Content Strategy Manager
Unit of Moment of Inertia is one kilogram per square meter (kg.m2). Moment of Inertia is a quantitative measure of a body's rotational inertia or angular mass of a body. It is the body's reaction or the opposition it faces to a shift in rotation around an axis, which might be internal or external. The ability of a substance that resists change in its state of rest or motion is called inertia. The moment of inertia (I) is based on the distribution of mass in the body with respect to the axis of rotation.
Key Terms: Unit of Moment of Inertia, SI unit of moment of inertia, Inertia units, Mass moment of inertia units, Rigid body, Moment of Inertia units
Units of Moment of Inertia?
[Click Here for Sample Questions]
For a point mass, the Moment of Inertia is the mass times the square of the perpendicular distance to the rotation reference axis and can be expressed as:
| I = mr2 |
Where,
- I = Moment of Inertia
- m = Mass
- r = Distance between Axis and Rotation Mass
The types of the unit of moment of inertia and their formula are as follows:
| Moment of Inertia Units | |
|---|---|
| Moment of Inertia SI Unit | kg.m2 |
| Area moment of inertia | mm4 or in4 |
| Mass moment of inertia | kg.m2 or ft.lb. s2 |
| Dimensional Formula | M1L2T0 |
Example of Moment of InertiaExample: Consider a tire and a uniform disc, both of which spin along the same axis and have the same mass. As you've seen, starting and stopping the tire is harder than starting and stopping the uniform disc. Why is this the situation despite the fact that both items have the same mass? Solution: That's because the force needed to cease a spinning body is proportional to the product of the mass and the square of the distance between both the axis of spin and the body's particles. Mathematically, it is represented as F = mr2 |
Read More: Angular Velocity Formula
Moment of Inertia of Rigid Body
[Click Here for Sample Questions]
The dynamics of rigid bodies rotating about fixed axes can be summarized in three equations:
- Angular momentum is L = Iω
- Torque is τ = Iα
- Kinetic energy is K = 1/2 Iω 2
The moment of inertia for rigid objects is as follows:
| Rigid object | Moment of Inertia |
|---|---|
| Solid cylinder | I = 1/2MR2 |
| Solid cylinder central diameter | I = 1/4MR2+1/2ML2 |
| Hoop about the symmetry axis | I = MR |
| Rod about centre | I = 1/2ML2 |
| Rod about end | I = 1/3ML2 |
| Thin spherical shell | I = 2/3MR2 |
| Hoop about diameter | I = 1/2MR2 |
Conversion Between Units of Moment of Inertia
[Click Here for Sample Questions]
The table below shows the conversion between moment of inertia units:
| Unit | kg. m2 | g.cm2 | lbmft2 | lbmin2 |
|---|---|---|---|---|
| kg. m2 | 1 | 1×107 | 2.37×10 | 3.42×103 |
| g.cm2 | 1×10-7 | 1 | 2.37×10-6 | 3.42×10-4 |
| lbmft2 | 4.21×10-2 | 4.21×105 | 1 | 1.44×102 |
| lbmin2 | 2.93×10-4 | 2.93×103 | 6.94×10-3 | 1 |
Parallel Axis Theorem
[Click Here for Sample Questions]
Parallel Axis Theorem is used to determine the moment of inertia of a rigid body’s area that remains parallel to the axis of a known moment body. The minimum rotational inertia for an axis in that direction in space is the moment of inertia of an object about an axis through its centre of mass.
The moment of inertia about a parallel axis through the centre of mass is determined by:
| I = Icm + Md2 |
Here,
- I = moment of inertia of the body
- Ic = moment of inertia about the center
- M = mass of the body
- d distance between two axes
Also check:
Radius of Gyration
[Click Here for Sample Questions]
If the moment of inertia (I) of a mass (m) body around an axis is represented as: I = Mk2. The radius of gyration of the body about the given axis is denoted by 'k'. It defines the radial range from a particular axis of rotation at which the body's total mass can be considered to be concentrated and rotational inertia maintains equality.
A solid sphere's gyration radius around its axis is: Mk2 = 2MR2/5, or k = \(\sqrt{\frac{2}{5}}\)× R
Factors that Influence Moment of Inertia
[Click Here for Sample Questions]
The following are the factors that determine the moment of inertia:
- The material's density.
- The body's shape and size
- Rotational axis (distribution of mass relative to the axis)
Also check: Class 11 Physics Notes
Dimensional Formula of Moment of Inertia
[Click Here for Sample Questions]
The mass product and the square of the spinning radius are used to determine the moment of inertia.
- Surface Energy = [ML0T-2]
- Angular Velocity = [M0L0T-1]
- Moment of Inertia = [ML2T-1]
- Gravitational Force = [MLT-2]
Moment of Inertia = Mass x (Radius of Gyration)2
Now, the Dimensional Formula of Mass = (M1L0T0)
Dimensional Formula of Radius of Gyration = (M0L1T0)
(Radius of Gyration)2 = M0 L2 T0
We get a Dimensional formula for the moment of inertia= M1L2T0 by swapping these variables in the equation above.
Also, read: Law of Inertia
Things to Remember
- Moment of inertia (I) is primarily determined by the mass distribution throughout the body in relation to the axis of rotation.
- The mass of the body, as well as its distribution with respect to the rotating axis, determines the moment of inertia.
- Moment of inertia is usually specified with respect to a chosen axis of rotation. It mainly depends on the distribution of mass around an axis of rotation.
- Moment of inertia units are of 2 types – Area moment of inertia and mass moment of inertia.
- The SI unit of moment of inertia is: kg.m2
- If the moment of inertia (I) of a mass (m) body around an axis is represented as: I = Mk2
Sample Questions
Ques. State the theorem of parallel axis. (2 Marks)
Ans. It says that a lamina's moment of inertia about any axis in its surface is equal to the sum of the lamina's moment of inertia around its centroidal axis parallel to the specified axis and the product of the lamina's area and the square of the perpendicular distance between the two axis.
Ques. What is the method of calculating the moment of inertia? (2 Marks)
Ans. Generally, the moment of inertia of every spinning item can be determined by estimating the difference between each component and the axis of rotation (r in the equation), squaring that value (that is the r2 term), and multiplying it by the mass of that specific object.
Ques. What are the factors that determine the moment of inertia? (2 Marks)
Ans. The following are the factors that determine the moment of inertia:
-
The material's density.
-
The body's shape and size
-
Rotational axis (distribution of mass relative to the axis)
Ques. What is the SI unit of moment of inertia? (2 Marks)
Ans. The SI units of moment of inertia is kg.m2
Moment of inertia of a body is an extensive or additive property of the body. For a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis).
Ques. Is angular momentum affected by the moment of inertia? (3 Marks)
Ans. Yes, angular momentum (L) is proportional to angular velocity (w), with the parameter being the moment of inertia (I).
In Newtonian rotational physics angular acceleration is inversely proportional to the moment of inertia of a body. You can think of the moment of inertia as the ability to resist a twisting force or torque. The angular momentum of a solid object is just Iω where ω is the angular velocity in radians per second.
Ques. How many types of the moment of Inertia units are there? (3 Marks)
Ans. The following are the two types of the moment of inertia units:
- Area moment of inertia
- Mass moment of inertia
The moment of inertia of a Mass (I) is defined as the sum of the products of the mass (m) of each particle of the body and the square of its perpendicular distance (r) from the axis and is mathematically represented as
I = mr²
Moment of Inertia of an Area (I) represents the distribution of points in a cross-sectional area with respect to an axis. It is also known as the second moment of area. For an elemental area dA in the XY plane the area moment of inertia is mathematically defined as Ix and Iy.
Ques. Define Radius of Gyration. (3 Marks)
Ans. The radius of gyration is the distance between two points where the entire area of a body is supposed to be focused from a certain axis.
For a bar cross-section with area S, the radius of gyration is given by
\(R_g = \sqrt {\frac{I_S}{S}}\)
where IS is the area moment of inertia (§B.4.8) of the cross-section about a given axis of rotation lying in the plane of the cross-section (usually passing through its centroid):
\(I_S =\int_S R^2 dS\)
where R denotes the distance of the differential area element dS from the axis of gyration.
Ques. What is the Difference Between Inertia and Moment of Inertia? (4 Marks)
Ans. Actually, inertia simply refers to the body's position, whether it's moving or stationary. The "Second Moment of Mass," also known as the "Moment of Inertia," is the object's measure of resistance to rotation with respect to an axis. It varies from axis to axis within the same body.
| Parameters | Inertia | Moment of Inertia |
|---|---|---|
| Definition | Inertia can be described as property or tendency of an object that resists any change to its state of motion. | Moment of Inertia is the measurement of an object’s resistance to change to its rotation. Moment of inertia is expressed with respect to a chosen axis of rotation. |
| Type | Natural tendency | Unit of measurement |
| Formula | Cannot be calculated using a formula. | Moment of Inertia = \(I_{xx} = \Sigma(A)(y^2)\) where A is the area of the plane of the object and y is the distance between the centroid of the object and the x-axis. |
Ques. A thin wire of length l and uniform linear mass density ρ is bent into a circular loop with a centre at O as shown in the figure. What is the moment of inertia of the loop about the axis XX’? (5 Marks)
Ans. Here, ρ = mass per unit length,
m = mass of the loop,
∴ m = ρl
R = radius of the ring or loop.
ID = M.I. of the ring about its diameter.
ID = 12mR2 = 12ρlR2
Since XX’ is parallel to the diameter of the loop,
∴ According to the theorem of parallel axes, M.I. about XX’ is given by
Ques. Equal torques are applied on a cylinder and a sphere. Both of them have the same mass and radius. Cylinder rotates about its axis and sphere rotates about one of its diameters. Which will acquire a higher speed and why? (3 Marks)
Ans. Speed of rotation acquired will rely on angular acceleration produced and is given by α = τI
In this, τ = is the torque, I is the moment of inertia.
As τ is the same in both the cases, thus,
Therefore, the sphere acquires greater speed than the cylinder as αs > αc.
Also check:
Also check:
Comments