# Eli5 moment of inertia

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I don’t even know really what intertia means let alone moment of it??
Plz help

In: 6 Inertia is the resistance to a change in motion. If you imagine a car, and a train in front of you, and you want to push them to get them both rolling up to 5 miles an hour, you can imagine it’s easier to do this with the car, than the train. We would say that the train has “a higher inertia” than the car does. Higher inertia means harder to accelerate.

But that’s straight-line motion, what about spinning motion? Here’s where “Moment of Inertia” comes into play. Imagine a basketball and a bowling ball, the same size and the same weight. Both balls have the same “center” of weight (I’m saying weight not mass because this is ELI5), which is just the center of the ball. BUT a bowling is full up of heavy stuff but a basketball is more like heavy, thin outer shell filled with air.

What we’re getting at here is that making the balls spin faster *isn’t* only about the where the center of weight is (it’s the same for both balls), *nor* how much the balls weigh (it’s the same for both balls) BUT you need to include some information about how the weight of the ball is distributed from the center. Since the basketball has all of it’s weight concentrated far from the center, it’s harder to spin vs. the bowling ball that has it’s weight evenly distributed.

A visualization for “why” might be imagine a see-saw. Making a see-saw balance isn’t just based on the weight of the people, but how far they are from the pivot. A small person can balance a big person if the small person is far from the pivot and the big person is right up on it. Similarly, a concentration of mass *far* from the center of the ball has a bigger effect than mass that’s closer to the ball.

In science speak, “Moment” refers to twisting and spinning forces and “inertia” means resistance to motion. So “Moment of Inertia” means an objects resistance to spinning.

Here’s one final parallel that might help. If you’re building a skyscraper, you want your steel beams to resist bending, right? So how do you shape them? Do you just make big steel cylinders? That’s not great because you’re making bowling-ball type shapes, all the weight is concentrated close to the center, you want to make a basketball type shape, put the weight far from the center. So you can make tubes, that would be better, but making round shapes is expensive and they are still pretty big. OR you could make an “I” shaped beam, two big thick heavy metal plates separated as far apart as possible with a thin central flange. That’s the strongest anti-bending shape possible AND is really cheap to make. And that, kids, is why we use I-beams to build buildings and not just hunks of steel or tubes. You’ve probably heard Newton’s first law of motion, that absent any outside force, an object in motion tends to stay in motion and an object at rest tends to stay at rest. This is the principle of inertia. It means that a force is required to change an object’s velocity.

Inertia is affected by the object’s mass. The classic formula F = ma, force equals mass times acceleration, can be written as a = F/m, acceleration equals force divided by mass. So the more mass something has, the less it will accelerate when a force is applied.

A “moment” in physics means the product of a physical quantity and a distance, but usually when we use the word “moment” we are specifically talking about torque: a product of force and distance. Torque (or moment) is the rotational equivalent of force; you apply it to cause an object to rotate; to change rotational velocity.

The moment of inertia is the rotational equivalent to mass. It tells you how much the object will resist rotational acceleration (usually called angular acceleration) for a given torque. So we can take the a = F/m formula and come up with its rotational analog. α is angular acceleration, 𝜏 is torque, and I is moment of inertia. So the formula becomes α = 𝜏/I. The angular acceleration equals torque divided by moment of inertia. Inertia is a measure of how hard it is to get something to change its velocity (that is, to speed up, slow down, or change direction). It’s essentially equivalent to mass – massive objects take more force to speed up, slow down, or to get them to change direction when they’re moving.

Moment of inertia is the same idea, but for rotation. Objects with a high moment of inertia take a lot of torque (the rotational equivalent of force) to speed up their rotation, slow down their rotation, or change the direction they’re rotating. Moment of inertia depends not only on how massive an object is, it also depends on how far away each part of the object is from the axis of rotation. More spread out mass means a higher moment of inertia, and more concentrated mass means a lower moment of inertia.

Moment of inertia is related to angular momentum. The angular momentum of an object is equal to its moment of inertia, multiplied by its angular velocity (like how linear momentum is equal an objects mass multiplied by its velocity). Angular momentum is conserved, which explains some things about the behaviour of spinning objects. The classical example here is an ice skater spinning on the spot, and then drawing their arms in to spin faster. They don’t change their mass, but by moving their arms inwards, they move their mass closer to the axis of rotation, which decreases their moment of inertia. Since angular momentum is conserved, their angular velocity has to increase to compensate, and they spin faster. So what’s the difference between mass and inertia when an object is simply just moving linearly and not rotating? Inertia is what makes heavier objects more difficult to get moving or to stop them from moving. For example, if you push on a bowling ball, it takes more effort to get moving than a soccer ball. Inertia is also the reason a bowling ball is more difficult to stop.

When things are moving in a straight line, inertia is simple. Heavier objects have more inertia.

But what about when things are rotating? That’s when things start to get interesting. The “moment of inertia” is what we use to describe inertia for rotating objects.

If you have an office chair nearby, sit down, extend your arms, and have someone spin you around. Careful, not too fast. Once you’re spinning, quickly pull your arms in, wait a moment, then extend them again.

You’ll notice that you rapidly speed up when you draw your arms in, then slow down when you extend them again. What’s up with that!?

The reason is really interesting, but also very simple. When you spin around with your arms extended, the mass in your hands and arms travels a path that is a circle. The linear distance traveled around one revolution is called the circumference. When you draw your arms inwards, that distance becomes shorter, but the mass in your arms has momentum from traveling a greater distance. By reducing the distance your hands will travel, you speed up your rate of rotation.

So the moment of inertia considers the mass of the rotating object, its angular velocity (same as RPM), and the distance of its mass from the center of rotation. All three of these values are “conserved” under the law of conservation of energy. So if you change any one of those variables, the others will adjust to compensate.