What is moment of inertia? A most widely asked question. When a hollow cylinder, solid cylinder, and sphere of the same mass are freely allowed to roll on a slope which one will reach down first and why?
Let’s go through some basics first!
Before understanding moment of inertia, you need to understand the concepts of the moment, inertia, center of gravity, centroid, and bending moment.
We will learn the basics of moment of inertia. At first, let me ask you one simple question, is torque and moment are same?
They are the same thing in physics, but they have significantly distinct meanings in mechanics.
They both have the same unit – Newton Meter (N.m).
Let’s see the difference between torque and moment!
|Torque is a force that may cause an item to revolve around an axis.||A force’s moment is a measure of its ability to cause a body to spin around a given point or axis.|
|It is used where rotation is involved.||It is used where no rotation is involved.|
|Applications Turbines, dynamo, shafts.||Applications Bending, bridges, structural designs.|
It is defined as the quantity indicated by a body resisting angular acceleration, which is the sum of the product of each particle’s mass with its square of the distance from the axis of rotation [Moment of inertia I = Σ miri2]
The moment of inertia ‘I’ of an element having mass ‘m’ positioned at a distance ‘r’ from the center of rotation equals,
In simpler words, it is the amount of torque required for a certain angular acceleration in a rotating axis.
Don’t be confused between inertia and moment of inertia. These are not the same!
|Inertia||Moment of inertia|
|Inertia is defined as an object’s property or inclination to resist changes in its state of motion.||The moment of inertia is a measurement of an object’s resistance to change in rotation. It is stated in relation to a certain axis of rotation.|
|When force is applied to an object it resists, that’s inertia.||When torque is applied it resists to bend, that’s a moment of inertia|
|There are 3 types, inertia of direction, rest, and motion||There are 3 types of moments – MI of area, MI of mass, and Polar MI.|
Let’s consider, a very small ball of negligible mass attached to a rod at the end and it is rotated about a fixed axis.
Based on Newton’s Second law of lenear motion,
Now, we can write from the definition of torque,
From the angular quantities, we also can write,
We have got two equations,
By equating these above equations,
This is the simple equation or formula for the moment of inertia, I=mr2
Let’s assume an object in which two steel balls are welded to a rod and the rod is attached to a bar. If we neglect the mass of rod on which the balls are welded and assuming the mass of balls as ‘m’ concentrated at a distance ‘r’ from the center.
The moment of inertia of the setup will be given by:
I = 2mr2
Let’s see what is the unit of MI?
We have already learned the formula of equation of MI, I=mr2
The unit of moment of inertia in SI unit
The unit of moment of inertia in FPS unit
Hence, the unit of area MI are kg.m2 in SI unit and lbf.ft2 in FPS unit respectively.
The dimension of MI is M.L2
There are so many examples for moment of inertia, a few of them as illustrated as below,
MI = (ML2)/12
MI = (ML2)/3
The mass MI, commonly known as rotational inertia, is a number used to calculate a body’s resistance to a change in rotation direction or angular momentum.
It essentially describes the acceleration experienced by an entity or solid when torque is applied.
The symbol IG is commonly used to represent the mass moment of inertia.
(a) F=ma analysis moment equation (ΣMG= IGα).
(b) Rotational kinetic energy (T = ½IGω2)
(c) Angular momentum (HG= IGω)
Practical example – we use ring shape wheels instead of disc shape wheels.
This theorem only applies to planar bodies. Bodies that are flat and have very little or no thickness.
The theorem states that the moment of inertia of a planar body at an axis perpendicular to its plane is equal to the sum of its moments of inertia along two perpendicular axes coincident with the perpendicular axis and lying in the plane of the body.
That means the Moment of Inertia Iz = Ix+Iy
The parallel axis theorem applies to bodies of any form and shape.
According to the parallel axis theorem, the moment of inertia of a body about an axis parallel to an axis passing through the center of mass is equal to the sum of the moment of inertia of the body about an axis passing through the center of mass, product of mass, and square of the distance between the two axes.
Essentially, the theorem can be mathematically states as: IXX= IG+Ad2
The radius of gyration is occasionally used to express a body’s moment of inertia about an axis. What exactly do you mean by “radius of gyration”?
The radius of gyration can be defined as the imaginary distance from the centroid at which the area of cross-section is thought to be focused at point to achieve the same moment of inertia.
To put it in other words, the radius of gyration specifies the distribution of the total cross-sectional area around its centroid axis. The more the area distributed away from the axis, the greater the buckling resistance.
A circular pipe is the most effective column section for buckling resistance as its area is scattered as far away from the centroid as feasible.
The radius of gyration is used to compare the behavior of various structural geometries under compression along an axis. It is applicable in the prediction of buckling in a compression beam or member.
J = Ix + Iy
To sum it up, the polar moment of inertia is the resistance provided by a beam or shaft when bent by torsion. This resistance is often caused by the cross-sectional area, and it should be highlighted that it is unrelated to the material composition.
If the polar moment of inertia is bigger, the torsional resistance of the body will be greater as well. To turn the shaft at an angle, more torque will be needed.
Let’s see the difference between area MI and mass MI, as follows:
|Area MI||Mass MI|
|Area MI is the measure of the capacity of a section to resist bending about the reference axis.||Mass MI is the measure of a capacity of a solid body to resist rotation (i.e angular acceleration) about the reference axis.|
|More the moment of inertia|
· Lesser will be the bending stress · Greater will be the beam stiffness
· Lesser will be the beam deflection
|Lower is the mass MI and hence lesser resistance against its rotation.|
|It is a property of two-dimensional sections or areas of planes.||It is a property of three-dimensional solid bodies.|
|Applications Design of beams and structures||Applications Design of gears, gear trains, and gearboxes. (rotating members) Also used in gyroscopic couple and its effect on automobile, ship, airplane.|
The moment of inertia is a computation of the amount of force necessary to spin an object. The value can be changed to enhance or decrease inertia. Athletes in sports like ice skating, diving, and gymnastics are continuously modifying their body structure. The MI increases as the radius from the axis of rotation rises, slowing the rotation.
If an athlete wishes to improve the speed of rotation, they must lower the radius by moving the segments of the body closer to the axis of rotation, hence lowering the radius and MI.
Moment of inertia is used in mechanics, physics and also in real life situations to achieve the desired rotational speed.
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