An introduction to physics in the context of everyday objects.

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From the course by University of Virginia

How Things Work: An Introduction to Physics

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University of Virginia

704 ratings

An introduction to physics in the context of everyday objects.

From the lesson

Seesaws

Professor Bloomfield illustrates the physics concepts of rotational versus translational motion, Newton's law of rotation, and 5 physical quantities: angular position, angular velocity, angular acceleration, torque, and rotational mass using seesaws.

- Louis A. BloomfieldProfessor of Physics

Why do the riders' distances from the pivot affect the seesaw's responsiveness?

Â The answer to that question is that the farther the riders masses are from the

Â pivot, the greater the seesaw's overall rotational mass and the slower its angular

Â accelerations. Two riders can balance the seesaw in a

Â variety of ways. To begin with, they can go to the ends of

Â the board and adjust their distances from the pivot carefully until it balances.

Â That is, until it experiences zero overall torque due to gravity.

Â I mean I'm pretty much there. Balanced seesaw.

Â But they can also come in close to the pivot and sit like this, with much smaller

Â lever arms to work with now. So that they're producing much small

Â torques as individuals. But once again, those two torques sum to

Â zero. And there's zero overall torque due to

Â gravity on this seesaw. So, there are a variety of ways to balance

Â the seesaw. And you might think that there's no

Â significant difference between those choices, but that's not true, there is a

Â significant difference. The farther these two riders sit from that

Â central pivot, and therefore the axis of rotation, the greater the seesaw's overall

Â rotational mass. Now, rotational mass is not something

Â completely independent of ordinary mass. They're related, just as forces and

Â torques are related. Every portion of the seesaw's ordinary

Â mass contributes to the seesaw's rotational mass.

Â And the amount of that contribution depends on where the portion of ordinary

Â mass is. More specifically, on how far that portion

Â of ordinary mass is from the axis of rotation.

Â And that dependence on axis of rotation is, is a strong one.

Â Small, modest changes in distance from the axis of rotation can lead to large changes

Â in the rotational mass contribution. The amount of rotational mass contributed

Â by a portion of ordinary mass. Is equal to the ordinary mass itself times

Â the square of the distance. Between that portion of ordinary mass and

Â the axis of rotation. So, taking a small portion here and

Â doubling its distance from the axis of rotation doesn't just double its

Â contribution to the rotational mass, it quadruples it.

Â That means that for the riders, when they sit in close, and their distance from that

Â central pivot and center of rotation, axis of rotation, is small, they contribute

Â very little to its rotational mass, the overall rotational mass of the seesaw and

Â riders. Even though these, these riders have large

Â masses, they're too close to the axis of rotation to contribute very strongly to

Â its rotational mass. But if they go out like this, to a large

Â distance, well then their contribution to rotational mass is huge.

Â They might be only ten times as far away from the pivot as before.

Â From that axis of rotation. But an increase of distance by a factor of

Â ten is an increase in contribution to rotational mass of ten times ten, or ten

Â squared, which is 100. So the rotational mass contribution of

Â these riders could easily be 100 times. That of these riders.

Â It's a big effect. To see how big, I've made these two rods

Â that look the same, and have the same the masses.

Â They contain the same materials actually. But the difference is that in one of these

Â rods, the mass is all in the middle, near my hand, under my hand, hidden from view.

Â And in the other bar, all of the mass is far from my hand, at the ends of the bar.

Â So, same mass, but in, in this case it's moved way out far from the center of

Â rotation, which will be here in the middle bar.

Â And the rotational mass of this bar is something like 30, 40 times that of this

Â one. Big difference.

Â To see that difference, since you can't hold the bars, let's go get some help.

Â Annie and Megan are going to help me here with these two bars.

Â Now, these bars have the same masses and the same weights.

Â So you could check that out. Just, just compare the weights, do they

Â feel the same? >> Yeah, definitely even, uh-hm.

Â >> Yeah, definitely. >> So, just by weighing them in your hands

Â you can't tell a difference between these two bars.

Â But that doesn't mean they're identical. So the difference is going to be subtle,

Â and this difference will show up maybe when you begin to try to rock them back

Â and forth. So I want, Annie grab it in the middle of

Â the bar, Megan same thing. And I'm going to count to three, and on

Â the number three. I want the two of you to rock it back and

Â forth as fast as you can. That is, make it undergo angular

Â acceleration first one way and then the other, back and forth as fast as you can.

Â >> Uh-hm. >> 'Kay.

Â >> One. Two.

Â Three. >> This is really difficult.

Â >> [laugh] So Annie's having no trouble here, and Meagan's really lagging behind.

Â Must be weak today, right? Forgot to eat your breakfast.

Â Okay, now swap bars. >> Okay.

Â >> Thank you. >> Now, I'll count again.

Â One, two, three. >> Oh.

Â [laugh]. >> Miraculous change.

Â >> What? >> Something's different about these two

Â bars. What do you think's different about the

Â bars? They have the same mass, what's different?

Â Megan? >> Maybe the distribution of the mass

Â within the bar? >> So the distribution of the mass in,

Â within the bar is different. Where is the mass in your bar, right now?

Â >> In the center. >> So your bar has almost all of the mass

Â in your hand. >> Yeah.

Â >> As a result, the moment of inertia, or the rotational mass of this bar is very

Â small. >> Yeah.

Â >> It's very easy to make this bar undergo angular salvation.

Â How about yours, Annie. Where do you think the mass is located?

Â >> I mean they must be at the end, right? >> Yeah, so all the mass in this bar are,

Â is at the ends where it contributes enormously to ro-, rotational mass.

Â So they have the same mass, it's just distributed differently.

Â In Annie's bar it's at the ends, in Megan's bar it's in the middle.

Â And they behave totally differently when you try to rock them back and forth.

Â >> Uh-hm. >> Show us a little more time.

Â >> Sure. [laugh].

Â >> I should make a face. >> Yeah, it's, it's a pretty dramatic

Â difference. Unfortunately you guys can't try it out.

Â >> [laugh]. >> But, but if you ever come by, test out

Â these bars and see how different they are. >> Okay.

Â >> Thanks. >> Sure.

Â >> So you see, if you place an object's ordinary mass far from its axis of

Â rotation, that object can have a surprisingly large rotational mass.

Â Now, the distance involved here is between ordinary mass and the rotational axis.

Â And you're often free to choose an object's rotational axis.

Â If you do that, and if you change your choice, you may well change the object's

Â rotational mass. That ability to change an object's

Â rotational mass is why rotational motion is so complicated to calculate

Â quantitatively. And that's a fact the keeps first year

Â physics graduate students rather busy. It's hard work.

Â I want to give you a, a taste of the issues without trying to overwhelm you

Â with them. But look at this rod.

Â This is the rod that's very hard to wobble back and forth like this because it has an

Â enormous rotational mass when you twist it back and forth about this axis.

Â The one pointing toward you through my hand or away from you through my hand.

Â About that axis, gigantic rotational mass. But what about this axis in which I'm

Â twisting it back and forth like, like a drill or a screwdriver?

Â It's easy. This, this direction has almost no

Â rotational mass. That's because all the portions of

Â ordinary mass are very close to this spindle-like axis about which I'm twisting

Â it. So, this, this rod here, has two very

Â different rotational masses, a huge one when you do this motion and a tiny one

Â when you do this motion. Now this is a you know, fun and games rod.

Â But something you're more familiar with is perhaps a tennis racket.

Â The tennis racket is a classic example of something that has three particularly

Â important rotational masses. It's smallest rotational mass is for this

Â rotation about it's There, the, the top bottom axis right now.

Â This motion, most of the mass is pretty close to that axis, the spindle about

Â which I'm twisting it, and therefore it has a relatively small rotational mass.

Â The next larger rotational mass is for this motion.

Â Sometimes referred to as the frying pan motion, when you're, you're flipping

Â pancakes. So this is the intermediate rotational

Â mass. And the biggest rotational mass is for

Â this rotation. In between these motions, life is

Â extremely complicated. And it's beyond the scope of this class as

Â something I don't like to deal with anymore.

Â I've done it. Been there, done that, I'll leave it.

Â But these three distinct rotational masses, this small one, bigger one,

Â biggest one, give the motion of a tennis racket or anything shaped like a tennis

Â racket the rotational motion's quite complicated.

Â To make things even worse, these are all motions about the center of mass.

Â These are all rotations in which the center of mass stays put.

Â What if you shift the rotation, so that you don't care about the center of mass of

Â the tennis racket. For example, when you're swinging a tennis

Â racket about your shoulder. In that case, you're shifting the mass of

Â the tennis racket even farther from the center of rotation, the axis about which

Â you're spinning it, and creating an even larger rotational mass for the tennis

Â racket. So the bottom line with all of this is

Â rotational mass depends on your choice of axis of rotation.

Â We're finally ready for the question I asked you to think about in the

Â introduction of this episode. To remind you, that question asked if you

Â and a child half your height lean out over a swimming pool at the same angle and let

Â go at the same moment, which of the two of you will hit the water first?

Â Despite a fair amount of rotational physics under our belts, that remains a

Â challenging question. So before I ask it, and leave you free to

Â answer it, I want to give you a little more background.

Â Get you all prepped for this question. First, what's the big picture issue?

Â What is going to determine who hits first? It's going to be angular acceleration.

Â The one of you that undergoes the fastest angular acceleration will tip, will, will

Â de-, develop the fastest angular velocity, will tip over the fastest and will hit the

Â water first. So look for big angular acceleration.

Â Second, what is the axis of rotation about which the two of you are going to be

Â rotating? It's not your centers of mass, it's going

Â to be here at your feet. That leaves two more issues.

Â One is the cause of angular acceleration and the other is the resistance to angular

Â acceleration. The cause is the neck torque on you.

Â The resistance is your rotational mass. And let's look at each one individually.

Â First Torque. The torque is due to gravity, and to make

Â our lives simple, let's compare the torques about this, about your feet,

Â that's the ax-, that's the axis rotation here, for, for you and the child.

Â Now, I've made life very simple by using exactly the same board material.

Â One is just half as long as the other, and this makes, you know.

Â This has all the physics in it, but no, none of the details.

Â Life is easier. So, you have twice the weight of the

Â child, that's no suprise. And that weight effectively acts at your

Â center of gravity, which is twice as far, it's right here in the middle.

Â It twice as far from the axis of rotation as for the child.

Â So, you're experiencing four times the gravitational torque of the ch-, of the

Â child. You have twice the weight acting at twice

Â the lever arm. 2 times 2 is 4.

Â Right? 4 times the torque.

Â That's the cause of angular acceleration. How about the resistance to angular

Â acceleration, the rotational mass? Well, as compared to the child who has

Â half the mass here, distributed around here with the center of mass being about

Â there. You have twice the mass distributed here

Â and there, there's a center of mass here. The center of mass has moved out by a

Â factor of two. You have twice as much mass, that is on

Â average at twice the distance from the axis of rotation, namely your feet.

Â Well, remember that the distance involved here In calculating, we-, it, the

Â contribution of mass depends, not on distance, but on distance squared.

Â So the rotational mass that you have is eight times that of the child.

Â Twice as much mass, at twice the distance and you, you square the distance.

Â So it's two times two times two, that's eight.

Â This has, this, you have eight times the rotational mass of the child.

Â With that as background now, answer the question.

Â Which of the two of you tips over fastest and hits the water first?

Â The child undergoes greater angular acceleration, develops a bigger angular

Â velocity and hits the water first. If you haven't already tried this

Â experimentally, give it a go. All you need is two sticks, one twice as

Â long as the other. I can show you what you'll see when you

Â try it. Here's you, here's the child, and we'll

Â put you both on your feet and tip you to the same angle, and then let go.

Â 3, 2, 1. No question, the child reached the water first.

Â It's a battle between torque and rotational mass.

Â In both cases about your feet. You have four times the gravitational

Â torque acting on you as, as the child has. So there's four times as much twist trying

Â to propel angular acceleration but you have eight times the rotational mass as

Â the child, resisting that angular acceleration.

Â Four times more impetus. Eight times more resistance, you get only

Â half the angular acceleration of the child.

Â So, the child undergoes twice you angular acceleration and just goes through the

Â whole rotational motion faster. And wins the race to the water.

Â Rotational motion clearly has some subtle complications, like a single object having

Â more than one rotational mass depending on your choice of axis of rotation.

Â But let's leave all those complications for the experts.

Â I chose seesaws as the topic for this episode because seesaws are comparatively

Â simple. That pivot fixes the axis' rotation so

Â that the seesaw can only rotate in one fashion.

Â And in general, it only has one rotational mass.

Â Things are simple. Nonetheless, the seesaw exhibits most of

Â the issues of rotational motion. Or at least the ones that I want to talk

Â about and try to convey to you. I've already done that now.

Â I've, I've shown you most of what happens in a rotating system like a seesaw with

Â one important exception. Energy.

Â As the seesaw rotates, the riders are exchanging energy.

Â And that, is the topic for the next video.

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