Introduction to Physics (PHSX 1010)
Adam Johnston

Course notes:
ROTATIONAL MOTION

At this point we move on to rigid bodies, center of mass, and rotational motion.  What you will see here is that for rotational motion, much of what we've already done still applies: velocity, acceleration, force, energy, momentum, etc.  But, we just have to be careful to apply these concepts in a different manner.

When we see a ball rolling across a table or a pair of string-connected tennis balls thrown across the room, there is lots of motion going on.  These rigid bodies are made up of many different pieces, and as they rotate and move across the room, each piece is traveling at a different velocity.  However, the center of mass of these objects follows a nice, smooth path.  The center of mass, or balance point, of an object is useful in characterizing its general motion from one place to another.

However, you might want to make something rotate, and you might want to describe all the intricacies of such motion.  First, let's think about what it would require to produce a change in this motion (to get it going).  For translational motion -- the motion of the center of mass -- Newton's second law says you need a force to produce a change in motion.  In a rotational sense, one needs to exert a torque.  That is,

torque = F * r

where F is the force applied in the direction perpendicular to the lever arm (wrench), and r is the length of the lever (length of the wrench).  Thus, you can get more torque in two different ways: Either increase the force, or make your lever bigger.  If you make your lever arm bigger, you will not need to produce as must force to get the same torque as a smaller lever with a bigger force.  This is why we like big wrenches.

The exciting thing about using Newton's laws is that we get to see how things will move and change their motion.  This also is true for rotational motions.  We should first ask what is required for a rigid body to be in equilibrium, or not changing its motion.  In the translational sense, the net force needs to be zero.  But here, in the rotational sense, the net torque needs to be zero.  That means any counter-clockwise torque needs to be balanced by a clockwise torque.  By doing this, you can get people on a teeter-totter to balance, or you can get a variety of objects to balance.  Some demonstrations of this were shown in class, and some more will be shown and described next time.

"Holey smokes!" yelled a student.  "It's the end of the fourth week!"  The class erupted into a sea of panic.  Where had the time gone?  Physics class was going to be over almost as quickly as it had started.  Upon realizing this, the students settled down and decided that they were going to make the most of today and every other day of physics class; for, in the words of Robert Frost, "nothing gold can stay."

Plus, there's an exam coming up next week.  A man in the third row began to weep softly to himself, sobbing into his notebook and smearing all of his notes.

Discussion continued to feature many different concepts.  Fortunately, you had seen them all before, just in the context of translational (back-and-forth) kinds of motion.  We just repeated all the same ideas, but this time but them in the context of a rotating body.

Maybe we could make a table (we didn't do it in class, but we could have) that shows the analogy between translational motion and rotational motion.  This isn't what you need to memorize, but it summarizes a bunch of concepts (for both kinds of motion) that you should be able to explain.  (A big problem with this version of this table is that I can't write out the Greek letters that we use in class and in the book, so you might need to do a little translating.)

What the heck does all that mean?  We covered such ideas in class.  Read on . . . 

We reviewed torque, showing how the torque produced varied both with the force involved as well as with the lever arm (demonstrated by trying to hold a weight at the end of a long rod).  Knowing that torques could be balanced, we also showed some examples of that taking place.  Such "equilibrium" situations could be summarized in a couple of different ways.  First, the torques balanced each other, making hammers suspend on hinges, bottles precariously balanced, and birds perched on their beaks.  (You had to be there to appreciate these demonstrations.)  Second, the center of mass of all these balanced systems was directly above (or below, in the case of the hammer) the point of rotation.  Actually, both of these statements are equivalent.  We didn't point it out in class, but we could have also mentioned that the total net force in all situations was also zero, since you didn't witness any acceleration up or down or left or right or back or forth.

Understanding that the property of inertia is important and depends on mass, it was interesting to see that the distribution of mass is an additional important factor when one is talking about rotational inertia.  The farther the mass is distributed from a point of rotation, the more difficult it is for the rotational motion to change.  Thus, a broom is most easily balanced when its bristle end is high in the air, and a hoop (with all the mass distributed to the outside) will lose the race to a solid disk (with the mass distributed more evenly) down a hill.  Likewise, it would be harder to stop the hoop, because it has a higher rotational inertia than the disk.

We could also observe that once something is actually spinning, it doesn't want to change the direction or orientation of the motion (in addition to the act of motion itself).  This is shown via the swinging vinyl record: When it is only swinging from a string, it flaps about; but if it is spinning while swinging at the end of the string, it holds its orientation.  (This is how people with lassos do all those groovy tricks, I think.)  In other words, there seems to be a conserved motion and direction to this motion, a conserved angular momentum.  This can be demonstrated in a couple of different ways.

First, Adam bravely stood upon the rotating platform of death, showing multiple ways of getting his motion to change.  One way, similar to riding a bicycle, was to have a wheel which was spinning and how changing the tilt of the spinning wheel resulted in a change in Adam's rotational motion.  That is, if twisting the wheel would produce a change in rotational momentum on its own, then there must be a corresponding and opposite change in momentum somewhere else, in order for the total momentum to be constant.  Thus, when you turn the wheel of your bicycle, a lean is also produced.  One of those momentum changes makes up for the other's momentum change.

More observable in everyday life, perhaps, is the idea behind how ice skaters get spinning so fast.  Stephanie stood upon the rotating platform of death, holding two dumbbells in her outstretched arms.  Adam started her rotational motion, and then the rest of the class went home for the weekend.  As best as we know, Stephanie is still spinning around, and she should still be there on Monday.

No, that's not true.  Instead, Stephanie's motion continued, the dumbbells rotating around with her.  Marvelously, when she pulled the dumbbells in towards her center of rotation, Stephanie's rotation sped up!  Nothing pushed on her from the outside, so her rotational momentum was conserved.  If rotational momentum is equal to rotational inertia times rotational velocity (I * rot. v), then any change in the rotational inertia must produce an opposite change in the rotational velocity.  In this case, the rotational inertia decreased (the masses were closer to the center of rotation), so the rotational velocity had to increase.  To slow herself back down, Stephanie increased her rotational inertia by putting the dumbbells away from her body, thus making the rotational velocity decrease.

Everyone cheered.  Then, everyone wanted to leave.  Adam started to cry, so people stayed a few more minutes.  Bless you, one and all.

It was also worth mentioning that rotation also is a way of having kinetic energy.  The more rotation something has, the more kinetic energy is involved in this rotation.  If energy goes into this kind of motion, then it won't be able to go elsewhere.  In the case of the two rolling objects down the hill, the hoop lost the race because more of its energy went into rotation (because of its higher rotational inertia), so less could go towards its motion down the hill. 

Another question to think about: What if one of the objects slid down a frictionless surface instead of rolled?  Would it win or lose or tie a race with a rolling object?  Why, in terms of energy?