Section 4 Seesaws
Seesaws are a simply toy that consists of a long board mounted on a central pivot. Two riders get on
opposite ends of that board, and adjust their positions until the seesaw balances at that point then,
they can begin to make the seesaw rock back and forth either by leaning toward or away from the
pivot, or by pushing on the ground with their feet.
Mechanically, a seesaw is a lever and fulcrum seesaws also work as a simple example of a
mechanical system with two equilibrium positions one side is stable, while the other is unstable
Levers can be used to exert a large force over a small distance at one end by exerting only a small
force over a greater distance at the other.
Classification
Industry
Weight
Fuel source
Components
Simple machine
Construction
Mass times gravitational acceleration
potential and kinetic energy {mechanical energy}
fulcrum or pivot, load and effort
http://en.wikipedia.org/wiki/Lever
When I was a kid, seesaws were everywhere. Any playground worth its' salt had a couple of them.
And either at recess or during a birthday party we'd clamber onto the seesaws, one at each end,
maybe one big kid, and one little kid. Maybe several at each end and we'd rock back and forth
furiously until our legs wore out, or one of us got hurt or we simply wanted to do something else.
Nowadays, seesaws become rarer and rarer it seems that either they're risky, or perhaps modern
children don't enjoy that sort of activity as much. Whatever the reason, they're missing an
opportunity to experiment with rotational motion, balance, levers, and mechanical advantage.
Seesaws turn out to be a wonderful context in which to explore the physics of rotational motion so
I've stuck with them, even as they've somewhat abandoned me. If you have a seesaw nearby, I urge
you to experiment with it although be safe, because there are ways in which you can get yourself
injured playing with a seesaw more on that later.
If you don't have a see-saw, well, you can make one yourself. All you need is some object to serve as
a central pivot and a board to balance on that pivot. You adjust the spacing’s just so and voila, a
seesaw. You can have it rock back and forth, just like the real thing. Actually, that simplicity the fact
that you can make something like this so easily, explains why seesaws have been around so long,
they're pretty simple to make.
As I suggested earlier, the story of seesaws is also the story of rotational motion, balance, levers, and
mechanical advantage. We'll study those concepts here in the context of seesaws and then use them
repeatedly as we continue to look at how things work. Before continuing, however, I want to ask you
a question to think about. Not to answer, but something you should have in mind as we work our
way through the story of seesaws. It's a difficult question, one that doesn't have an obvious answer
so it's a good prelude to the rest of the story here on seesaws.
Suppose you and a child half your height lean out over a swimming pool at the same angle so here's
you, here's the child, and you're both leaning out over the swimming pool at the same angle. If you
both let go at the same moment, so that you begin to rotate into the pool which of the two of you
reaches the water first? To help guide us through the science of see saws, we'll pursue six, how and
why questions.
How does a balanced seesaw move? Why does a seesaw need a pivot? Why does a lone seesaw
rider plummet to the ground? Why do the riders' weights and positions affect the seesaw's motion?
Why do the riders' distances from the pivot affect the seesaw's responsiveness? How do the
seesaw's riders affect one another? There is one video sequence for each of those questions and a
summary sequence at the end and now onto the first question.
Part 1.
How does a balanced seesaw move, the full answer to that question will require some careful
explaining, but a short answer is that a balanced seesaw rotates steadily about a fixed axis. Now it's
tempting to think that I've just asked a trick question, that a balanced seesaw doesn't move at all. In
fact, that it's horizontal and motionless. But the real answer to that question is more subtle. Yes, a
balanced seesaw can be horizontal, and it can be motionless but it doesn't have to be, what a
balanced seesaw does exhibit, however, is rotational inertia. If I set it spinning, it rotates steadily
about a fixed axis. Up until now, I've talked about a type of motion that takes you from place to
place.
So, in the episodes on skating, falling balls, and ramps we went somewhere from place to place. In
this episode on seesaws, we don't go anywhere. Seesaws are installed in playgrounds and they stay
there indefinitely what seesaws do, do, however is rotate. So, the world of motion can divide into
two main types the motion of translation, of going somewhere and the motion of rotation, spinning
in place and see-saws. They're about spinning in place.
In the episode on skating, we saw that a skater exhibits translational inertia. The inertia of going
places and associated with that translational inertia was Newton's first law of translational motion.
Namely, that an object that's free of external forces, moves at constant velocity. In this episode on
seesaws, we're looking at objects that can exhibit rotational inertia. When they're at rest, they stay
at rest. When they're rotating, they continue to rotate. Associated with rotational inertia is another
Newton's first law, but now it's the Newton's first law of rotational motion.
In a draft form Newton's first law of rotational motion states that a rigid object that is wobbling and
that is not experiencing any outside influences, rotates about a fixed axis turning equal amounts in
equal times. That law has a couple of extra words in it, it refers only to rigid objects and objects that
are not wobbling. So Newton's First Law of Rotational Motion has relatively limited applicability.
What can you do? Rotational motion simply is more complicated than translational motion and
therefore the Newton's 1st Law in the world of rotation is fairly limited. There are lots of things that
don't follow Newton's 1st Law of rotational motion because they either change shape, or because
they're wobbling.
http://physics.bu.edu/~okctsui/PY105%20Lecture_notes/PY105_Fall2009_files/PY1056pm_NewtonLaw-Rotation.pdf
To perfect the draft of Newton's First Law of Rotational Motion, we need to identify the external
influences, and we need better language to describe rotation about fixed axes turning equal
amounts and equal times. I'm going to start with the second task. In the previous episodes I
described translational motion and identified several physical quantities that are useful for that
description. Among those physical quantities were position and velocity. In describing rotational
motion, there are analogous physical quantities.
There is a physical quantity describing angular position. Angular, rotational, it doesn't matter, but
there, it's technically called angular position and there is a physical quantity describing how angular
position changes with time and it are called angular velocity. So, those are the two quantities I want
to introduce. 1st, angular position, angular position is an objects orientation, and instead of
illustrating angular position using the seesaw, I'm going to illustrate it using my body. So, angular
position will describe how I'm oriented. I'm going to start with a zero of angular position, which is
the starting point, this will be my 0 of angular positioning, the orientation that we all agree is the
starting point, facing you.
If I change my angular position, that means that I'm facing some other direction, like this or this or
like this and like that, and so on. Well, how do you describe, technically, quantitatively, those various
other orientations? How do you do it? Actually, you need an amount and a direction. You need a
vector and here's how the vector. So angle position is a vector quality and here's how it works. First,
the amount is an angle the angle through which you have to rotate to go from the zero, namely
facing you, to the orientation that you're trying to describe.
For example this, this is the one I'm going to try to describe. Facing like this and the angle that I have
to rotate through to go from the zero to this is 90 degrees from there to there, that's 90 degrees,
you know what, 90 degrees right? So this angle position is 90 degrees but that's not enough. This is
90 degrees and so is this and so is this, alright? So there are a bunch of 90 degree angles positions.
We need a direction as well and the direction of an angular position is the axis about which the
rotation occurs.
For example, to rotate to this 90 degree angular position, I need to rotate about a vertical axis as
though I were a toy top being spun. So I'm being spun, there I go. So this is 90 degrees about a
vertical axis but there's an ambiguity. This is 90 degrees about a vertical axis and so is this. They're
both 90 degrees about a vertical axis how do you distinguish them?
Well, physicists and mathematicians distinguish them using a convention known as the right-hand
rule and the right-hand rule says that if you take your fingers of your right hand and curl them in the
direction in which the rotation occurs. For example, if I'm going from this to this, the rotation is like
that. Then look at my thumb the thumb of my right hand, points in the official direction of that
rotation, downward. So in going from 0 to this, I rotated 90 degrees downward.
On the other hand, if I go from this to this, my fingers have to be pointing the other way my thumb is
now pointing up, this orientation this angular position is 90 degrees up. So the ambiguity is solved by
the right hand rule. 90 degrees down and 90 degrees up. How about this that is 90 degrees toward
you and this is 90 degrees towards me final word about angles. The angle part of anchor position can
be measured in various units. Up until now, I've been using the unit known as the degree. It's a
familiar unit of angle; this is zero degrees, 90 degrees, 180 degrees, 270, 360.
Another possible unit of degree is the rotation full rotation, this is zero, quarter rotation, half, three
quarters, full rotation but the unit that mathematicians and physicists normally use to describe
angles, is neither of those two. It's the radian and there are two pi radians in a full rotation where pi
is the mathematical constant, three point one four one five nine and so on and that is the natural
unit of angles.
There are reasons why it's particularly useful in physics whether you use it or not, doesn't matter.
Pick your unit of angle and stick with it, you're fine. So you can describe this angular position as. 90
degrees down or quarter rotation down or pi over two radians down. They're all the same. That's
angular position, but that by itself doesn't help us re-draft Newton's first law of rotational motion.,
we need to look a little deeper we have to look at how angular position is changing with time,
because when something is actually rotating its angular position is evolving, changing with time and
we need the next physical quantity which is angular velocity.
Angular velocity is the rate at which angular position is changing with time so right now my angular
position is not changing with time, so my angular velocity is zero but if I begin to spin, then my
angular velocity is no longer zero. For example, if I turn like this I am now turning about 90 degrees
per second and I am the same right hand rule applies. I'm turning such that my fingers curl like this
and my thumb points out. This is an angular velocity of 90 degrees per second down. Also pie over
two radians per second down.
Let me stop and show you 90 degrees up, here it is. Alright, I could show you 90 degrees toward you,
90, yeah 90 degrees per second toward you but that's, I'm going to run out of ability to do this. But
you get, I hope you, I hope you get the point. That angle velocity describes how an object is rotated,
that is how fast it's going through angles, and also the axis about which it's spinning and finally, the
right, using the right hand rule, the specific direction of its spin around that axis. So you should be
able now to distinguish 90 degrees per second down from 90 degrees per second up. That now, that
physical quantity, angle velocity will be useful in redrafting Newton's first law, rotation motion.
Because we can rewrite the turning about, rotating about fixed axis, turning equal amounts in equal
times as having constant angular velocity. If I'm turning 90 degrees per second down and staying
that way my angular velocity is constant. Alright, that brings us to this, to the other task. Which is
identifying the external influences that show up in Newton's first law of rotational motion and those
external influences are twists, technically, they're known as torques.
A torque is the influence that causes, that upsets rotational inertia and therefore, violates Newton's
first law of rotational measure. We'll look more at torques but just so that you know what a torque
is. Let me show you what happens when I exert a torque on this seesaw to do it, I twist the see-saw.
So I'll grab the see-saw from the front, and I will twist and suddenly, it changed its angular velocity. It
started with an angular velocity of zero, let's get zero there and it's now, for the present a rigid
object that's not wobbly, it's obeying Newton's first law of rotational motion but if I come in with an
external influence of the right type namely, a torque while I'm exerting that torque, it is not
following Newton's first law of rotational motion it changed its angular velocity.
We can now state Newton's first law of rotational motion in all its glory. A rigid object that is not
wobbling and that is free of external torques rotates at constant angular velocity.
That brings us to a question what influence or effect causes the earth to rotate steadily? Turning
once every, approximately 24 hours the Earth is rotating because it exhibits rotational inertia. It's
experiencing essentially no torques, and therefore, it rotates according to Newton's 1st Law of
Rotational Motion, namely it's a rigid object that is not wobbling, it is not experiencing any external
torques, so it rotates with constant angular velocity. That angular velocity is approximately one
rotation per 24 hours about the north poles so that the rotational axis points from the center of the
earth up to the North Pole and that's the way the earth rotates.
So we see a balanced seesaw is not necessarily motionless or horizontal what we can say about that
balanced seesaw however, is that it exhibits rotational inertia. If it's motionless, it remains
motionless if it's rotating, it continues to rotate because it's a rigid object that's not wobbling, it
exhibits a particularly simple type of rotational motion namely, constant angular velocity. So right
now, the con, the angular velocity of this balanced seesaw is 0 but if I twist it, and during the twist
it's not rotationally inertial and so I'm violating Newton's 1st Law of rotational motion by doing the
twist.
Here we go, I'll give it a twist and now it's once again rotationally inertial it's obeying Newton's 1st
Law of rotational motion it's a rigid object that is not wobbling, it's free of external torques so, it
rotates at constant angular velocity, apart from some air resistance problems here. The point is it's
rotating right now, not because something is twisting it, but because nothing is twisting it. It is its
nature, and the nature of objects in our universe to keep rotating in the absence of twists. They keep
going that's rotational inertia.
So, in, in a normal seesaw that perpetual rotation isn't possible, because during the rotation even
when it's balanced initially it eventually touches the ground and at those moments when it touches
the ground, the ground exerts torques on the seesaw. It twists the seesaw and therefore takes it out
of the, out of Newton's First Law of Rotational Motion, violates Newton's First Law of Rotational
Momentum and new things happen and those new things, basically the consequences of torques, a
subject for the next video.
Part 2.
Why does a seesaw need a pivot? The answer to that question is that the pivot prevents the seesaw
from undergoing translational motion, while leaving it free to undergo rotational motion. Without a
pivot to support its weight and that of its riders, the seesaw would fall and while two children might
find it exciting to jump out of an airplane seated at opposite ends of unsupported seesaw that idea is
unlikely to be popular with their parents.
The physics will be fabulous but I'm not going to film it I'm going to leave that for children who enjoy
extreme recess. Instead, I'm going to show you how an unsupported and riderless seesaw moves.
Basically, I'm going to throw the seesaw through the air and we'll watch its motion for obvious
reasons, I'm not going to use a large seesaw I'm not even going to use one as big as this but even so,
even with a small seesaw board, or a pretend seesaw board, I need more room, so, let's go outside
and have some fun. I'm going to throw a riderless, unsupported seesaw, well that sure was quick.
But this is video so I can show you that throw again and this time I can slow it down to one tenth its
original speed. Moreover, I can make the images of the seesaw linger on the screen so that you can
see all the previous images as the seesaw goes through its travels. Now, because the camera takes
30 frames per second, those images will be separated from one another by a thirtieth of a second.
Here we go the same throw, slowed down to one tenth its original speed, with all the previous
images of the seesaws lingering on the screen. Seeing all those image of the seesaw is pretty, but
how do we make sense of the seesaw's motion?
It turns out that he seesaw is doing two things at once, it's translating and it's rotating. Translational
motion, its center of mass is traveling in the arc of a falling object as though it were a tiny ball
located at the center of mass, that's traveling in the arc that we're familiar with for falling balls. At
the same time the seesaw, which is an extended object, is rotating about its center of mass, its
natural pivot and it's doing these two things at once, the translation motion of a falling object
located right at its center of mass, and the rotational object motion of an extended object rotating
about its natural pivot, its center of mass.
I'm going to show you that same video again, same throw, once again, at one tenth normal speed
with all the previous images of the seesaw board. In view, but this time, I'm going to show you the
arc of a falling object, and I picked the arc just right so that the seesaw's center of mass will travel
along that arc, as the seesaw rotates about its own center of mass, located on that arc.
A seesaw has its center of mass located pretty much in its geometrical center. So, the motion we
saw had the center of the board travelling in the arc of a falling object as the rest of the board
rotated about its geometrical center but not all objects have their centers of mass at their
geometrical centers.
For example, a mallet, nearly all of the mass of this mallet is in its head. The handle is almost
nothing. So, when I throw this mallet the head will travel in the arc of a falling object because the
center of mass is almost dead center in that head. So you'll see that center of mass travel in the arc,
and that's the head at the same time, the handle which is almost an insignificant contribution of
mass, will rotate about the center of mass, and the arc will look a little different.
So, now I'm going to throw the mallet. Here we go. This rubber mallet has most of its mass in its
head. Once again, that was very quick. So I'm going to show you the same throw but this time, I'm
going to slow it down to one tenth its original speed and I'm going to let all the previous images of
the mallet linger on the screen. So you can watch the path the mallet takes and its orientation while
it's taking that path.
It's already pretty obvious that the mallet is following the arc of a falling object as it's rotating but
just to make that crystal clear. I'm going to show you the same throw again, one-tenth its original
speed, with all the images of the mallet lingering on the screen, but this time I am going show the
arc of a falling object that travels along in the path taken by the mallet's center of mass. So you see,
when you throw something, and it becomes a falling object. That is, it's experiencing only one force,
its weight its motion is actually fairly simple. The object's center of mass travels in the arc of a falling
object as though it were a simple thing like a falling ball at the same time, the rest of the object may
be rotating about that center of mass the object's natural pivot. So the object is doing two things at
once it's translating in the arc of a falling object as it's rotating in the manner of an object that's just
simply free to rotate about its own natural pivot is center of mass.
This may look like an ordinary beach ball, but it's not. Watch how it moves. It's hard to catch. Why
does this beach ball move in such a crazy manner? This beach ball has its center of mass located far
from its geometrical center. There's a container over here on the side of the beach ball that's full of
water so that most of the mass of the ball is located here where I can touch. As a result, the center
of mass is here on the surface of the ball and when I throw the ball, and it becomes a falling object,
it's that center of mass that travels in the arc of a falling object the rest of the ball comes along for
the ride and it rotates about its center of mass, it's natural pivot and therefore about one surface,
one side of the ball.
So that wobbly motion you're seeing is the ball rotating about the side of the ball, it's natural pivot
where the center of mass is located. You can begin to locate a small object's center of mass by
setting it on a surface to support its weight and then giving it a spin. It naturally spins about its
center of mass. So, what I can say in, for this basketball is that the center of mass of the basketball
lies somewhere on this rotational axis. It's spinning about a line, passing from top to bottom of the
ball, and the center of mass is located on that line.
I can't tell you for sure where along that line is unless I rotate the ball and spin it again, rotate the
ball and spin it again, but eventually, I could pin down the fact that for a basketball, the center of
mass is pretty much dead center in the geometrical center. That's true of a basketball but not so true
for knife.
How do you find the center of mass of a knife? Give it a spin. It's right about there; it's somewhere,
that's the point that's staying put as it spins. It's spinning about that point. So I can tell you the
center of mass is somewhere between my two fingers. To pin it down further, I'd have to spin the
knife about another axis. Can I do it? Ooh. I can so now I've really pinned down the center of mass of
the knife. It's really right there in the middle of this metal piece.
Well, for a seesaw, you do the same thing so here's a seesaw board. You give it a spin. The point
that's trying to stay put is right about here. So that's the center of mass somewhere between my
fingers and that's where the pivot goes when you make the seesaw into a real rideable seesaw and I
can pull one of those up, here it is. The rideable seesaw, if you're very, very small is supported right
at its center of mass, and therefore pivots naturally about that point. So we're supporting it right at
its center of mass, and allowing it to undergo rotational motion about its own natural pivot.
When examining rotational motion, it's technically necessary to specify the center of rotation. That
is, the point about which all the physical quantities of rotational motion are defined. We're free to
choose that center of rotation but some choices are better than others. For example, if I'm rotating
like this and we want to describe my rotation as simply as possible using the physical quantities of
rotational motion, the most obvious choice for a center of rotation about which to build our
language is my center of mass, 'cause that's the point about which I'm pivoting. So, in this case, we
define the center of rotation as located at my center of mass.
So for example, my angle of velocity now is about 90 degrees per second up remember the right
hand rule, about my center of mass. So about my center of mass, is pinning down the center of
rotation about which our language is built. But if I'm rotating not about my center of mass, but about
my thumb, watch this. Here we go. I can pivot about things other than my center of mass. I need
help to do that. But I can do it, and I'm now pivoting about my thumb. So, it makes good sense to
define that as our center of rotation and to say that I am currently rotating. My angular velocity is
about 90 degrees again, up, about my thumb that's the center of rotation.
Well by now I hope you can see that while choosing a center of rotation is necessary to define the
physical quantities of rotational motion, stating that center of rotation explicitly every time you use
one of those physical quantities is a nuisance, and I'm going to stop doing it. Instead I'm going to
assume that the center rotation that we have in mind is obvious, unless it's not, in which case I will
say it. So, for this case of a seesaw mounted with a pivot passing right through its center mass, its
own natural pivot that is an obvious choice for our center of rotation. Right here in the middle of the
board where the pivot passes through the center of that board the board's center mass. That's the
obvious choice for the center of rotation and it will assume for the remainder of this story, that
every physical quantity of rotation is defined about that point so instead of adding language now to
all our physical quantities of rotation for the seesaw. Let's put some riders on it and that's the job for
the next video.
Part 3.
Why does a lone seesaw ride plummet to the ground? The answer to that question is that the lone
rider produces a torque on the seesaw and causes it to undergo angular acceleration. The seesaw
rotates such that the rider descends toward the ground and hits it. There are several ways of
examining this situation. So I'm going to follow the path that I think is most straightforward. The
rider's weight gives rise to a torque on the seesaw and since the seesaw is no longer rotationally
inertial, its angular velocity is no longer constant. Instead, that angular velocity changes with time,
and the rider soon plummets to the ground but those observations give rise to two more questions.
How does the seesaw respond to torques, and what is the origin of this particular torque? So let me
start by looking at the seesaws response to torques. When the seesaw is experiencing no outside
torques it's covered by Newton's First Law of Rotational Motion. So it rotates at constant angular
velocity, like this but once there is torque acting on the seesaw, the seesaw is no longer covered by
Newton's First Law of Rotational Motion and its angular velocity is no longer constant, instead, its
angular velocity begins to change with time.
The seesaw undergoes angular acceleration, angular acceleration is another vector physical quantity
of rotational motion, and it is the rate at which angular velocity is changing with time. Like ordinary
acceleration, translational acceleration, it's a subtle quantity. It's hard to see. You have to look
carefully. It takes three glances to see acceleration and it takes three glances to see angle
acceleration. So I'm going to illustrate angle acceleration with my body and hope that you can see it
happening.
So here we go, let me start, motionless, rotationally motionless. That is my angle velocity is zero. If I
change my angular velocity, during the time over which that angular velocity's changing, I am
undergoing angular acceleration. So here we go. I'm going to undergo angular acceleration, and then
I'm going to stop undergoing angular acceleration, and watch what happens. Here goes the angular
acceleration; it's going to be up, right hand rule again. I'm going to rotate. Here we go, okay, I did it,
it’s over I am now coasting rotationally, at constant angular velocity but when I first got started I was
undergoing angular acceleration. If I don't undergo angular acceleration again, I'm going to keep
spinning here forever, and this will make me very dizzy. So I'm going to undergo angular acceleration
downward in a moment. Ready? Get set, whoop, there I did it.
So during those two moments when I changed, extended moments. When I changed my angular
velocity I did it by way of angular acceleration. I'll show it to you again. I'm going to do an
intersection upward for about a quarter of a second and then, I'm going to do an intersection
downward for about a quarter of a second and come to stop. Ready? There. Now, I'm coasting and
now. So, the angular acceleration portion of that situation was during the changes in my angular
velocity.
Coming back to the seesaw then, the angular acceleration is absent now. Ready, get set, there it is.
There were a lot of angular accelerations there at the bottom, but they initially kicked in, the first
angular accelerations kicked in when the rider got on the seesaw. Right now, so we see, a torque
causes a seesaw to undergo an angular acceleration. But what if there's more than one torque,
acting on that seesaw at the same time?
In that case, those torques add together to become a net torque and the net torque is what causes
the angular acceleration. So for example if I've got two riders hopping on to the seesaw at once, the
seesaw can't respond with several separate angular accelerations at the same time, it only has one.
Instead it responds to the net torque, produced by those two riders. Well, if net torque causes
angular acceleration, the question comes up is how much angular acceleration?
It turns out that the seesaw's angular acceleration is proportional to the net torque acting on it. So if
a gently net torque acts on a seesaw. It undergoes a small angular acceleration but if a large net
torque acts on the seesaw it undergoes a large angular acceleration but there's a second factor
involved in determining the seesaw's angular acceleration, the seesaw's rotational mass.
Rotational mass is the measure of an object's rotational inertia, its resistance to undergoing angular
acceleration. Now traditionally that physical quantity is called moment of inertia and it has various
complexities to it. They're beyond the scope of our little discussion here not relevant really to
seesaws. So rather than trying to have you remember a name like moment of inertia, with its
complexities, I'll make our lives simpler by simply calling it rotational mass that conveys the
characteristic that it's a mass like thing, it's a resistance to acceleration of some form, in this case,
rotational acceleration.
So, this seesaw has a certain rotational mass, a certain resistance to angular acceleration. So if I
exert a certain torque on it. It responds with a specific angular acceleration and I go back to, to
putting a rider. So, my little rubber stopper rider is here. If I put a certain rider on this seesaw and let
it undergo angular acceleration, well, it undergoes rather rapid angular acceleration, and down goes
the rider. But I can increase the rotational mass of this seesaw by adding a second board. When I do
this, I'm increasing the rotational inertia of the seesaw.
Try to glue it and tape it in place and now, it's less responsive to the same torques as before. In this
case, if I put 1 rider on it undergoes angular acceleration, but not as much. Overall, the seesaw's
angular acceleration is proportional to the net torque acting on the seesaw and also inversely
proportional to the seesaw's rotational mass.
Those two observations form the basis for Newton's Second Law of Rotational Motion, which states
that an object's angular acceleration is equal to the net torque acting on that object divided by that
object's rotational mass.
I'm going to ask a question about angular acceleration, but I'm going to do it in the context of a
bicycle wheel that I can hold in my hands. At present the bicycle wheel is motionless and I'm going to
do three things to it, in sequence. First, I'm going to start it spinning. Second, I'm going to turn the
wheel all the way around like this, so it's spinning in the opposite direction and now as a third thing,
I'm going to stop it from spinning. The question is, during which of those 3 steps was the bicycle
wheel undergoing non- zero angular acceleration? All three steps involved angular acceleration of
the wheel. When I started it spinning it went from having an angular velocity of zero to having an
angular velocity toward you.
Remember the right hand rule here when I pivoted it around like this. I reverse the direction of the
wheels angular velocity from toward you to toward me. That's angular acceleration. I had to, to
make the wheel undergo angular acceleration to reverse its direction of, of rotation and finally,
when I stop the wheel from spinning, I take its angular velocity from toward me to zero. So, all three
steps require the wheel to undergo angular acceleration.
We see that a seesaw responds to a net torque by undergoing angular acceleration. Why then does
a low rider sitting at the end of the seesaw board. Exert a torque on that board. After all, the rider
has a weight, which is a force and if I hold the seesaw in place now, the rider and seesaw are pushing
on each other with forces.
The seesaw to support the rider's weight and the rider pushing back on the seesaw in response it is
all forces out here. Where does the torque come from? Well it turns out that forces and torques are
related and that a force can produce a torque and a torque can produce a force.
To see how that all works, let's go experiment with a door because doors are a wonderful example
of rotational motion and the use of a force to produce a torque. So here I am outside the physics
building, opening and closing doors in a light rain. The things we do for science doors are a nice
example of rotational motion after all, they don't go anywhere. They simply rotate open and closed
about their hinges they have all of the characteristics we've come to expect of rotating objects. They
have angular positions they have angular velocities, and they even have angular accelerations but
that brings us to the issue at hand, which is when you open a door you do it by exerting a force on
the door handle and yet the door undergoes angular acceleration.
Well, angular accelerations are produced by torques, not by forces. So how is it that a force exerted
on the door handle produces a torque on the door? To show you how that works, I first have to
define a center of rotation. Now, the obvious place to put the center of rotation is somewhere along
the hinge line because that's the line about which all the door's rotation occurs. But I have to be
more specific than that because center of rotation is actually a point not a line. So I'm going to put
the center of rotation in line horizontally with the door handle for reasons that we'll come too
eventually and that's going to be our center rotation right there on the hinge line aligned nicely with
the door handle.
Having done that then, let's look at ways in which not to produce a torque about that center
rotation. Starting with a force, so these are all the unsuccessful ways to try to open a door, some of
which you may have encountered by accident. So, first unsuccessful way to produce a torque,
starting the force, is to push the door handle toward the center of rotation. So I'm pushing right at
that center of rotation. No effect, I'm producing no torque. How about reversing my force? Instead
of pushing toward the center of rotation, let me pull away from the center of rotation. Also no luck
doesn't do anything so we see that pushing toward or away from the center of rotation is
unsuccessful.
How about pushing on the center of rotation let me come over here to the center of rotation and
push right on it. I'll try to pull right on it, all the kids of forces, none of it works. So you can't move
the door by exerting your force toward, away from or on the center of rotation. Okay. Now it's time
to be successful. We can only take so much frustration so now I'm going to exert a force out here on
the door handle, not toward or away from the center of rotation, but at right angles to a special line.
It's actually a vector, it's called the lever arm and this is what the lever arm is. The lever arm is going
to be the vector that extends from the center of rotation to the point at which I'm going to exert my
force namely on the door handle.
So there is a vector that points along this line to this point here it has a length of about one meter
like that and it's direction is exactly to your left and I'm going to exert my force not along that vector
or, you know, with it or against it, but at right angles to it, perpendicular to that lever arm. I'm going
to exert my force toward you, and watch what happens the door undergoes angular acceleration
and begins to rotate open.
That is how to produce a torque starting with a force if you'll exert your force at a lever arm from the
center rotation, that is the vector that extends from a center of rotation to where you exert your
force and you exert your force at perpendicular to that lever arm. Then you produce a torque and
the torque has a specific direction. Its direction follows yet another right hand rule. If you take your
right hand and extend your index finger in the direction of the lever arm towards your left, right now
and then you sweep the index finger of your right hand in the direction of the force which is towards
you, so that's the sweep.
Look what my thumb is doing my thumb is pointing up. That is the direction of the torque I produced
in pulling toward you with, on the door handle. The lever arm is that direction. The force is toward
you. The torque I exert is up, and so it causes upward angular acceleration in the door which swings
open.
Now the amount of that torque that I produce depends on two things. One is how much force I
exert. The torque is proportional to the force I exert. A gentle force produces a gentle torque. A big
force produces a big torque. So, that's the first observation. Second observation is the length of the
lever arm matters. The torque I produce is proportional to the length of that lever arm. Here I have a
lever arm about that long, but if I go inside and I push near the hinges, I can make the lever arm very
short; and watch what happens. That was hard, so, I'm exerting my force here, very close to the
pivot therefore at a very short lever arm, and I'm obtaining a very small torque until I really crank up
my force.
We can combine these observations to relate the force to the torque it produces, quantitatively.
That torque is equal to the lever arm times the force. Where only the component of force that is
perpendicular to the lever arm is included and where the torque is in the direction determined by
the right hand rule. So in this case if the lever arm is pointing to your left and the force is pointing
toward you the torque is up.
Now this door is complicated because it has a closing mechanism, like many doors. It has a system to
try and keep that door closed when you leave it alone. So it's not free to exhibit rotational inertia
and has all kinds of its own trouble and I had to overcome that resistance the mechanism trying to
keep the door closed. That's a lot easier to overcome if I'm out here with a big lever arm. I can exert
relatively gentle force on the door handle and get the door to open despite the closing mechanism.
If I try to push very close to the hinges, that closing mechanism is hard to beat and you may have
had this experience that if you go to a door that isn't very well labelled and you have to push it open,
you can't tell which side of the door has the hinges. If you push near the hinge side of the door, the
door doesn't open very easily. It's very resistant to opening because you're producing so little torque
with your force. You need to go out to the other side of the door where you have a big lever arm to
work with and therefore can really create a lot of torque with a small force.
To produce a torque with a force then, all we need is a lever arm for an unconstrained seesaw like
this one that can rotate in any possible direction, the options are limitless. I'm going to choose as our
center rotation The seesaw's center of mass just for convenience here, right about there and now,
let me show you a couple of torques. Things you've seen before, maybe some you haven't. If I come
out here to a lever arm Towards your left and then I push down with my force, which is at right
angles to that lever arm or, in fact I don't have to be perfectly right angles I have to just, just have to
have some component that's at right angles to the lever arm and I push down, I cause an angular
acceleration toward you. Right hand rule again.
On the other hand, if I come out to a lever arm, same lever arm but I push my force towards you.
Watch what happens. I cause angular acceleration up and if I come out to a level arm toward you,
very short one but its there, and push down, I caused angular acceleration to your right. I flipped the
board like that. Well. This is exciting, but very complicated. There are too many options with our
unconstrained seesaw. So fortunately, we're going to focus on a constrained seesaw, one that has a
pivot shot through the center that forces it to, to rotate in a very simple manner. This seesaw board
down here cannot do this kind of rotation, or this kind of rotation and so, it operates in a more
simple fashion, like this and it still exhibits the same sorts of behaviours.
To produce a torque on this seesaw, I come out to a lever arm and push at right angles or partly at
right angles to that lever arm down and I cause angular acceleration toward you because my torque
was toward you. We've seen how to produce torques with forces in the context of doors, in the
context of seesaws but what about another important household use of torques, putting in or taking
out screws or bolts? You rotate a bolt into place and you rotate it the other direction to take it out of
place. Well suppose you have a big bolt like this. That has rusted in place, and you're trying to get it
out but it won't turn when you grab it with your hand and try to twist. You need more torque. So, in
that case you get a wrench.
This is a device, and you will have to figure out how it works. This is a device that when you put it on
the head of the, the bolt it allows you to produce more torque and by now, you should be thinking
about how this works but what if this is really, really stuck? And you need a bigger wrench? Well,
that's already a pretty big wrench, you think and you're probably thinking that I'm going to go over
and get this wrench to show you the bigger wrench but no I have in mind this wrench and so we take
this wrench, put it on our stuck bolt and low and behold, it's a lot easier.
To produce a large torque on that bolt and remove it from wherever it's stuck. The question then, is
this, why is using this larger wrench more effective it, why does it enable you to remove that bolt
when this wrench didn't to the job? This wrench has a longer handle, and it provides a longer lever
arm with which to produce a torque using a force. So when you come out here to the end of the
handle and push perpendicular to that handle and therefore perpendicular to the lever arm your
force produces a larger torque as compared to this wrench.
There's just not as much length here to work with. It's got a shorter lever arm and so when you push
on the handle of this wrench with that shorter lever arm your force produces less torque so we see,
whenever a lone rider goes out to a lever arm on the seesaw and sits down, the rider's weight gives
rise to a torque on the seesaw that causes it to undergo angular acceleration, such that the rider
ends up pretty much sitting on the ground.
The rider's weight is a force and that weight causes the rider to push on the seesaw with a force, but
the force acting at a lever arm from the center rotation produces the torque that causes all this to
happen. Pretty much the only place a single low rider can sit or stand and not produce a torque on
the seesaw is exactly on top of the pivot, which is kind of an interesting place to stand and I must
admit to having done that myself, from time to time but it's much more fun to have two riders on a
seesaw, and that is the subject for the next video.
Part 4.
Why do the rider’s weight and positions affect the seesaw's motion? The short answer to that
question is that they affect the net torque on the seesaw, and therefore the seesaw's angular
acceleration. In most cases, the riders of a seesaw position themselves so the net torque on the
seesaw is zero or very nearly zero. As a result, the angular acceleration of the seesaw is either zero,
that is, it’s coasting rotationally or it's just got the smallest amount of angular acceleration.
Well, that then requires a longer explanation how does that come about? You put riders on the
seesaw why don't they produce enormous net torques? So we know that if we put one rider on the
seesaw because of the rider's weight, the rider pushes down on the seesaw over here, on your left.
That's to the lever arm from the pivot it produces a torque and boom, the seesaw undergoes rapid
angular acceleration such that the rider drops to the ground.
But what if we put two riders on the seesaw simultaneously? And what I'm going to do is I'm going
to position them very carefully and look. The seesaw is experiencing very little angular acceleration,
so the net torque on it is either zero or very near zero. How did that happen? Aren't these riders
producing big torques on the seesaw? There are two of them, glad you asked that question.
Here's the story, this is now the longer explanation to the question that's prompted this video. That
rider because the rider's weight is pushing down on the board, over here to your left, the lever arm
that riders force is using to produce a torque, points towards your left. Here it is, and using the right
hand rule now, we can see the direction of the torque produced by that rider. The torque, we follow
the lever arm and we roll, I roll my finger down in the direction of the force and my thumb is
pointing toward you. That is the direction of a torque, produced by this seesaw rider.
Let's come over to this seesaw rider, I need my right hand again. I can't swap hands or I'll get the
wrong answer. So, that rider by virtue of his or her weight is pushing down on the seesaw board. The
lever arm with which that rider is producing a torque now, points to your right. So, there it goes and
now I turn my, my index finger in the direction of the force and lo and behold, the torque produced
by that rider is away from you these torques are in opposite directions, this rider is producing a
torque toward you while the other rider is producing a torque away from you. When we add
torques, which are the two torques acting on this seesaw, they sum to zero or very nearly zero and
that's how it is that when I let go of this board and allow it to show you its angular acceleration
there's almost zero.
If there is a little bit, and there is, I can adjust the distance of one of the riders from the pivot. This
riders producing a little too much torque and now I move it toward the pivot, still a little too much
torque. So I move it a little closer to the pivot, and now that rider's producing almost just the right
torque. Let me move the rider in a little closer and now this rider's producing too little torque. I have
been I have adjusted the rider's positions that are the lever arms they're using, to show you that we
can go all the way from. Almost perfect balance, and I'll talk about balance in a minute, with that
rider dominating a little bit, to almost perfect balance with that rider dominating a little bit and
everything in between, including in principle, perfect balance where there's zero net torque on the
seesaw.
Actually, balance is an interesting concept. The balance that we talk about in the context of a
seesaw, and many other objects that teeter back and forth like a seesaw, is a situation where gravity
produces no torque on the object. So, when this seesaw is balanced it’s experiencing zero torque
due to gravity. I can come in and, and change things. I'm, I'm here, and very carefully adjusting
positions in order to try get this situation. This seesaw is almost perfectly balanced meaning it's
experiencing almost zero torque due to gravity and that is the normal situation for a seesaw, and
riders.
They like that situation because a balanced seesaw is free of torque, this assumes nothing else is
exerting torques on it, and it will turn at constant angular velocity. It is an object that obeys
Newton's First Law of Rotational Motion and it's not wobbling, it's rigid, assuming the riders don't
change their positions and therefore, in the absence of any torque, and there's no gravitational
torque on a balanced seesaw, it turns a constant negative velocity.
So, the, the reason the riders have to adjust their positions very carefully and it, and their weights
are important as well, is because they are trying to sum their torques to zero, and how they place
themselves matters. If, for example, the riders have, essentially identical weights, and these two
riders do they need to sit at equal distances from the pivot because the torque they produce, after
all, is the product of the force they exert on the seesaw times the lever arm they have to work with.
There are some subtleties in here with regard to the angles involved between the lever arm and the
force but in this situation we can really ignore those. The forces and lever arms are essentially at
right angles to each other, and our lives are simple. So these two identical riders, seated at identical
distances from the pivot, produce identical but oppositely directed torques, and the seesaw
balances. What if we have a heavier rider around, though? So instead of this rider, we bring up one
and this is made of steel. This is heavy stuff. So I'm going to put this rider in. If I put this rider out at
the same distance as the rider on your right it completely dominates, and I run the risk of tossing this
rider.
This is one of the flaws with seesaws, is it's easy for one of the riders to become an astronaut, when
a very heavy rider gets on the seesaw and does that to it but this rider cannot sit that far out from
the pivot. Too much lever arm for a large force, and therefore this rider dominates it, and produces a
torque that, that one cannot compensate with. Comp-, compensate for. So I have to bring the
heavier rider in close. How close? Pretty close. I'm almost at balance. There we are this is balance.
Alright it's as close as I'm going to get and, again, the net torque on the seesaw is zero, or pretty
close to zero and, you'll notice that, that now the lever arm with which this rider is producing the
torque, is quite short because this one weighs a lot more, so a big downward force, short lever arm
and that is balancing, or cancelling out the torque due to this one, which is in the opposite direction,
but it's produced by a by a smaller force acting at a larger lever arm.
So this is common in, in playing on a seesaw when you have two children of, of significantly different
weights. They have to sit at different distances from the pivot. The heavier child sits close to produce
a certain torque, and the lighter child sits far from the pivot to produce an equal amount of torque
but in the opposite direction. Well, that brings us to a question and the question is this. Can two
riders, and we can adjust their weights as you like. Ever sit on the same side of the seesaw, and still
balance the seesaw?
Two riders cannot sit on the same side of the seesaw, and expect the seesaw to balance. That's
because those two riders produce torques in the same direction about the pivot. Their forces are in
the same direction, their lever arms are in the same direction, so their torques are in the same
direction and when you add those torques, they sum to something larger than each one individually.
So you get a lot of torque on the seesaw, and it’s terribly unbalanced. In order to balance the
seesaw, the two riders, or however many you want to put on the seesaw, have to distribute
themselves on opposite sides of the pivot so that their, their torques cancel one another and
eventually, if you do it all right, they sum to zero and the seesaw is rotationally inertial. It has zero
net torque on it and no angular acceleration. It coasts rotationally. There are two seemingly different
ways to think about the balanced see saw situation.
The first way is the way we've been doing. Where this rider produces a torque, that rider produces a
torque, the two torques sum to zero and as a result the seesaw experience zero torque due to
gravity, It's balanced. The second way to think about this situation is in terms of a concept known as
the center of gravity. Now center of gravity is the effective location of an object's weight I have one,
you have one and these riders have one. Even the seesaw board has one.
This rider's center of gravity that is where its effective weight is located it’s pretty much at its center,
same with that rider. The seesaw board's center of gravity, the location, the effective location of its
weight, is at its middle. Right there and that might make you think that center of gravity, which is
here, and center of mass, which is here, are the same idea.
Center of mass, center of gravity, aren't they the same? They're not they happen to coincide for all
objects here near the earth's surface. Celestial objects violate this concept for complicated reasons
that I'll leave for another day. But small objects do have their center of gravity at the same locations
as their center of mass, but they're different concepts. Center of mass is the effective location of an
object's mass it's natural pivot. We watch centers of mass in action when I threw various wobbling
objects or sticks and so on through the air and you'd watch. The center of mass was traveling in the
arc of a falling object. That's the mass moving and the inertial properties of the object in play.
So, center of mass is all about inertia in motion center of gravity is about forces and its forces of
gravity it's got to do with gravity. If there's no gravity around, center of gravity means nothing. So it's
the effected location object's weight the fact that weight is proportional to mass here near the
earth's surface, means that center of gravity and center of mass share the same location. But they're
different concepts and so if you're dealing with the inertial aspects of an object, you're probably
paying attention to the center of mass. Mass. If you're dealing with the gravitational or weight
aspect of an object, you're probably dealing with center of gravity.
So, back to the situation here we have objects with various centers of gravity and that brings us to
an observation that this entire structure two riders in a seesaw is, we can consider it as a single
object. Where is its center of gravity? That composite object and it turns out that this overall object's
center of gravity is located right above that Pivot and its being pulled straight down, like the centers
of gravity are pulled straight down. They're gravity after all, right? The forces of gravity are toward
the center of the earth. It's being pulled straight down right towards the pivot, the center of rotation
and as we've seen before, forces that act toward the center of rotation produce no torque about the
center of rotation.
So, this seesaw is balanced for two, you know, in two ways you can think of it one is in terms of the
individual riders producing torques that sum to zero. The other way, which is kind of cool, is that the
riders and seesaw together have a center of gravity located vertically above the pivot and therefore,
the force of gravity acting on this entire structure acts right toward the pivot, and produces no
torque. It's along the lever arm and produces no torque.
So Annie and Megan here are riding a real seesaw, not one of the little things I have in my lab and
they're balanced right now. Can you show us this? It takes delicate adjustment, but Megan's
distance is just right from the pivot, the pivot's right here Andy's distance is just right from the pivot
They've adjusted it, so the net torque on this thing is, is as close to zero basically, as they can get it.
But this is a boring way to ride seesaws, if you just sit here balancing, I guess it's not too boring. It's
kind of exciting, trying to keep it balanced but they can unbalance it In order to rock back and forth
in one of two ways. They can either push on the ground with their feet. So Meagan, why don't you
push on the ground and that extra force produces another torque, which causes Annie to rotate
down. Now Annie can push down on the ground and cause Meagan to rotate down. So, they're
causing angular accelerations back and forth by exerting new torques on it.
The other way they can unbalance this Is by leaning so each one of them has a center of gravity
that's located somewhere sort of mid-body but if they lean, they can shift the location of their
center of gravity and therefore. Exactly where they're exerting the forces on the seesaw board, and
cause it again to experience a net torque so it undergoes angular acceleration. So, if you both lean
towards Annie, what happens?
It goes down, Annie goes down, because basically the lever arm With which she's working gets
longer and the one that Megan's working with gets shorter. So the torque is this way, toward me.
But how if we lean, everybody lean towards Megan, now the lever arms get longer and shorter in
the opposite direction and then that torque is toward you. So, they can rock back and forth, so, this
is how a seesaw works.
Okay, you guys can go at it, all right, here we go. Either way this is what makes seesaw fun right, is all
the adjustments of the torque so that you undergo angular acceleration in opposite directions, back
and forth. Seesaws are not the only structures in our world that need to balance. Mobile sculptures
do as well. This mobile sculpture is entitled, happy hanging hardware and I built it out of a torque
wrench, a ball peen hammer, and a metal file.
Amazingly enough, each of these components is rotationally inertial. You don't see any of them
undergoing angular acceleration, after all and that brings us to a question. For all of the components
of a mobile structure, to be rotationally inertial, how must those components be arranged? Each
component of this mobile structure has its center of gravity at or below the point at which that
component is supported in effect, the pivot, about which that component could rotate.
This is actually a relatively complicated concept though because there are three components here
which aren't the individual tools. First component, the, the simplest, is the, is the file that file has its
center or gravity at or below this support point which is the loop of string going around it. That's the
pivot about which the file can rotate and so, the file has its center or gravity at or below that pivot
and therefore. Gravity produces no torque on the file, its rotational inertial.
So far, so good the ball peen hammer isn't an object by itself it's not the component by itself. Rather,
the ball peen hammer and the file together are the next component of this mobile and that
combined object. Ball peen hammer and file has its overall center of gravity at or below its support
point. This loop of string and lastly, the torque wrench and everything below it has its combined
center of gravity at or below this support point the support point that is acting on the torque
wrench.
So, each of these components, the file and the hammer and file and the wrench, hammer and file,
each of those components has it's center of gravity directly below its support and therefore, gravity
pulling down on the center of gravity produces no torque on that component about its pivot. It
doesn't undergo any angular acceleration then due to gravity, it's balanced and so the file is
balanced. The hammer and file are balanced. The wrench, hammer, and file are balanced. The entire
mobile then, is balanced, and it's all rotationally inertial.
So we see that objects that can rock or tip are only rotationally inertial if you balance them carefully.
Sometimes that's what you want, like with a mobile. Sometimes that's almost what you want, like
with a seesaw, where getting it perfectly balanced is interesting, but kind of unexciting in the long
run and you want to unbalance it a little bit to get some action happening. We'll talk more about
balance in the episode on bicycles, but for now. It's clear that in the context of seesaws, balance and
near balance are the name of the game.
Part 5.
Why do the riders' distances from the pivot affect the seesaw's responsiveness? The answer to that
question is that the farther the rider’s masses are from the pivot, the greater the seesaw's overall
rotational mass and the slower its angular accelerations.
Two riders can balance the seesaw in a variety of ways. To begin with, they can go to the ends of the
board and adjust their distances from the pivot carefully until it balances. That is, until it experiences
zero overall torque due to gravity. I mean I'm pretty much there, balanced seesaw. But they can also
come in close to the pivot and sit like this, with much smaller lever arms to work with now so that
they're producing much small torques as individuals but once again, those two torques sum to zero
and there's zero overall torque due to gravity on this seesaw.
There are a variety of ways to balance the seesaw and you might think that there's no significant
difference between those choices, but that's not true, there is a significant difference. The farther
these two riders sit from that central pivot, and therefore the axis of rotation, the greater the
seesaw's overall rotational mass. Now, rotational mass is not something completely independent of
ordinary mass they're related, just as forces and torques are related. Every portion of the seesaw's
ordinary mass contributes to the seesaw's rotational mass and the amount of that contribution
depends on where the portion of ordinary mass is. More specifically, on how far that portion of
ordinary mass is from the axis of rotation and that dependence on axis of rotation is a strong one.
Small, modest changes in distance from the axis of rotation can lead to large changes in the
rotational mass contribution. The amount of rotational mass contributed by a portion of ordinary
mass is equal to the ordinary mass itself times the square of the distance between that portion of
ordinary mass and the axis of rotation. So, taking a small portion here and doubling its distance from
the axis of rotation doesn't just double its contribution to the rotational mass, it quadruples it.
That means that for the riders, when they sit in close, and their distance from that central pivot and
center of rotation, axis of rotation, is small, they contribute very little to its rotational mass, the
overall rotational mass of the seesaw and riders. Even though these, these riders have large masses,
they're too close to the axis of rotation to contribute very strongly to its rotational mass. But if they
go out like this, to a large distance, well then their contribution to rotational mass is huge.
They might be only ten times as far away from the pivot as before from that axis of rotation but an
increase of distance by a factor of ten is an increase in contribution to rotational mass of ten times
ten, or ten squared, which is 100. So the rotational mass contribution of these riders could easily be
100 times that of these riders, it's a big effect.
To see how big, I've made these two rods that look the same, and have the same the masses. They
contain the same materials actually but the difference is that in one of these rods, the mass is all in
the middle, near my hand, under my hand, hidden from view and in the other bar, all of the mass is
far from my hand, at the ends of the bar. So, same mass, but in this case it's moved way out far from
the center of rotation, which will be here in the middle bar and the rotational mass of this bar is
something like 30, 40 times that of this one, big difference.
To see that difference, since you can't hold the bars, let's go get some help. Annie and Megan are
going to help me here with these two bars. Now, these bars have the same masses and the same
weights. So you could check that out. Just compare the weights, do they feel the same? So, just by
weighing them in your hands you can't tell a difference between these two bars but that doesn't
mean they're identical.
So the difference is going to be subtle, and this difference will show up maybe when you begin to try
to rock them back and forth. So I want, Annie to grab it in the middle of the bar, Megan same thing
and I'm going to count to three, and on the number three. I want the two of you to rock it back and
forth as fast as you can. That is, make it undergo angular acceleration first one way and then the
other, back and forth as fast as you can.
So Annie's having no trouble here, and Meagan's really lagging behind. Must be weak today, right,
forgot to eat your breakfast, okay, now swap bars. Now, I'll count again. One, two, three, miraculous
change, something's different about these two bars. What do you think's different about the bars?
They have the same mass, what's different, maybe the distribution of the mass within the bar? So
the distribution of the mass in, within the bar is different. Where is the mass in your bar, right now?
In the center, so your bar has almost all of the mass in your hand as a result, the moment of inertia,
or the rotational mass of this bar is very small. It's very easy to make this bar undergo angular
salvation how about yours, Annie. Where do you think the mass is located? I mean they must be at
the end so all the mass in this bar is at the ends where it contributes enormously to rotational mass.
So they have the same mass, it's just distributed differently. In Annie's bar it's at the ends, in
Megan's bar it's in the middle and they behave totally differently when you try to rock them back
and forth. Yeah, it's a pretty dramatic difference. Unfortunately you can't try it out but if you ever
come by, test out these bars and see how different they are.
So you see, if you place an object's ordinary mass far from its axis of rotation, that object can have a
surprisingly large rotational mass. Now, the distance involved here is between ordinary mass and the
rotational axis and you're often free to choose an object's rotational axis. If you do that, and if you
change your choice, you may well change the object's rotational mass. That ability to change an
object's rotational mass is why rotational motion is so complicated to calculate quantitatively and
that's a fact the keeps first year physics graduate students rather busy. Its hard work, I want to give
you a taste of the issues without trying to overwhelm you with them but look at this rod. This is the
rod that's very hard to wobble back and forth like this because it has an enormous rotational mass
when you twist it back and forth about this axis the one pointing toward you through my hand or
away from you through my hand about that axis, gigantic rotational mass.
But what about this axis in which I'm twisting it back and forth like, like a drill or a screwdriver? It's
easy. This direction has almost no rotational mass. That's because all the portions of ordinary mass
are very close to this spindle-like axis about which I'm twisting it. So, this, this rod here, has two very
different rotational masses, a huge one when you do this motion and a tiny one when you do this
motion.
Now this is you know a fun and games rod but something you're more familiar with is perhaps a
tennis racket. The tennis racket is a classic example of something that has three particularly
important rotational masses. Its smallest rotational mass is for this rotation about its top bottom axis
right now. This motion, most of the mass is pretty close to that axis, the spindle about which I'm
twisting it, and therefore it has a relatively small rotational mass. The next larger rotational mass is
for this motion. Sometimes referred to as the frying pan motion, when you're, you're flipping
pancakes.
So this is the intermediate rotational mass and the biggest rotational mass is for this rotation. In
between these motions, life is extremely complicated and it's beyond the scope of this class as
something I don't like to deal with anymore. I've done it. Been there, done that, I'll leave it. But
these three distinct rotational masses, this small one, bigger one, biggest one, give the motion of a
tennis racket or anything shaped like a tennis racket the rotational motion's quite complicated. To
make things even worse, these are all motions about the center of mass. These are all rotations in
which the center of mass stays put. What if you shift the rotation, so that you don't care about the
center of mass of the tennis racket?
For example, when you're swinging a tennis racket about your shoulder in that case, you're shifting
the mass of the tennis racket even farther from the center of rotation, the axis about which you're
spinning it, and creating an even larger rotational mass for the tennis racket. So the bottom line with
all of this is rotational mass depends on your choice of axis of rotation. We're finally ready for the
question I asked you to think about in the introduction of this episode.
To remind you, that question asked if you and a child half your height lean out over a swimming pool
at the same angle and let go at the same moment, which of the two of you will hit the water first?
Despite a fair amount of rotational physics under our belts, that remains a challenging question. So
before I ask it, and leave you free to answer it, I want to give you a little more background. Get you
all prepped for this question.
First, what's the big picture issue? What is going to determine who hits first? It's going to be angular
acceleration. The one of you that undergoes the fastest angular acceleration will tip, will develop the
fastest angular velocity, will tip over the fastest and will hit the water first. So look for big angular
acceleration. Second, what is the axis of rotation about which the two of you are going to be
rotating? It's not your centers of mass; it's going to be here at your feet. That leaves two more
issues. One is the cause of angular acceleration and the other is the resistance to angular
acceleration.
The cause is the neck torque on you, the resistance is your rotational mass and let's look at each one
individually, first torque. The torque is due to gravity, and to make our lives simple, let's compare
the torques about this, about your feet, that's the axis rotation here, for you and the child. Now, I've
made life very simple by using exactly the same board material. One is just half as long as the other
and this makes, you know. This has all the physics in it, but none of the details. Life is easier.
So, you have twice the weight of the child, that's no surprise and that weight effectively acts at your
center of gravity, which is twice as far, it's right here in the middle its twice as far from the axis of
rotation as for the child. So, you're experiencing four times the gravitational torque of the child. You
have twice the weight acting at twice the lever arm. 2 times 2 are 4. Right? 4 times the torque.
That's the cause of angular acceleration. How about the resistance to angular acceleration, the
rotational mass? Well, as compared to the child who has half the mass here, distributed around here
with the center of mass being about there. You have twice the mass distributed here and there,
there's a center of mass here. The center of mass has moved out by a factor of two. You have twice
as much mass that is on average at twice the distance from the axis of rotation, namely your feet.
Well, remember that the distance involved here in calculating it, the contribution of mass depends,
not on distance, but on distance squared.
So the rotational mass that you have is eight times that of the child. Twice as much mass, at twice
the distance and you square the distance. So it's two times two times two, that's eight. This has you
have eight times the rotational mass of the child. With that as background now, answer the
question. Which of the two of you tips over fastest and hits the water first? The child undergoes
greater angular acceleration, develops a bigger angular velocity and hits the water first.
If you haven't already tried this experimentally, give it a go. All you need is two sticks, one twice as
long as the other. I can show you what you'll see when you try it. Here are you, here the child and
we'll put you both on your feet and tip you to the same angle, and then let go. 3, 2, 1. No question,
the child reached the water first. It's a battle between torque and rotational mass in both cases
about your feet. You have four times the gravitational torque acting on you as, as the child has. So
there's four times as much twist trying to propel angular acceleration but you have eight times the
rotational mass as the child, resisting that angular acceleration.
Four times more impetus, eight times more resistance you get only half the angular acceleration of
the child. So, the child undergoes twice you angular acceleration and just goes through the whole
rotational motion faster and wins the race to the water. Rotational motion clearly has some subtle
complications, like a single object having more than one rotational mass depending on your choice
of axis of rotation.
But let's leave all those complications for the experts. I chose seesaws as the topic for this episode
because seesaws are comparatively simple. That pivot fixes the axis' rotation so that the seesaw can
only rotate in one fashion and in general, it only has one rotational mass. Things are simple.
Nonetheless, the seesaw exhibits most of the issues of rotational motion. Or at least the ones that I
want to talk about and try to convey to you. I've already done that now. I've shown you most of
what happens in a rotating system like a seesaw with one important exception, energy. As the
seesaw rotates, the riders are exchanging energy and that, is the topic for the next video.
Part 6.
How do the seesaw's riders affect one another? The answer to that question is that they support
one another and they exchange energy as the seesaw rotates back and forth. Let's start with the
support issue and to do this, I'm going to treat the seesaw as a facilitator rather than the object of
our main attention. I'm going to define the center of rotation as lying in the middle of the pivot. So
the axle rotation for this entire story is right here at the pivot of the seesaw but, otherwise, the
seesaw is just helping the red rider exert a torque on the purple rider about the pivot and the purple
rider exert a torque on the red rider about the pivot.
Well, you might wonder why the red rider should ever care about exerting a torque on the purple
rider. That's because, the purple rider is already experiencing a second torque, a torque due to
gravity. Gravity is pulling downward on that purple rider, at a level arm from the pivot, the axle
rotation. So gravity by itself is exerting a torque on the purple rider, and that torque is towards me,
that am away from you. To keep that purple rider from undergoing angle acceleration and
plummeting toward the ground, the red rider comes to the rescue. Okay? The red rider is exerting a
second torque on the purple rider, and that torque is toward you.
So the second torque acting on the purple rider cancels gravity's torque on the purple rider. This is
all about that pivot, and prevents the purple rider from undergoing angular acceleration. The red
rider really is supporting the purple rider. How about the other way around? Well, the red rider is
also experiencing a gravitational torque, forces down lever arm is toward your left. So, the
gravitational torque on the red rider is toward you by itself that would cause the poor red rider to
undergo angle acceleration, boom, and drop toward the ground.
But once again, the purple rider comes to the rescue and now, the purple rider is exerting a torque
on the red rider that is this way, toward me it cancels the gravitational torque on the red rider and
saves the red rider. So the long and short of it is, the red rider is keeping gravity from twisting the
poor purple rider to the ground and the purple rider is stopping gravity from twisting the poor red
rider to the ground. They really are supporting one another. So we have two torques between the
riders the red rider is exerting a torque on the purple rider and the purple rider is exerting a force on
the red rider.
How do that torques compare? They turn out to be exactly equal in amount, but opposite in
direction. That's an example of Newton's third law of rotational motion, which observes that for
every torque one object exerts on a second object, the second object exerts an equal, but oppositely
directed torque back on the first object.
That's how our universe works again, it's never violated, and it’s always there. So the red rider, when
it exerts a torque on the purple rider, causes the purple rider to exert an equal, but oppositely
directed torque back on the red rider every time and that ultimately allows this whole thing to
balance smoothly. At the same time, the purple rider rescues the red rider from the grips of gravity.
The red rider rescues the purple rider from the same clutches, and so, the two, two objects here
both reach the balance condition at the same time. They all end up with no overall torque due to
gravity.
This mutual support idea still works when one of the riders weighs far more than the other. If I
replace the red rider with this silver one, I can't put that heavy silver rider all the way at the end of
the board. I have to come in close to balance the seesaw, so right about here and look what has
happened. That rider weighs far more than the red rider did, so, if I put it out far, it would produce
way too much torque to support the purple rider. It would turn the purple rider into an astronaut.
So, by pulling the silver rider in closer to the pivot, I have compensated for the silver rider's greater
weight by shrinking the lever arm that the silver rider uses to produce a torque about the pivot. As
result, the silver rider is still just perfectly supporting the purple rider against the purple rider's
gravitational torque.
So, that part, part of the balance still works. How about the purple rider's support of the silver rider?
Well, it still works there too. The gravitation torque acting on this silver rider was reduced by
bringing the silver rider in close to the pivot. Gravity is pulling down with a big force on that heavy
silver rider, but since that force has only a short level arm with which to work, it doesn't produce all
that much torque on the silver rider about the pivot. As a result, the purple rider is still able to
support the silver rider. The silver rider may be heavier, but because the silver rider is closer to the
pivot, it doesn't take as much torque to support this silver rider against the torque produced by
gravity itself. I haven't talked about units in rotational motion, because they're somewhat
complicated and they'll distract us from more important issues.
But let me take a moment to mention the units of torque. In the SI or metric system, the unit of
torque is the newton meter and it corresponds to a force of 1 newton exerted at a lever arm of 1
meter from the center of rotation. In the English system of units, the unit of, of torque is the foot
pound. It corresponds to a pound force, exerted one foot from the center rotation. I introduced the
units of torque in part so that I can ask a question. Here is a wood screw and I will screw it into the
wood by exerting a torque on it, using my right hand, that is down towards the wood and now, I will
unscrew the screw by exerting a torque on it that is up out of the wood. If in both cases, inward
torque and outward torque, I exert a torque in the amount of 1 newton meter.
What torque does the wood screw exert back on me? If I exert a torque of 1 newton meter on the
wood screw, it has to exert a torque of 1 newton meter back on me in the opposite direction. That's
Newton's third law of rotational motion. So as I exert a torque of 1 newton meter downward into
the wood block, it exerts a torque of 1 newton meter up out of the wood block on me and if I exert a
torque of 1 newton meter out of the block, upward, on the wood screw, it exerts a torque of 1
newton meter downward on me, every time.
Now, let's watch the riders exchange energy as the seesaw rotates I've pretty well balanced the
board here, I have the red and purple riders at about the right differences from the pivot, so that the
seesaw balances. It experiences zero net torque and therefore rotates at constant angular velocity.
I'm going to hold the riders in place, so that as the seesaw rotates, they don't slide about and
possibly fall to the floor. But what I want you to do is watch the altitudes of the two riders. I'm going
to make the seesaw rotate at a constant angular velocity toward you, that is, like this, and the red
rider descends as the purple rider rises.
That means the red rider's gravitational potential energy is decreasing, while the purple rider's
potential energy is increasing. Where did the energy from the red rider go and what provided the
energy to the purple rider? What a mystery. I wonder as you might suspect, the red rider is
transferring energy to the purple rider by way of the seesaw. There are number of ways of following
that energy transfer, so I'm going to pick what I think is the most straightforward path. It involves
the work that the red rider does on the seesaw and the work the seesaw does on the purple rider.
I'm going to make the seesaw rotate very slowly at constant angular velocity toward you, and as I do,
I'll tell you how the transfer is occurring. Let me, let me get it started here.
Okay, it's started. We're going very slowly so I have time to talk. The red rider is pushing down on
the seesaw as the seesaw moves down in the direction of that downward force. So the red rider is
doing work on the seesaw. At the same time, the seesaw is pushing up on the purple rider and the
purple rider is moving up in the direction of that upward force. So the seesaw is doing work on the
purple rider. The job is complete, the red rider is, is transferring energy away to the seesaw and the
seesaw is transferring energy away to the purple rider. Hope you could follow that. So, energy is
flowing through the seesaw from the red rider to the purple rider, pretty neat, as the rotation occurs
in the opposite direction. So I'm going to make it rotate a constant angle velocity toward me. In this
case, the purple rider's pushing down on the board as the board moves in the direction of that push.
Therefore, the purple rider is doing work on the seesaw the seesaw is pushing up on the red rider, as
the red rider moves up in the direction of the seesaw's push. So, the seesaw is doing work on the red
rider.
Again, energy is going from the purple rider into the board and from the board into the red rider.
Because the seesaw is balanced, all of the gravitational potential energy lost by the rider that's
moving downward becomes gravitational potential energy in the rider that's moving upward. The
transfer of gravitational potential energy to gravitational potential energy is perfect. If the seesaw
isn't balanced, well then, it undergoes various types of accelerations and kinetic energy becomes
part of the equation.
But this arrangement where a balanced seesaw is transferring energy and particularly gravitational
potential energy from the descending to the rising rider works even if the riders have different
weights. If I replace the red rider with the much heavier silver rider and balance the seesaw again, I
achieve that balance by putting the heavier silver rider close to the pivot. Now, it's pretty well
balanced. As the seesaw rotates, at constant angle velocity which is what it normally does, the silver
rider who is close to the pivot, doesn't move up or down very much anymore. Right, they're close to
the pivot. They only move up and down, oh, no, this far, as the purple rider is moving up and down,
you know, much farther, more than a hand-width, that is the secret or maybe the, the explanation,
behind the perfect transfers of energy again.
If I again rotate this balanced seesaw your velocity toward you, the silver rider is descending. It's
pushing down on the seesaw board with a large force, because it weighs a lot. But that seesaw that
it's touching isn't moving very far. A little bit, but not all that far. So it's doing work of a reasonable
amount on the seesaw. At the same time, the seesaw is pushing up on the purple rider with a more
gentle force. But the purple rider is moving a longer distance. So the work the seesaw is doing on the
purple rider is the same as the work the silver rider is doing on the seesaw. To make sure that's clear
again. The silver rider is exerting a large downward force on the, on the seesaw as the seesaw is
descending a short distance. At the same time, the seesaw is exerting a small, upward force on the
purple rider as the purple rider moves upward a large distance.
The product of force time’s distance travelled in the direction of that force, the products are the
same. The work done by the descending silver rider on the seesaw is the same as the work done by
the seesaw on the rising purple rider. Energy is transferred perfectly. This arrangement of the
seesaw where a heavy rider can lift a light rider or a light rider can lift a heavy rider highlights the
fact that the seesaw is an example of a lever.
One of the simple machines like all simple machines, the lever allows you to change the amount and
or direction of the force you're using to perform a particular task. To make that clearer in this case,
let me get rid of one riders, purple rider take a rest, and now, I'm going to lift the heavy rider with
my hand by exerting a force directly on the seesaw. I can lift the heavy rider and, in this case, the
force that I'm exerting on this side of the seesaw was relatively small much less in amount than the
weight of the heavy rider. But as I lift the heavy rider, I have to move the seesaw I'm pushing on
much farther than the seesaw moves the heavy rider. So I'm doing work as I push down on my side
of the seesaw and I'm doing it by exerting a small force over a large distance. Whereas, the seesaw is
doing work on a heavy rider consisting of a, of a large upward force exerted as the, as the, heavy
rider moves a short distance.
So, the work I do on the seesaw is the same as the work the seesaw does on the heavy rider. But the
relationship of force and distance has changed and that's typical of many levers in our world, which
brings me again to a question. Let me get out a familiar lever and ask you about it. It's an all too
familiar situation. You take a nail, you pound it into a piece of wood and then you change your mind
and you want to take the nail back out. So you grab the nail with your hand, you pull as hard as you
can.
No luck. Can't get that nail out what are you going to do? You take a claw hammer. That is, a
hammer that has a groove in it, meant for grabbing the heads of nails and you grab the head of the
nail with that groove. Tip the hammer and it plucks the nail right out of the wood. So the question is
this, what aspect of what I did to the hammer is the same as what the hammer did to the nail? The
hammer acts as a lever and the work I did on my part of the hammer was the same as the work the
claw did on the nail, as the whole system rotated about an effective pivot where the, where the top
of the hammer touched the wood. So I was doing my work with a small force exerted over a long
distance as the claw was doing its work with a large force exerted over a small distance.
This is why I was able to pull out the nail. It was stuck tight and it needed a huge force to pluck it out
of the wood. I couldn't provide that big force with my own hands directly. But with the help of the
hammer, I could pull it off, because I have mechanical advantage associated with this lever. I'm
moving the level far from the pivot, so I can use a gentle force and it's moving the nail close to the
pivot, so it provides a huge force. I don't get something for nothing, though. I have to exert my force
over a long distance, whereas the claw exerts its force over a short distance. Nonetheless, the
mechanical advantage provided by this lever allowed me to do something I couldn't normally do,
namely pull a nail out of a piece of wood.
As you can see, the rider's interaction on the seesaw is pretty sophisticated. Not only are they
supporting one another, but they're exchanging energy as the seesaw tips back and forth. The
seesaw is just a kid's toy and yet, it exhibits all this really amazing rotational behaviour and in
particular, it acts as a lever allowing one rider to lift the other rider, allowing a heavy rider to lift a
light rider and a light rider to wow pretty much something for everybody.
Summary
A seesaw is a pretty amazing toy and a beautiful example of rotational motion. It rotates about its
own center of mass, its own natural pivot so that, the only reason I need this physical pivot passing
through it is to prevent the seesaw from falling. If I could turn off gravity, I wouldn't need that
physical pivot to make the seesaw do exactly what it's doing. Well, there is gravity so you need the
central pivot and the pivot also constrains the seesaw so it can't do rotations that we don't want. It
can only do what you see it doing at the moment.
Although it can also do that in the, in the other direction the seesaw is balanced as well, meaning
that it's experiencing no torque due to gravity. That's because the center of gravity of the seesaw is
located right at the pivot as well basically vertically in line with the pivot. In fact it's on the pivot and
as a result, that, the force of gravity produces no torque on the seesaw about its pivot. No torques.
So, the seesaw is turning according to Newton's First Law of Rotational Motion. It's a rigid object
that is not wobbling and it's not experiencing any external torques.
So, it rotates at constant angular velocity. Well, when you put riders on the seesaw, life can get more
complicated, which we've seen. Those riders produce their own torques on the seesaw, and they
move in various interesting ways. If they're not balanced, the seesaw undergoes angular
acceleration in various ways. But once they are balanced, and again there is no gravitational torque
on the seesaw and its riders, the seesaw is once again rotationally inertial. It turns according to
Newton's first law of rotational motion. If they come out of balance, and the seesaw undergoes
angular acceleration, it does so according to Newton's second law of rotational motion.
The angular acceleration of the seesaw is equal to the net torque acting on the seesaw divided by its
rotational mass and depending on where the, the riders sit on the seesaw, they can go from making
the, giving the seesaw a small rotational mass to giving it a large rotational mass. Lots of flexibility
and as they're tipping back and forth, they're exerting torques on one another, these two riders and
that's according to Newton's Third Law of Rotational Motion. If the red rider exerts a torque on the
purple rider, the purple rider has to exert an equal but oppositely directed torque back on the red
rider, every time and as they tip up and down, they do work on each other. They transfer energy to
each other with the help of the seesaw board. So there's really a, a lot going on here in the seesaw
and we'll use many of these same concepts later on as we continue to look at how things work. .