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This Snack explores the forces that make some turning objects harder to start and stop than others.
By interchanging the marked and unmarked pipes, you can set up your finished apparatus so its paired arms can be used in three different ways: long and empty (LE), short and empty (SE), and short and full (SF). The investigations below use different arrangements of these weighted arms to explore aspects of rotational inertia.
Investigation 1: SE—LE—axle—LE—SE
Arrange the apparatus so that the long and empty arms are on the inside, near the axle, and the short and empty arms are on the outer ends, connected with the slip connectors. Twirl the arms to wind the string onto the arm holder so the bottle is lifted off the table and brought to the top, or some other repeatable position. Let the bottle fall, and time how long it takes for the weight to reach the table. Record the time, run the experiment again, and record the time again. Hopefully, the two times will be close. If not, check to see what was different between your trials, and keep adjusting until you get a consistent result.
Investigation 2: SF—LE—axle—LE—SF
Reconfigure the arms so that the long and empty sections are in the center and the short and full sections are on the outside. What do you think will happen to the time necessary for the bottle to reach the bottom if we increase the weight of the arms? Do you think the time will be about the same, shorter, or longer? Wind up the bottle to the same spot as last time and let it fall, recording the time with each trial, and doing enough trials to get a consistent result. What was the result? Replace the short and empty arms at the end with the short and full arms. As before, wind up the bottle and let it fall, recording the time with each trial, and doing enough trials to get a consistent result. Compare your findings.
Investigation 3: LE—SF—axle—SF—LE
Reconfigure the arms so that the short and full sections are on either side of the axle and the long and empty sections are on the outside. What do you think will happen to the time with the apparatus in this configuration? Again, wind up the bottle to the same spot as last time and let it fall, recording the time with each trial, and doing enough trials to get a consistent result.
There are still more possibilities. Give them a try. Keep track of your results. What factors are important?
When you wind up the bottle and let it fall, it applies a kind of turning force called a torque on the cross arm, and that force tries to make the arm turn. The arm, however, has a resistance to rotation, called rotational inertia (sometimes called moment of inertia) in the same way that an object in linear motion resists speeding up or slowing down.
Going from Investigation 1 to Investigation 2 was probably not that surprising. In Investigation 2, the arms had more mass, and thus more rotational inertia, so were harder to rotate.
Going from Investigation 2 to Investigation 3 can sometimes be surprising. The mass stayed the same, so why was the rate of rotation so much slower in Investigation 3 than in Investigation 2? The rotation was slower because the rotational inertia went up, even though the amount of mass didn’t change.
While the linear inertia of an object only depends on the object’s mass, its rotational inertia depends not only on the mass, but also on the distribution of the mass. In addition to the rotational inertia being proportional to the mass, the rotational inertia is proportional to the square of the distance between the mass and the axis of rotation.
The distribution of the mass is, in a sense, more important than the mass. If the mass is tripled, the rotational inertia is tripled. But if the distance between the mass and the axis of rotation is tripled without increasing the mass, the rotational inertia is nine times larger. As a result, when the mass was closer to the center in Investigation 3, the rate of rotation increased by a lot.
Rotational inertia is important in many situations. For example, when some people lose their balance, they extend their arms outward. While part of this helps recover balance, another part increases rotational inertia, moving the mass of a person’s body away from the axis of rotation. Increasing their rotational inertia makes it harder for them to rotate, so they have more time to get their feet back under them.
Rotational inertia is also important in sporting events. Gymnasts and dancers often pull themselves inward when trying to turn. By having a smaller rotational inertia, they can increase their rotation more easily. Similarly, sprinters will pull their trailing leg close to their body, turn their leg, and then fling it forward. By turning their leg while it is close to their body, it is easier to turn.
For baseball players, aluminum bats can be made hollow, and thereby lighter than wooden bats. They will have less rotational inertia than wooden bats, and for the same force. A batter can also get an aluminum bat into position faster than a wooden bat. For this reason, they are banned in Major League Baseball, but players cheat from time to time by hollowing out their wooden bats.
On the racing track, many driving enthusiasts replace their stock steel wheels with alloy wheels. The alloy wheels have less mass, reducing their rotational inertia, and making them easier for the engine to turn. Some cars have smaller than typical wheels to reduce the rotational inertia, which also makes them easier to turn.
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Attribution: Exploratorium Teacher Institute