For many, moment of inertia seems like one of those topics that one must go over because they have a physics exam. But it’s actually an interesting concept.
For example, our early ancestors found the spear particularly useful when hunting for food in the Savannah. Although they never studied physics, they realized it was so much easier and far more effective to throw a spear spinning along its length rather than throwing it like a boomerang.
Little did they know that moment of inertia had something to do with that.
What is Moment of Inertia?
In simple terms, moment of inertia is a measure of difficulty in rotating or spinning an object.
You see, different objects have different difficulties in rotation. For example, it is much easier to rotate a spinning top than a screwdriver. It is easier to rotate an empty bottle than a bottle full of water and so forth.
Unlike linear motion, which is motion along a straight line, understanding rotational motion is a bit difficult, and sometimes, confusing – like going around in circles.
But don’t worry, this article will keep things as simple as possible. We won’t even use mathematical equations. This article is intended for readers trying to get their heads around the basic concept rather than those looking to answer a physics paper tomorrow morning.
Who discovered moment of inertia?
A Dutch scientist named Christiaan Huygens danced around the concept while experimenting with oscillating pendulums in 1673. However, the term itself was first used by a Swiss mathematician, Leonhard Euler about a century later in a published work.
Introducing moment of inertia
Moment of Inertia belongs to a class of mechanics called Rotational Dynamics, the branch of physics relating to rotational motion and its causes.
The study of rotational motion is quite useful since there are rotating bodies all around us.
The earth you are standing on, for example, is one of them, and so is the moon, the stars, and the sun. Almost every form of transport on Earth is facilitated by a spinning wheel of some sort. There are even people who feel like they are spending their life spinning around in circles. Needless to say, there is an entire branch of physics dedicated to studying that kind of motion.
Why moment of ‘inertia’?
You know what happens to passengers when a bus driver suddenly slams the brakes? Yeah, that’s inertia, the property of a body to resist change in its state of rest or motion.
We usually think of inertia as relating to bodies moving in a straight line (linear motion). But as it turns out, rotating bodies also tend to resist motion.
Back when DVD players were a thing, you could notice the CD inside the player rotate for a few seconds after the device is turned off. This was a clear example of rotational inertia, the CD inside the player ‘persists’ to rotate even after the power is cut off. Of course, the disk comes to a screeching halt shortly afterward thanks to the external influence of friction.
Another example is the spinning wheel used in lottery shows. In this setting, precise measures have been taken to reduce friction in so that the wheel keeps rotating for a considerable time even after the player stops spinning it. They are essentially making use of the principle of inertia to elicit suspense from players.
Finally, consider a spinning top. A simple child’s toy that seems to defy common sense when you set it in motion. A spinning top keeps rotating long after you give it a twist. Because of the smaller contact between the top and the ground, there is very little friction and it continues spinning because of inertia.
Objects that are rotating tend to keep on rotating and those that aren’t rotating, tend to remain at rest. The property of a body to resist changes to its rotational state of motion is called rotational inertia, also known as moment of inertia.
Calculating moment of inertia
Assume you have three cups: the first cup is full of water, the second cup is full of porridge, and the third cup is full of wet concrete. Which of the contents in the cups will be easier to stir? Why do you think so? Water is lighter than all three, so it will be the easiest to stir, and wet concrete being the heaviest, will be the hardest to stir. So, setting a body in rotational motion (in this case, stirring) depends on the heaviness of the body. Let’s call that heaviness, mass.
Distribution of mass
As a second example, assume you have three containers of varying diameters. The first one is a tall bucket, the second is a washing basin, and the third is a round bathtub. Which of the containers above do you think will be easier to stir if it had water in it? Why do you think so?
Clearly, the taller the container, the easier it is to stir the contents inside it. For tall, narrow containers like coffee cups, the contents inside them are forced to remain closer to the axis of rotation and therefore become easier to stir. However, for wider containers, the contents are free to wander off far from the axis of rotation and this makes stirring a bit difficult.
This also adds to the reason of why cups are often wider at the top than at the bottom. This is because when you dip down your spoon to stir, the bottom part being smaller in diameter is easier to stir.
So, setting a body in rotational motion (in this case, stirring) depends on the relative distances of the contents of the body from the axis of rotation. This property is called mass distribution.
The product of these two parameters, (i.e. mass and mass distribution) gives the measure of difficulty in rotating the body, which is moment of inertia. As promised, I won’t put the mathematical equation here, but you are more than welcome to check it out yourself.
Dynamics of rotational motion
The easiest way to study rotational motion is to contrast it with linear motion, which is motion in a straight line.
For instance, we say that an object is in linear motion if its position changes over time. In the same way, we say that an object is in rotational motion if it sweeps out an angle about a given axis over time.
Also, physical quantities in linear motion are closely related to physical quantities in rotational motion. The table below summaries the relationship:
Linear motion and Rotational motion parallels
| Linear motion | Rotational motion |
| Displacement | Angular displacement |
| Velocity | Angular velocity |
| Acceleration | Angular acceleration |
| Force | Torque |
| Linear momentum | Angular momentum |
| Kinetic energy | Angular kinetic energy |
| Mass | Moment of inertia |
Like translational motion, in which bodies can travel in different directions, objects can also rotate in different directions:
A pencil, for example, can rotate about an axis along its length or about an axis perpendicular to its length, or about an axis at the end of its length. Or it can rotate haphazardly in all sorts of directions – like what happens when you fidget nervously in an exam.
However, despite the similarities, the two kinds of motion are different and quantities cannot cross over from one motion to another. For example, linear momentum cannot be converted to angular momentum and vice versa. And there is also the concept of torque in place of force and moment of inertia in place of mass when talking about rotational motion.
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