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Moment Of Inertia Of A Rod

Moment Of Inertia Of A Rod

Edited By Vishal kumar | Updated on Jul 02, 2025 05:34 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. When a thin rod's linear mass density, or mass per length, remains constant throughout its length, it is said to be uniform.

This Story also Contains
  1. Definition of Moment of Inertia
  2. Radius of Gyration (K)
  3. Moment of Inertia of a Rod
  4. Summary
Moment Of Inertia Of A Rod
Moment Of Inertia Of A Rod

In this article, we will cover the concept of the moment of inertia of a rod. 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. Over the last ten years of the JEE Main exam (from 2013 to 2023), five questions have been asked on this concept. And for NEET one question was asked from this concept.

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

Definition of Moment of Inertia

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

Formula of Moment of Inertia
Moment of inertia of a particle I=mr2

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

Moment of Inertia For System of Particle
I=m1r12+m2r22+………mnrn2=∑i=1nmiri2

(This is Applied when masses are placed discreetly)

Moment of Inertia For Continuous Body
I=∫r2dm

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

- Dimension =[ML2]
- S.I. unit =kg−m2

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=IM
or, I=MK2

  • 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-
  • S.I. unit: Meter.

Moment of Inertia of a Rod

Let I=Moment of inertia of a ROD about an axis through its centre and perpendicular to it, to calculate I (Moment of inertia of rod).

Consider a uniform straight rod of length L, mass M and having centre C.

mass per unit length of the rod:
λ=ML

Take a small element of mass dm with length dx at a distance x from point C.

dm=λ⋅dx=ML⋅dx⇒dI=x2dm

Now integrate this dl between the limits x=−L2 to L2

I=∫dI=∫x2dm=∫−L2L2MLx2∗dx=ML∫−L2L2x2dx=ML212

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Solved Examples Based on Moment of Inertia of a Rod

Example 1: The moment of inertia of a uniform cylinder of length l and radius R about its perpendicular bisector is I. What is the ratio lR such that the moment of inertia is minimum?

1) 32
2) 32
3) 1
4) 32

Solution:

The length of the Cylinder =l
Radius of cylinder =R
Moment of inertia =l
Moment of inertia of uniform rod of length(I) -
I=Ml212

wherein

About axis passing through its centre & perpendicular to its length.

We know that a solid cylinder is about an axis that is perpendicular bisector.

I=mR24+ml212=m4[R2+l23]I=m4[πlR2πl+l23]=m4[vπl+l23] putting πlR2=vdIdl=0⇒m4[−vπl2+2l3]=0⇒−vπl2+2l3=0vπl2=2l3⇒v=2πl33πR2l=2πl33⇒R2=2l23l2R2=32lR=32

Hence, the answer is the option (1).

Example 2: A thin uniform bar of length L and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m are moving in the same horizontal plane from opposite sides of the bar with speeds 2v and v respectively. The masses stick to the bar after collision at a distance L/3 and L/6 respectively from the centre of the bar. If the bar starts rotating about its centre of mass as a result of the collision, the angular speed of the bar will be :

1) v5L
2) 6v5L
3) 3v5L
4) v6L

Solution:

Moment of inertia of uniform rod of length (l) -
I=Ml212

About axis passing through its centre \& perpendicular to its length.
Law of conservation of angular moment -
τ→=dL→dt

If net torque is zero
i.e. dL→dt=0
L→= constant

angular momentum is conserved only when external torque is zero.

The centre of mass of the system from o
=8m×0+m(L/3)−2m(L/6)8m+m+2m=0

So, the centre of mass is at "O".
From the conservation of angular momentum;
Li=LfLi=m⋅(2v)∗(L/3)+2mv∗(L/6)=mvLLf=[(8m)⋅L212+m⋅(L/3)2+2m⋅(L/6)2]ω

=[23mL2+mL29+mL218]ω=(12+2+118)mL2ω=56mL2ω56mL2ω=mvL∴ω=6v5L

Example 3: A uniform thin rod of length 4l, mass 4m is bent at the points as shown in the figure. What is the moment of inertia of the rod about the axis passing through point O & perpendicular to the plane of the paper?

1) ml23
2) 10ml23
3) ml212
4) ml224

Solution:

Moment of inertia of uniform rod of length (l)
I=Ml212

About axis passing through its centre \& perpendicular to its length.
The given structure can be broken into 4 parts
For AB:I=ICM+m×d2=ml212+5m4l2;IAB=43ml2

For BO: I=ml23
For composite frame : (by symmetry)
I=2[IAB+IOB]=2[4ml23+ml23]=103ml2.

Example 4: Three identical rods, each of length l are joined to form a rigid equilateral triangle. Its radius of gyration about an axis passing through a corner and perpendicular to the plane of the triangle is:

1) l
2) l/2
3) l/2
4) l/3

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

I=MK2K=IM

MI of system w.r.t an axis perpendicular to the plane and passing through one centre

=ml23+ml23+[ml212+m[3l2]2]=2ml23+10ml212=1812ml2NOW3mk2=32ml2k=l/2

Example 5: A uniform straight rod of length L, mass M has Moment of inertia about an axis through its centre and perpendicular to it as I. If we take the new rod of length 2L, and mass (0.5M), Then the Moment of inertia of the new rod about an axis through its centre and perpendicular to it will be:

1) I

2) 2I

3) 3I

4) 4I

Solution:

The moment of inertia of a Rod is given as I=ML212 Now take M′=0.5M and L′=2L
so, I′=M′L′212=12M×4L212=2I

Hence, the answer is the option 2.

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 the formula for the moment of inertia of a rod when the axis is through the end?

The formula for the moment of inertia of a rod when the axis is through the end isI = \frac{1}{3} ML^2.

2. Is there any difference between the moment of inertia and rotational inertia?

No

3. Is the moment of inertia a scalar or a vector quantity?

Scalar quantity

4. Does the moment of inertia change with the change of the axis of rotation?

Yes

5. Why is the moment of inertia important in understanding a rod's rotational motion?
The moment of inertia is crucial for understanding a rod's rotational motion because it determines how easily the rod can be rotated. A larger moment of inertia means the rod is more resistant to changes in its rotational motion, similar to how mass resists changes in linear motion.
6. What's the difference between the moment of inertia of a rod and its angular momentum?
While both involve rotation, moment of inertia and angular momentum are different concepts. Moment of inertia (I) is a property of the object that resists changes in rotational motion. Angular momentum (L) is the rotational equivalent of linear momentum, given by L = Iω, where ω is the angular velocity.
7. Can the moment of inertia of a rod be negative?
No, the moment of inertia of a rod (or any object) cannot be negative. It's always a positive quantity because it represents the resistance to rotational acceleration, which is always in opposition to the applied torque.
8. How does the rotational kinetic energy of a rod relate to its moment of inertia?
The rotational kinetic energy of a rod is directly proportional to its moment of inertia. The formula is KE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. A larger moment of inertia results in more rotational kinetic energy for the same angular velocity.
9. What role does the moment of inertia play in the rotational period of a rod?
The moment of inertia plays a crucial role in determining the rotational period of a rod. A larger moment of inertia results in a longer rotational period for a given applied torque. This is analogous to how greater mass results in slower linear acceleration for a given force.
10. What is the moment of inertia of a rod?
The moment of inertia of a rod is a measure of its resistance to rotational acceleration. It depends on the rod's mass distribution and the axis of rotation. For a uniform rod, it's calculated as (1/12)ML^2 for rotation about its center, where M is the mass and L is the length of the rod.
11. How does the moment of inertia of a rod change if its mass is doubled?
If the mass of a rod is doubled while keeping its length constant, the moment of inertia also doubles. This is because the moment of inertia is directly proportional to the mass of the object.
12. Why is the moment of inertia of a rod different for different axes of rotation?
The moment of inertia of a rod varies for different axes because it depends on how the mass is distributed relative to the axis of rotation. Mass farther from the axis contributes more to the moment of inertia, following the r^2 term in the formula I = Σmr^2.
13. How does the moment of inertia of a rod compare when rotating about its center vs. its end?
The moment of inertia of a rod rotating about its end is greater than when rotating about its center. Specifically, it's four times larger. This is because more mass is distributed farther from the axis of rotation when the rod rotates about its end.
14. What's the significance of the (1/12) factor in the moment of inertia formula for a rod?
The (1/12) factor in the formula I = (1/12)ML^2 for a rod rotating about its center comes from the integration of mass elements along the rod's length. It represents the average distribution of mass relative to the center of rotation.
15. How does the moment of inertia of a hollow rod compare to a solid rod of the same mass and length?
A hollow rod has a larger moment of inertia than a solid rod of the same mass and length. This is because the mass in a hollow rod is distributed farther from the axis of rotation, increasing the r^2 term in the moment of inertia calculation.
16. Why is the moment of inertia of a rod different from that of a point mass?
The moment of inertia of a rod differs from a point mass because the rod has an extended mass distribution. While a point mass concentrates all its mass at a single point, a rod's mass is spread out along its length, resulting in a non-zero moment of inertia even when rotating about its center.
17. How does changing the length of a rod affect its moment of inertia?
Changing the length of a rod has a significant impact on its moment of inertia. The moment of inertia is proportional to the square of the rod's length. So, doubling the length increases the moment of inertia by a factor of four.
18. What happens to the moment of inertia if a rod is cut in half?
If a rod is cut in half, its moment of inertia about its center decreases by a factor of 8. This is because the moment of inertia depends on both the mass (which is halved) and the square of the length (which is reduced to 1/4). So, (1/2) * (1/4) = 1/8 of the original value.
19. How does the concept of parallel axis theorem apply to calculating the moment of inertia of a rod?
The parallel axis theorem states that the moment of inertia about any axis parallel to an axis through the center of mass is equal to the moment of inertia about the center of mass plus Md^2, where M is the total mass and d is the perpendicular distance between the axes. This theorem is useful for calculating the moment of inertia of a rod about different axes.
20. How does the material of the rod affect its moment of inertia?
The material of the rod doesn't directly affect its moment of inertia. The moment of inertia depends on the mass and its distribution, not the specific material. However, different materials with different densities will result in different masses for the same volume, indirectly affecting the moment of inertia.
21. How does the moment of inertia of a rod relate to its angular momentum in the absence of external torques?
In the absence of external torques, the angular momentum of a rod remains constant (conserved). The relationship is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. If the moment of inertia changes, the angular velocity must change proportionally to maintain constant angular momentum.
22. What's the relationship between a rod's moment of inertia and its rotational kinetic energy?
The rotational kinetic energy of a rod is directly proportional to its moment of inertia. The formula is KE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. A larger moment of inertia means more energy is stored in the rotation for the same angular velocity.
23. Can a rod have zero moment of inertia?
In practice, a real rod cannot have zero moment of inertia. However, in theoretical physics, we sometimes approximate very thin rods as having negligible moment of inertia when rotating about their long axis. This is an idealization and not physically realizable.
24. How does the moment of inertia of a rod compare to that of a disk of the same mass and radius?
For rotation about their respective centers, a rod has a smaller moment of inertia compared to a disk of the same mass and radius. The rod's moment of inertia is (1/12)ML^2, while the disk's is (1/2)MR^2. This is because the disk has more of its mass distributed farther from the axis of rotation.
25. How does adding mass to the ends of a rod affect its moment of inertia?
Adding mass to the ends of a rod significantly increases its moment of inertia. This is because the moment of inertia depends on the square of the distance from the axis of rotation (I = Σmr^2). Mass at the ends is farther from the center, contributing more to the total moment of inertia.
26. Why is the moment of inertia of a rod important in designing things like pendulums or metronomes?
The moment of inertia of a rod is crucial in designing pendulums or metronomes because it affects the period of oscillation. A larger moment of inertia results in a slower swing, allowing for precise timing adjustments by changing the rod's mass distribution.
27. How does the moment of inertia of a rod relate to its angular acceleration?
The moment of inertia of a rod is inversely proportional to its angular acceleration for a given torque. This relationship is described by the rotational form of Newton's Second Law: τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration.
28. Can you explain why a rod's moment of inertia changes when it rotates about different parallel axes?
A rod's moment of inertia changes for different parallel axes due to the parallel axis theorem. As the axis of rotation moves away from the center of mass, the moment of inertia increases by Md^2, where M is the mass and d is the distance between the new axis and the center of mass axis.
29. How does the concept of radius of gyration relate to a rod's moment of inertia?
The radius of gyration (k) is related to the rod's moment of inertia by the equation I = Mk^2, where M is the total mass. It represents the distance from the axis of rotation at which all the mass could be concentrated to give the same moment of inertia as the actual distribution.
30. Why is it easier to balance a longer rod on your finger than a shorter one?
It's easier to balance a longer rod because it has a larger moment of inertia. This means it resists changes in its rotational motion more strongly, making it more stable and slower to tip over. The larger moment of inertia gives you more time to make small corrections to keep it balanced.
31. How does the moment of inertia of a rod affect its angular momentum conservation?
The moment of inertia of a rod is crucial in angular momentum conservation. When a rod's moment of inertia changes (e.g., by changing its shape), its angular velocity must change to conserve angular momentum (L = Iω). A larger moment of inertia results in a smaller angular velocity for the same angular momentum.
32. What's the relationship between a rod's moment of inertia and its rotational energy?
The rotational energy of a rod is directly proportional to its moment of inertia. The formula for rotational kinetic energy is KE = (1/2)Iω^2, where I is the moment of inertia and ω is the angular velocity. A larger moment of inertia means more energy is stored in the rotation for the same angular velocity.
33. How does the distribution of mass along a rod affect its moment of inertia?
The distribution of mass along a rod significantly affects its moment of inertia. Mass located farther from the axis of rotation contributes more to the moment of inertia due to the r^2 term in the formula I = Σmr^2. A rod with more mass concentrated at its ends will have a larger moment of inertia than one with mass concentrated in the center.
34. Why do figure skaters pull in their arms to spin faster, and how does this relate to a rod's moment of inertia?
Figure skaters pull in their arms to decrease their moment of inertia, which is analogous to shortening a rod. By reducing their moment of inertia while conserving angular momentum (L = Iω), their angular velocity increases, making them spin faster. This demonstrates the inverse relationship between moment of inertia and angular velocity when angular momentum is conserved.
35. How does the moment of inertia of a rod compare when it's rotating about its long axis versus perpendicular to its long axis?
The moment of inertia of a rod is much smaller when rotating about its long axis compared to rotating perpendicular to its long axis. This is because when rotating about the long axis, all the mass is close to the axis of rotation, while rotation perpendicular to the long axis involves mass distributed farther from the axis.
36. What's the significance of the moment of inertia in calculating the torque needed to rotate a rod?
The moment of inertia is crucial in calculating the torque needed to rotate a rod because it determines the rod's resistance to angular acceleration. The relationship is given by τ = Iα, where τ is the torque, I is the moment of inertia, and α is the angular acceleration. A larger moment of inertia requires more torque to achieve the same angular acceleration.
37. How does the concept of rotational inertia relate to a rod's moment of inertia?
Rotational inertia is another term for moment of inertia. Both refer to an object's resistance to changes in its rotational motion. For a rod, this resistance depends on its mass distribution relative to the axis of rotation, as described by the moment of inertia formula.
38. Why is the moment of inertia of a rod important in understanding the physics of a simple pendulum?
The moment of inertia of a rod is important in understanding simple pendulums because it affects the pendulum's period of oscillation. A larger moment of inertia results in a longer period. This is why the length of a pendulum (which affects its moment of inertia) is crucial in determining its swing time.
39. How does the moment of inertia of a rod change if it's bent into a V-shape?
When a rod is bent into a V-shape, its moment of inertia decreases compared to when it's straight, assuming rotation about an axis through the vertex of the V. This is because some of the mass is brought closer to the axis of rotation, reducing the overall r^2 term in the moment of inertia calculation.
40. What's the difference between the moment of inertia of a uniform rod and a rod with varying density along its length?
A uniform rod has a consistent mass distribution, resulting in a predictable moment of inertia formula. A rod with varying density will have a different moment of inertia because the mass distribution is not uniform. Areas of higher density farther from the axis of rotation will contribute more to the total moment of inertia.
41. How does the moment of inertia of a rod affect its precession when it's spinning?
The moment of inertia of a rod affects its precession rate when spinning. Precession is the slow rotation of the spin axis itself. A larger moment of inertia results in a slower precession rate for a given applied torque, as described by the equation ωp = τ / (Iω), where ωp is the precession rate, τ is the torque, I is the moment of inertia, and ω is the spin rate.
42. Why is understanding the moment of inertia of a rod important in designing rotating machinery?
Understanding the moment of inertia of a rod is crucial in designing rotating machinery because it affects the machine's dynamic behavior. It influences factors like start-up torque requirements, braking distances, and vibration characteristics. Proper consideration of moment of inertia helps in designing more efficient and stable rotating systems.
43. What's the significance of the perpendicular axis theorem in calculating the moment of inertia of a rod?
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. For a rod, this theorem can be used to relate its moments of inertia about different axes.
44. How does the moment of inertia of a rod affect its stability when balanced vertically?
The moment of inertia of a rod affects its stability when balanced vertically. A larger moment of inertia (like in a longer rod) provides more resistance to toppling, making it more stable. This is why it's easier to balance a long rod on your finger than a short one.
45. Why is the moment of inertia of a rod important in understanding the physics of a compound pendulum?
The moment of inertia of a rod is crucial in understanding compound pendulums because it determines the pendulum's period of oscillation. Unlike a simple pendulum where all mass is considered at a point, a compound pendulum (like a swinging rod) has its mass distributed, and this distribution, represented by the moment of inertia, affects its motion.
46. How does adding a small mass to the end of a rod affect its moment of inertia?
Adding a small mass to the end of a rod significantly increases its moment of inertia. This is because the moment of inertia depends on the square of the distance from the axis of rotation (I = Σmr^2). Even a small mass at the end of the rod is at a large distance, contributing substantially to the total moment of inertia.
47. How does the concept of torque relate to changing the angular velocity of a rod with a given moment of inertia?
Torque is

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