Twirling Tester

1. Attach the steel rod to the ring stand with a right-angle clamp.
2. Cut two pieces of PVC so that when they are attached to the cross connector they almost cover the steel rod. (For example, if the bar is 25 cm long, cut one piece about 3 cm and the other piece about 12 cm. The cross piece is about 7 cm.) Connect as shown, and slide the assembly onto the steel rod (see below). This will be the arm holder.

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3. Attach the other right-angle clamp to the end of the rod, but be sure that the arm holder can turn freely.
4. Cut two pieces of PVC about 20 inches (50 cm) long. The exact length is less important than having them be the same length. Attach slip connectors to the ends of the rods.
5. Insert the two long PVC pieces into the two free arms of the cross connector (see photo in step 2).
6. Cut four shorter pieces of PVC to be all the same length, about 10 inches (25 cm) long. Again, the exact length is less important than having them be all the same length. Attach an end cap to one end of each PVC piece, leaving the other end open.
7. Take two of the 10-inch (25-cm) pieces of PVC you just cut and place an end cap at one end of each, leaving the pipes open at the other end. Set them aside for use in your investigations.
8. Place an end cap at one end of the other two pieces of 10-inch (25 cm) PVC pipe, leaving each pipe open at the other end. Then, working with one pipe at a time, use the funnel to fill each with sand to about 1 inch (2.5 cm) from the top. Fill the rest of the space with hot glue so the sand is trapped inside. Wait for the glue to cool and set. Try to put the same amount of sand in both pieces (see photos below).

     

9. When the glue has dried at one end of each sand-filled pipe, you need to seal the other ends. Flip the pipes over and remove the end cap. Some sand will probably fall out. Shake out a bit more sand so there’s room for the same amount of glue as used at the other end of the pipe. Seal with hot glue. Wait for the glue to set, and then replace the cap.

   

10. Mark these two pieces with tape so you know they have sand in them. Then push one marked pipe into the slip connectors on each PVC arm of the apparatus to make a long cross, as shown below. Leave them open at the other end, and don’t put tape on them.
11. Fill the bottle with water and screw on the cap. Tie the string to the mouth of the bottle, and then tape it to the PVC support arm, about two inches from the ring stand. The string should be just long enough for the bottle to rest on the base of the ring stand when fully elongated.
12. Your finished apparatus should look like this. Make sure that everything can turn freely.

    

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.