In this article, you will learn a complete overview of the moment of inertia such as its definition, formula of different sections, units, depending factors, calculation, and many more.
In physics and mechanics, the moment of inertia plays an important role in the analysis of the rotational motion of objects.
Moment of inertia concerned with the dynamics.
It is used to calculate the torque required to rotate an object about a given axis, as well as the angular acceleration of the object under the influence of a given torque. It is also used to determine the stability of an object when it is rotating, as well as to predict the motion of objects under the influence of external forces or torques.
We have also discussed the polar moment of inertia, area moment of inertia, first moment of area in our previous article here we will learn only about the moment of inertia.
What is a Moment of Inertia?
Moment of inertia, also known as rotational inertia, mass moment of inertia, angular mass, and second moment of mass is a measure of an object's resistance to change in its rotational motion.
It is defined as the sum of the products of the mass of each particle in an object and the square of its distance from the object's axis of rotation.
The greater the moment of inertia, the more difficult it is to change the object's rotational speed or direction.
The moment of inertia of an object depends on the object's shape, size, and mass distribution.
For example, a solid cylinder has a different moment of inertia than a hollow cylinder of the same size and mass. In general, objects with more mass concentrated towards their center of mass have a smaller moment of inertia than objects with mass distributed more evenly.
It is typically denoted by the symbol I.
Formula of Moment of Inertia
In the expressions for torque and angular momentum for rigid bodies (which are considered bulk objects), we have come across the term Σmr².
This quantity is called the moment of inertia (I) of the bulk object. For point mass m, at a distance r, from the fixed axis, the moment of inertia is given as, mr².
Moment of inertia for a point mass,
I = mr²
Where,
m = Mass of the body
r = Distance from the axis of the rotation.
So,
Moment of inertia for bulk objects,
I = Σmr²
In translational motion, mass is a measure of inertia; in the same way, for rotational motion, a moment of inertia is a measure of rotational inertia.
In general, mass is an invariable quantity of matter (except for motion comparable to that of light).
But, the moment of inertia of a body is not an invariable quantity.
It depends not only on the mass of the body but also on the way the mass is distributed around the axis of rotation.
To find the moment of inertia of a uniformly distributed mass; we have to consider an infinitesimally small mass (dm) as a point mass and take its position (r) with respect to an axis.
The moment of inertia of this point mass can now be written as,
dI = (dm)r²
We get the moment of inertia of the entire bulk object by integrating the above expression.
I = ∫ dI = ∫(dm)r²
I = ∫r²dm
We can use the above expression for determining the moment of inertia of some of the common bulk objects of interest like rods, rings, discs, spheres, etc.
Unit of Moment of Inertia
In the SI system,
As we know, the moment of inertia
I = Σmr²
So,
I = kg-m²
I = kg-m² or kg-mm²
Similarly in the CGS system, it will be,
I = kg-cm²
So, the unit of moment of inertia is, kg-m² or kg-mm² or kg-cm² and Its dimension is ML².
Factors on which Moment of Inertia Depends
The moment of inertia of a body depends on several factors, including:
- Shape of the Body
- Size of the Body
- Distribution of Mass within the Body
- Orientation of the Body
- Axis of rotation
- Material Properties of the Body
Shape of the Body
The moment of inertia is generally smaller for a body with a more compact and symmetrical shape and larger for a body with a more elongated or irregular shape.
Size of the Body
The moment of inertia is generally larger for a larger body than for a smaller body with the same shape and mass distribution.
Distribution of Mass within the Body
The moment of inertia is generally larger for a body with a more centralized mass distribution and smaller for a body with a more evenly distributed mass.
Orientation of the Body
The moment of inertia of a body may change depending on the orientation of the body relative to the axis of rotation.
Axis of rotation
The moment of inertia of a body depends on the location of the axis of rotation relative to the body.
For example, the moment of inertia of a body rotating about an axis passing through its center of mass is generally smaller than the moment of inertia about an axis that does not pass through the center of mass.
Material Properties of the Body
The moment of inertia of a body may be affected by the material properties of the body, such as the density and stiffness of the material.
Moment of Inertia for Different Shapes
Moment of Inertia of a Uniform Rod
Let us consider a uniform rod of mass M and length l as shown in the figure.
Let us find an expression for the moment of inertia of this rod about an axis that passes through the center of mass and is perpendicular to the rod.
First, an origin is to be fixed for the coordinate system so that it coincides with the center of mass, which is also the geometric center of the rod.
The rod is now along the x-axis. We take an infinitesimally small mass dm at a distance x from the origin.
The moment of inertia dl of this mass dm about the axis is,
dI = dm.x²
As the mass is uniformly distributed, the mass per unit length of the rod is,
λ = M/L
The dm mass of the infinitesimally M small length as,
dm = λ.dx = (M/L).dx
The moment of inertia I of the entire rod can be found by integrating dI,
I = ∫ dI = ∫dm.x² = ∫ {(M/L).dx}.x²
As the mass is distributed on either side of the origin, the limits for integration are taken from -L/2 to L/2.
So,
I = (M/L) ∫ x².dx (from -L/2 to L/2)
After calculating this,
I = (ML²)/12
Moment of Inertia of a Uniform Ring
Let us consider a uniform ring of mass M and radius R. To find the moment of inertia of the ring about an axis passing through its center and perpendicular to the plane, let us take an infinitesimally small mass (dm) of length (dx) of the ring. This (dm) is located at a distance R, which is the radius of the ring from the axis as shown in the figure.
The moment of inertia dl of this small mass dm is,
dl = dm.R²
The length of the ring is its circumference (2πR).
As the mass is uniformly distributed, the mass per unit length λ is,
λ = mass/length = M/2πR
The mass (dm) of the infinitesimally small length is,
dm = λ.dx = (M/2πR).dx
Now, the moment of inertia of the entire ring is,
I = ∫ dI = ∫dm.R² = ∫ {(M/2πr).dx}.R²
I = (MR/2π) ∫dx
To cover the entire length of the ring, the limits of integration are taken from 0 to 2πR.
I = (MR/2π) ∫dx (From 0 to 2πR)
After integrating,
I = M.R²
Moment of Inertia of a Uniform Disc
Consider a disc of mass M and radius R. This disc is made up of many infinitesimally small rings as shown in the figure.
Consider one such ring of mass dm and thickness dr and radius r.
The moment of inertia dl of this small ring is,
dI = (dm).r²
As the mass is uniformly distributed, the mass per unit area,
σ = mass/area = (M/π.R²)
Where,
σ = Surface mass density
The mass of the infinitesimally small ring is,
dm = σ.2πrdr = (M/π.R²).2πrdr
Where the term 2πrdr is the area of this elemental ring 2πr is the length and dr is the thickness.
dm = (2M/R²).rdr
So,
dI = (dm).r²
dI = (2M/R²).r³dr
The moment of inertia of the entire disc is,
I = ∫dI
To cover the entire length of the disc, the limits of integration are taken from 0 to R.
I = ∫(2M/R²).r³dr (From 0 to R)
After resolving,
I = 1/2(MR²)
Similarly, we can find,
Moment of Inertia of a Uniform Solid Sphere
I = (2MR²)/5
Moment of Inertia of a Uniform Hollow Sphere
I = (2MR²)/3
Moment of Inertia of a Solid Cylinder
I = (1/2).m.r²
Moment of Inertia of a Hollow Cylinder
I = (1/2).m(R² + r²)
Moment of Inertia for Solid Rectangular Plate
I = (1/12).M(h² + w²)
Moment of Inertia of about x' and y' Axis
Sometimes it is necessary to calculate the moment of inertia of a body with respect to the x' and y' axes.
We can calculate the moment of inertia for the x' and y' axes using two theorems.
- Parallel Axis Theorem
- Perpendicular Axis Theorem
Parallel Axis Theorem
The parallel axis theorem states that The moment of inertia of a plane section about any axis parallel to the centroidal axis is equal to the moment of inertia of the section about the centroidal axis plus the product of the area of the section and the square of the distance between the two axes.
Mathematically, It can be written as,
Iₓ, = Iₓ + Ad²
Where,
A = area of the body
Iₓ = Moment of inertia about point x
d = Perpendicular distance between the two lines x and x'.
Iₓ, = Moment of inertia about point x'
A similar method can be applied to calculate a moment of inertia about y' axis.
Perpendicular Axis Theorem
The theorem states that the moment of inertia of a plane laminar body about an axis perpendicular to its plane is equal to the sum of moments of inertia about two perpendicular axes lying in the plane of the body such that all three axes are mutually perpendicular and have a common point.
This theorem is also called the polar axis theorem.
Mathematically,
Iz = Iₓ + Iᵧ
Where,
Iz = Moment of Inertia about the Z axis
Iₓ = Moment of Inertia about the X axis
Iᵧ = Moment of Inertia about the Y axis
Calculation of Moment of Inertia
Question
A solid cylinder has a radius of 5 cm and a height of 10 cm. It has a mass of 2 kg. Calculate the moment of inertia of the cylinder about its axis of symmetry.
Solution
Given,
r = 5 cm
h = 10 cm
m = 2kg
As we know,
The moment of inertia of a solid cylinder about its axis of symmetry is given by the formula:
I = (1/2).m.r²
Where I is the moment of inertia, m is the mass of the cylinder, and r is the radius of the cylinder.
After putting the above values, we get:
I = (1/2) × 2 × 5²
= 50 kg-cm²
The moment of inertia of the cylinder is 50 kg-cm².
So here you have to know all aspects related to the moment of inertia.
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