{"id":660,"date":"2026-07-27T22:55:39","date_gmt":"2026-07-27T22:55:39","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=660"},"modified":"2026-07-27T22:55:40","modified_gmt":"2026-07-27T22:55:40","slug":"moment-of-inertia","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/mechanics\/moment-of-inertia\/","title":{"rendered":"Moment of Inertia: Formula for Common Shapes"},"content":{"rendered":"\n
Definition<\/div>

\nMoment of inertia is the rotational equivalent of mass: it measures how strongly an object resists a change to its spin about a chosen axis. It equals the sum of each mass element times the square of its distance from that axis, giving I = k m r squared for common shapes, in kilogram metres squared.\n<\/p><\/div>\n\n

Hold a hammer by its handle and swing it. Now flip it round, grip the head, and swing again. Same hammer, same mass, same muscles \u2014 yet one swing feels sluggish and the other whips through the air.<\/p>\n\n

Nothing about the hammer changed except where the heavy end sat relative to your wrist. That single observation is the whole of this topic: when things rotate, where<\/em> the mass sits matters more than how much<\/em> there is.<\/p>\n\n

What Is Moment of Inertia?<\/h2>\n\n

Moment of inertia is a measure of how much an object resists being spun up or slowed down about a particular axis. Push a shopping trolley and its mass fights you; twist a heavy door and something else fights you \u2014 that something is its moment of inertia.<\/p>\n\n

Mass tells you how stubborn an object is when you try to move it in a straight line. Moment of inertia tells you how stubborn it is when you try to turn it. The two are cousins, not twins.<\/p>\n\n

The critical difference is distance. Every scrap of material in a rotating body sweeps out a circle, and the wider that circle, the faster that scrap has to travel for a given rate of spin. Material near the rim therefore costs far more effort to accelerate than material near the hub.<\/p>\n\n

That cost does not grow in proportion to distance. It grows with the square<\/strong> of distance \u2014 move a bolt twice as far from the axis and it contributes four times as much.<\/p>\n\n

Why the axis is part of the answer<\/h3>\n\n

Here is the point that trips up more students than any other: an object does not have<\/em> a moment of inertia. It has one for each axis you might spin it about.<\/p>\n\n

A metre rule spun about its middle and the same rule spun about its end are two completely different problems, with answers differing by a factor of four. Quote a moment of inertia without naming the axis and you have said almost nothing.<\/p>\n\n

The Moment of Inertia Formula<\/h2>\n\n

For a single point mass, the definition is as simple as physics gets.<\/p>\n\n

I = m r\u00b2<\/div>\n\n