With a lacrosse stick, does the lever 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 you push on 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.

What animals have elastic energy-recapture mechanisms?

Some insects use an elastic material to store energy for leaps. They use their muscles to store energy in this elastic material and then release that energy suddenly to hop up into the air. In effect, they are stretching a rubber band and then letting it go.

If you had an object in an empty sphere with a radius of a few miles, surrounded by equally distributed and very concentrated mass, what effects of gravity would the object feel?

As long as the mass isn't so concentrated that the laws of general relativity become important, the object won't feel any gravity at all. The forces from opposite sides of the surrounding mass will cancel exactly. For example, if you were at the center of the earth in a large spherical opening, you would be perfectly weightless. The force from the north side of the earth would balance the force from the south side. This effect is quite remarkable and depends on the fact that gravity becomes weaker as the inverse square of the distance separating two objects. That way, even if you aren't in the exact center of the earth, the forces still cancel.

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.

Shouldn't the seesaw be completely horizontal in order to be balanced? Why can it be balanced if it's not (horizontal)?

A balanced seesaw is simply one that isn't experiencing any torque (the net torque on it is zero). Because there is no torque on it, it isn't undergoing any angular acceleration and its angular velocity is constant. If it happens to be horizontal and motionless, then it will stay that way. But it could also be tilted or even rotating at a steady rate.

I don't understand center of gravity in balls (spherical objects).

A spherical object normally has its center of gravity (and center of mass) at its exact center. That is the point about which all of the weight (and all of the mass) is balanced in the ball. You can think about gravity as acting on the center of gravity, there in the center of the ball. You can also think of the mass as being placed at the ball's the center of mass. When you spin a ball that's isolated, it will rotate about its center of mass; its natural pivot.

Why when you drop a ball from 1 meter in the air, does it take 4.9 seconds to hit the ground, and not 9.8 seconds to reach the floor?

I think that this question means to be: "Why when you drop a ball for 1 second, it falls 4.9 meters, rather than 9.8 meters?" That being the case, here is the answer: After 1 second of fall, the ball has accelerated to a velocity of 9.8 meters/second. That's because it accelerated downward at 9.8 meters/second2 for a period of 1 second, giving 9.8 meters/second as the final downward velocity. But during that 1 second, it also dropped a certain distance. Since its speed gradually increased, it didn't spend the whole second at 9.8 meters/second downward. Instead, it had an average velocity of 4.9 meters/second downward. As a result, it descended 4.9 meters during that first second of fall.

Is the equation: Work=Force*Distance, the same for angular (rotational) motion as it is for linear "transformation"? Also is torque a "measurement of" what?

There is a relationship between work and torque that is useful in rotational activities. It is Work=Torque*Angle (where angle is measured in radians). Thus as you twist something and it rotates in the direction of that twist, you do work on it. The more torque, the more work. The farther it rotates, the more work. As for torque, it is a measurement of how much twist is being exerted. There is a Newton's second law for rotations: Torque=Moment of Inertia*Angular Acceleration. This law says that the more torque you exert on an object, the greater its angular acceleration. It also says that the larger that object's moment of inertia, the less angular acceleration it experiences. This equation sets the scale for torque (just as the equivalent law sets the scale for force).

How can cats turn their bodies around to land on their feet if they fall from high buildings 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 the 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.

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. We'll discuss the effects of changing moments of inertia in the next class.