The rotational world is a different world

The world of physics can be roughly divided into the world of lines and the world of curves, or more succinctly, the linear world and the rotational world.

By now you've probably studied linear velocity, acceleration, momentum and the relationships between those and energy (kinetic energy, potential energy, work).

While there are many parallels or analogies between these two worlds, there are also some significant differences. Whole books, for example, have been written about the concept of angular momentum and the interactions between two rotating things – and how to tease them apart to figure out what's going on – and to make predictions.

For example, you've probably heard the myth that sinks (or toilets) drain counterclockwise in the northern hemisphere and clockwise in the southern. While this is false (the forces involved are far too small to move water on that scale), it is true in another sense: large storm systems in the northern hemisphere almost always circulcate in a counterclockwise direction and those in the southern hemisphere spin in the opposite direction.

This is due to a coupling – or interaction between – the linear motion of the storm and the spinning motion of the nearly-spherical Earth. The force involved is called the coriolis force.

Source: Wikipedia Commons

Mass vs. moment of inertia

One of the most important concepts that is quite different in the rotational world is that of mass. In most linear problems, we can make a good first pass by approximating that all of the mass of an object can be "collapsed" to a single point, the center of mass, in order to make calculations easier. For example, if we're calculating the velocity of a cubic block sliding without friction down a ramp, we lose nothing at all by assuming that the mass is all collapsed to a point.

"Moment of inertia"

On the other hand, where the mass is and how it is distributed throughout a rotating object matters a lot. The figure below shows three round objects rotating around the same axis, which passes through the center of mass of each. If each has the same mass, we can easily see that the mass of each is distributed quite differently. It's uniform on the left, and nonuniform, to different degrees, in the middle and on the right.

In rotation, it's not just the mass, but how the mass is distributed.

Because of this, these objects have significantly-different rotational properties. For example, if we were to let each go at the top of a ramp at the same time, the solid wheel would reach the bottom first, followed by the middle wheel, then the one on the right — even though they have the same mass.

Mathematically, we quantify this property of mass distribution in an object by calculating the moment of inertia. So that's one big difference between the linear and rotational worlds: mass → moment of inertia.

In the rotational world, mass becomes moment of inertia

The wheel on the left has a smaller moment of inertia than the ones on the right, and the rightmost wheel has the largest moment of inertia. This gives it greater angular momentum, as we'll learn in another section. You can think of it as rotational inertia. Given a certain amount of twisting force, the wheel on the right will have more of a tendency to keep going, despite a little friction, than the one of the left.


The table below lists all of the analogies between properties and measurements in linear and rotational motion. Each concept has an analog in the other "world." It's a useful thing to refer to when working through rotational problems. Click on the table to download a .pdf copy of it.

Notice that for the concept of radial or centripetal acceleration, there is no linear analog. This is the "center-seeking" force that keeps things moving in a curve. Think of a rock tied to a string and twirled overhead. The only thing keeping the rock moving in a curve is the inward pull of the string.


through rotational topics

Use the buttons on the right to navigate to topics (pages) in rotational motion. You'll want to learn about center of mass before moment of inertia. Rotation concepts includes angular velocity, angular acceleration and centripetal acceleration. Angular momentum is one of the most important rotational concepts, and torque – the twisting force – is what produces angular momentum.

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