33. How does the principle of least action relate to the moment of inertia in rotational systems?
The principle of least action in rotational systems is closely tied to the moment of inertia. In rotational mechanics, the action is related to the rotational kinetic energy, which depends on the moment of inertia. The principle states that the path taken by a rotating system between two points in time is the one for which the action is stationary (usually a minimum). For rotational systems, this often means that

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