## When you push on a rotating object, when are you doing work?

### When you push on a rotating object, when are you doing work?

That’s an interesting question and requires two answers. First, if you push on a part of the rotating object and that part moves a distance in the direction of the force you exert, then you do work on it. In principle, it is possible to identify all the work that you do on the rotating object via this approach.

However, it is also possible to determine the work you do entirely in terms of physical quantities of rotation. If you exert a torque on the rotating object and it rotates the an angle in the direction of your torque, you again do work on the object. That’s the rotational version of the work formula: whereas force time distance is the translational work formula, torque times angle is the rotational work formula.

An important complication arises, however, in that you must measure the angle in the appropriate units: radians. The radian is the natural unit of angle and is effectively dimensionless (no units after it). When you multiple the torque times the angle in radians, the resulting units are those of work and energy. If you use a non-natural unit of angle, such as the degree, then you’ll have to deal with presence of the angle unit in your result.

## What is torque?

#### What is torque? — JPT, Calgary, Alberta

A torque is a physicist’s word for a twist or a spin. When you twist the top off a jar, you are exerting a torque on the jar and causing it to undergo an angular acceleration—it begins to rotate faster and faster in the direction of your torque. Similarly, when you spin a toy top, you do this by exerting a torque on the top and it again undergoes an angular acceleration.

## We know that spinning objects on earth can lose their spin (angular momentum) du…

#### We know that spinning objects on earth can lose their spin (angular momentum) due to friction (fluid or sliding) with the air or ground. However, if an object is set spinning in space, will it lose its initial angular momentum eventually or will it spin forever assuming no outside forces (e.g., gravity) act upon it? If it does come to rest, how does the earth maintain its spinning motion? — RD, Kingwood, TX

If a spinning object is truly free of outside torques—the influences that affect rotation—then it will spin forever. Angular momentum is a conserved quantity in our universe, meaning that it can’t be created or destroyed and can only be transferred between objects. Thus if you set an object spinning (by exerting a torque on it) and then leave it entirely alone, it will not be able to change its angular momentum. The earth is a good example of this situation—it’s almost free of torques and so it spins steadily about a fixed axis in space. Its angular momentum is essentially unchanging.

Since gravity acts at the center of rotation of a freely falling object (which is that object’s center of mass), gravity exerts no torque on freely falling objects. Because of that fact, even objects in orbit around the earth are essentially free of torques and satellites that are set spinning when they’re launched continue to spin steadily for centuries. The space shuttle astronauts encounter this result each time they release or catch a satellite. If they set it spinning when they let go of it, it will still be spinning when they retrieve it years later.

## When a lacrosse stick acts as a lever, does it convert a big force to a small on…

#### When a lacrosse stick acts as a lever, does it convert a big force to a small one or vice versa?

The lacrosse stick converts a big force into a small one. As you flip the stick, you do work on it—you push part of it forward while that part moves forward. You use a large force and the place on which you push moves forward a small distance. The stick, in turn, does work on the ball. It exerts a small force on the ball but moves that ball through a large distance. The products of force times distance are essentially equal (the stick itself takes some of the energy). The result is a very fast moving lacrosse ball that sails across the field.

## When you exert a torque on a merry-go-round, how does it exert one on you? I hav…

#### When you exert a torque on a merry-go-round, how does it exert one on you? I have to exert a lot of torque to get it going but it doesn’t feel like torque is being exerted back on me.

When you spin a merry-go-round, you exert a torque on it and it exerts a torque back on you. If you were free to rotate, this torque on you would be quite apparent. Suppose that the merry-go-round was located on an ice skating rink and that you were attached to the central pivot of the merry-go-round by a strap that went around your waist. As you spun the merry-go-round clockwise, you would begin to spin counter-clockwise. In fact, because your moment of inertia is much smaller than that of the merry-go-round, you would experience a much larger angular acceleration and would end up spinning much faster than merry-go-round. The reason that you don’t rotate like this after spinning a playground merry-go-round is that your feet touch the ground. As the merry-go-round exerts its torque back on you, you exert that same torque on the ground. The result is that the earth undergoes angular acceleration in the opposite direction from that of the merry-go-round. Because the earth’s moment of inertia is so huge, you can’t tell that it undergoes angular acceleration at all. It really does, just as the earth undergoes acceleration when you jump-you push down hard and the earth as it pushes up hard on you and you both accelerate away from one another. Since the earth is much more massive than you are, it doesn’t accelerate nearly as much as you do.

## You said that when you were spinning around in circles, you were actually causin…

#### You said that when you were spinning around in circles, you were actually causing the earth to move, but it was too tiny a motion to notice. If everyone on the planet got together in one area and started spinning around at exactly the same time and with the same angular velocity, could the effect of the people causing the earth to move be noticed?

I don’t think that it would be possible to detect any change in the earth’s rotation. The earth has a mass of about 6,000,000,000,000,000,000,000,000 kg, which is about 20,000,000,000,000 times the mass of all the people on earth. The earth’s moment of inertia is even more different than that of the people because much of the earth’s mass is located far from its rotational axis. So if all of the people gathered together and started spinning one way, the effect on the earth would be to make it spin the other way about 1/1,000,000,000,000,000,000 as much. The result might be that the day would change lengths by about a trillionth of a second. (1/1,000,000,000,000 s). That change is less than the natural fluctuations in the earth’s rotation rate, so no one would ever notice. You might find it interesting that the earth’s rotation rate changes slightly with the seasons because of snow in the mountains. When there is lots of snow in the northern hemisphere (during its winter), the earth’s moment of inertia increases just enough to slow its rotation. The day is a tiny bit longer than during our summer. People might be able to duplicate this effect by all climbing to the tops of mountains.

## Can you give me an example of when the angular acceleration is in a different di…

#### Can you give me an example of when the angular acceleration is in a different direction from the torque applied?

When an object isn’t symmetric, it can rotate in very peculiar ways. If you throw a tennis racket into the air so that it is spinning about an axis that isn’t along the handle or at right angles to the handle, it will wobble in flight. Its axis of rotation will actually change with time as it wobbles. If you were to exert a torque on this wobbling tennis racket, its angular acceleration wouldn’t necessarily be along the direction of the torque.

## Given a lever long enough, could you move the world?

#### Given a lever long enough, could you move the world?

Yes. Of course, you would need a fixed pivot about which to work and that might be hard to find. But you could do work on the world with your lever. If the arm you were dealing with was long enough, you could do that work with a small force exerted over a very, very long distance. The lever would then do this work on the world with a very, very large force exerted over a small distance.

## How can cats turn their bodies around to land on their feet if they fall and how…

#### How can cats turn their bodies around to land on their feet if they fall and how can people do tricks in the air when they are skydiving if you’re supposed to “keep doing what you’ve been doing” when you leave the ground?

Cats manage to twist themselves around by exerting torques within their own bodies. They aren’t rigid, so that one half of the cat can exert a torque on the other half and vice versa. Even though the overall cat doesn’t change its rotation, parts of the cat change their individual rotations and the cat manages to reorient itself. It goes from not rotating but upside down to not rotating but right side up. Overall, it never had any angular velocity. As for skydiving, that is mostly a matter of torques from the air. As you fall, the air pushes on you and can exert torques on you about your center of mass. The result is rotation.

## Is moment of inertia determined only by mass, as inertia is in translational mot…

#### Is moment of inertia determined only by mass, as inertia is in translational motion?

No, moment of inertia embodies both mass and its distribution about the axis of rotation. The more of the mass that is located far from the axis of rotation, the larger the moment of inertia. For example, a ball of dough is much easier to spin than a disk-shaped pizza, because the latter has its mass far from the axis of rotation.