Study Guide

# Rotation In the Real World

• ### Engineering

There are three things that really set mankind apart from the remainder of the animal kingdom: 1) opposable thumbs, 2) the collected works of William Shatner, and 3) tools.* The ability to use tools has allowed man to build, create, and otherwise accomplish feats no human being could do alone, largely through a phenomenon called mechanical advantage.

Mechanical advantage is kind of a blanket term for any sort of force amplification created by a simple machine—think crowbars, levers, gears—that outputs much more force than a person could create on his or her own. We've seen this already with levers, where a light object can balance a heavier one by increasing its distance from the fulcrum (Archimedes' Law of the Lever). Gears can accomplish the same thing based on their size or number of teeth.

When gears mesh together, their teeth interlock and their motion is inextricably connected:

This interlocking motion means the tangential velocity of any point on either of two interlocking gears must be the same. (If this wasn't true, the tip of the gears' teeth would move at different speeds and the gears wouldn't mesh at all.) Identical tangential velocities but different radii means that two interlocking gears must have two different angular velocities, since . The larger gear must rotate slower in order to mesh with the smaller gear.

Imagine you start turning the smaller gear (maybe you're pedaling a bike, or cranking a winch, or maybe you just like gears). You're applying some torque τsm to the smaller gear, which has radius rsm and turns at speed ωsm. Conservation of energy tells us that the amount of energy per second you put into the gear—that is, the power you supply—must be the same as the amount of energy per second output by the larger gear, with radius rlg and angular velocity ωlg. The power you input to the small gear is P = τsmωsm, while the power output by the larger gear is exactly the same: P = τlgωlg. Then we have τsmωsm = τlgωlg, or:

We know ωlg < ωsm, so this tells us τlg will always be bigger than τsm—in other words, the larger gear will always output more torque than input to the smaller gear, but will rotate more slowly.

So suddenly, just by turning a gear, you can make more torque than you ever could on your own. Shatner would be proud.

*Language, too, maybe. But mostly thumbs, Shatner, and tools.

• ### Machines

Rotational motion is ubiquitous in modern machine design—a lot of which boils down to the fact that electric motors rotate, and don't translate. But what if you're designing a machine that needs to move some other way? Say, follow a non-circular path, even a linear path?

Mechanical engineers have gotten around this limitation through the use of four-bar linkages. A four-bar linkage is a collection of four* bars joined together at the ends. One bar, or link, is connected to a motor or a crank or something that turns it around in a circle (the rocker), one bar is fixed, and the other two move, following a path determined by their lengths.

With this relatively simple set up, any number of complex paths can be created, allowing for all sorts of movement. Some neat examples6:

Crank-Rocker: takes small circular motion and amplifies it, though only in a partial arc (not a full circle).

Parallel Four-Bar: creates identical motion at two (or more) points from a single input.

Chebyshev Linkage: allows for approximations of straight line movement with rotating inputs.

(See it in action here.)

*Sometimes three, with the ground or whatever the mechanism is mounted to serving as the fourth "bar."

### Flywheels and Energy Storage

Nine out of ten dentists agree that the wheel was a pretty great invention. But modern scientists and engineers aren't just content to let a millennia-old technology like that sit around gathering dust. No, the modern wheel has many more uses than chariot races and wheat grinders (NASCAR races and blenders, to say the least).

One application of the simple fact that once you get a large wheel turning, it's hard to stop: the flywheel. The idea's been around for a while—just look at an old potter's wheel:

The potter kicks a pedal attached to the wheel, and it starts to spin. Once up to speed, it will keep the lump of clay on the table spinning with minimal additional effort—just the occasional kick to fight frictional losses.

An industrial flywheel, used for energy storage the same way your car uses a battery, is basically the same thing: a large rotating wheel or cylinder that is spun up to high speed like an electric motor.7 Once up to speed, the flywheel's enormous moment of inertia ensures that a) it's very hard to stop, and b) it can store a tremendous amount of energy, as given by .

An additional benefit of the flywheels is their ability to be used as a backup generator, much like an emergency diesel engine.8 If something disastrous were to happen to the electricity grid, physics means the flywheel keeps on spinning. You can continue to pull energy out of it until it slows to a halt.

Because flywheels can provide the same security as a backup generator—with the additional benefit of having zero startup time (the wheel's already spinning when the power goes out, so there's no need to fire up the generator)—they're an intriguing option for buildings that have to hedge against any sort of power fluctuation, such as data centers, which store all the information for large companies' websites. Any time a data center goes down, taking with it your Facebook photos or tweets or, you know, maybe \$2 million in profit, people get a little angry. Flywheels provide a totally non-polluting option for ensuring the world can keep watching cat videos, no matter what's going on outside their houses.