Physics and ice skating share a lot of commonalities. If you watched the Winter Olympics recently, you probably saw Japanese skater Yuzuru Hanyu jump and spin with incredible speed.

    His performance is a perfect example of how the moment of inertia affects things. In this article, we're going to take a closer look at what the moment of inertia is and how to calculate it.

    To get accurate conversions between moments of inertia, try this free conversion calculator. With over 10 units, it takes the confusion out of converting to kg·m2, lb·ft2, and more.


    What is moment of inertia?

    The moment of inertia is simply how much an object doesn't want to rotate.

    The factors that influence this resistance are:

  • the mass of the object
  • its radius from the center
  • the type of object (different objects have different moments)

  • How do mass and radius effect the moment of inertia?


    If the mass of an object is greater, it will have more resistance to rotation. Conversely, a light object will spin with more freedom.


      Think about putting a pencil through the center hole of a ruler. If you push the edge of the ruler, it'll spin about the tip of the pencil.

      But, why doesn't the ruler spin forever with infinite speed? Of course, there is something holding the ruler back.

      Now, imagine if you had a really big, dense ruler, more like a meter stick. It'd be three times the length and five times as thick. You can expect it'd be rotating differently.

      So would the stick be rotating faster or slower?

      Because the meter stick has more mass, it would be rotating slower than the ruler.


    If the radius of an object is greater, it will have more resistance to rotation. Conversely, an object will spin faster when the radius of the object is shorter.


      Think about tetherball. If you've never played, it's a game where you hit a ball back and forth around a pole, which connects to the ball with a rope.

      When the game starts, the rope isn't wrapped around the pole. Hitting it means the ball will fly out very far from the pole, taking its sweet time to go all the way around.

      As you continue playing, the rope forms more loops around the pole. The ball comes closer and closer inward and takes less time to revolve.

      If you're losing--meaning you let your opponent wrap the ball around the pole--it'll be very hard to recover, since the ball is now spinning much faster.

      This distance between the ball and the pole is called the radius.

    What are the different moments of inertia?

    As mentioned earlier, different objects will have different moments. Here are some of those moments.

    Point Mass

    This is where we assume the object behaves much like a tetherball. It goes around a "pole", whether it's real or imaginary.

    Rods (In The Center)

    If we're rotating a rod around, not all parts of the rod are the same distance from the rotation point.

    The beginning of the rod is much closer, so it's easier to rotate that side, but the end of the rod has more resistance.

    Rods (From The Edge)

    Like putting a pencil at the edge of a ruler instead of at the center, a rod will rotate differently depending on where the point is.

    Cylinders, Spheres

    These guys are hefty and weird. Like rods, they have different mass distribution.

    It always depends on how the mass is laid out in comparison to the radius.

    What is the standard unit for moment of inertia?

    If you're literally anywhere except America, expect to use kg·m2 (kilogram-meter squared).

    Here in the USA, we use slug square foot (slug·ft2). One slug is about 32.2 pounds.

    How do you convert moment of inertia units?

    It's a matter of knowing how many units of one go into another.

    In the metric system, that's fairly easy. It's always in groups of ten, as:

    • There are 1000 millimeters in a meter.
    • There are 100 centimeters in a meter.
    • Meter is the basic unit.
    • Kilometer is 1000 meters.
    • Etc.

    The same pattern holds true for grams, kilograms, and so on. But converting to imperial units can be a bit more difficult.

    We recommend using a conversion calculator to ensure that you are using the correct ratios.

    Example: kg·m2 to g·mm2

    So let's say you wanted to go from 16 kg·m2 to g·mm2.

    We know there are 100 grams in a kilogram and 1000 millimeters in a meter. So:

    16 kg·m2

    = 16 * (100 g / kg) * (1000 mm / m)2

    = 1600 g * (1000 mm)2

    = 1,600,000,000 g·mm2

    Those fractions are saying there are 100 grams per kilogram and 1000 millimeters per meter.

    Naturally, the easiest way to do this is with a table or a reliable calculator.

    Check out our Moment of Inertia Converter to help you develop a better intuition of this concept.

    What is the formula for moment of inertia?

    The formulas are very logical. It comes naturally from what we just learned about rotation.

    Point Mass Formula

    Here it is:

    I = m · r2


    • I is the moment of inertia
    • m is the mass
    • r is the radius

    The moment of inertia depends on the radius by a degree of the mass. Naturally, this equation changes for different objects.

    Rod (In The Center) Formula

    The equation for a rod about its center is:

    I = 1 / 12 M · L2


    • M is mass
    • L is length (because if it's from the center, the length serves the same purpose as a radius).


    Let's say we had a rod that weighed 12kg and was 4 meters long. We'd find the rotational inertia like this:

    I = 1/12 * (12kg) * (4 meters)2

    I = (1 kg) * 16 m2

    I = 16 kg·m2

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