Good morning. We have already discussed Uniform Circular Motion. For example , the motion of
the small, yellow chip on top of the record; the chip is an object moving with a constant
angular velocity. Because its angular velocity is constant, its change in angular velocity
is zero, the angular acceleration of the chip is zero, the tangential acceleration
of the chip is also zero, and its motion is often called Uniform Circular Motion.
Right. ♪ Flipping Physics ♪
Yeah. When the angular velocity of an object is
not constant, like the ball on the end of this string which is moving in a vertical circle,
the change in angular velocity of the object does not equal zero,
so the object does have angular acceleration, the object therefore also has tangential acceleration,
and the motion of the object is called Non-uniform Circular Motion because the
angular velocity of the object is not constant. The angular velocity of that ball is not constant?
Yeah, it sure looks constant.
Right, I know the ball looks like it could be moving at a constant angular velocity, however,
when we analyze the forces acting on the ball, we will be able to show that the angular velocity
of the ball cannot be constant, and this must be Non-uniform Circular Motion. Bobby,
please tell me what forces are acting on the ball. Uh, can we slow the video of the ball down a bit?
(Sure Bobby. Here the video shows the ball moving 40 times slower than real speed.)
Thanks. the force of gravity is always acting downward on the ball. And the
string causes a force of tension on the ball and that force of tension acts well,
it always acts inward toward the center of the circle the ball is moving through.
Bobby, those are correct. Now, the mass of the ball is always the same, so the force of
gravity acting on the ball is constant. However, notice that I have illustrated that the magnitude
of the force of tension acting on the ball is not constant. To understand why that is,
realize because the ball is moving in a circle, there must always be a centripetal force acting
on the ball. In other words, there must always be a net force acting inward on the ball.
Because the force of gravity changes direction relative to the “in-direction” as the ball moves
around the vertical circle, the force of tension needs to increase and decrease to
maintain that net inward force. That is why the force of tension has a maximum magnitude when
the ball is at the bottom of its path where the force of gravity is pointed directly out away
from the center of the vertical circle which is opposite the direction of the force of tension.
And the force of tension has a minimum magnitude when the ball is at the top of its path where the
force of gravity is pointed directly in toward the center of the vertical circle which is the
same direction as the force of tension. Oh yeah. We already talked about this a
bit with the water in a bucket moving in a vertical circle. Right mr.p?
Billy, that is correct, we discussed this same concept in that lesson , however,
today we are getting into more of the details. For example, in order to understand how these forces
affect the motion of the ball, we need to break forces into components, something we did not
do in the “Analyzing Water in a Bucket Revolving in a Vertical Circle” lesson.
Bo, in what directions do you think we should break our forces into components in?
The x and y directions? Don’t we usually break forces into components in the x and y directions?
Not when an object is moving in a circle. Right. Uh. We need to break forces
into components in the “in-direction”. Yeah, mr.p said “in” twice in the question.
He thinks it’s funny. (Yes, I do)
Right. (think it’s funny.)
So, we break forces into components in the “in-direction” and 90 degrees to that or well,
tangent to the circle the ball is describing. So, the “tangential direction”?
Right. Tangential direction because it is tangent to the circle and
therefore in the direction the ball is moving. Exactly. Let’s break forces into components in the
“tangential direction” and the “in-direction”. However, rather than calling it the “in-direction”
let’s refer to it this time as the “radial direction”.
Radial direction as in in the direction of the radius, sure.
Correct Bo. Sometimes this is referred to as the “radial direction”, so I thought we would use that
terminology today. Because the force of tension is always directed radially inward, we do not need
to find any components of the force of tension. However, the force of gravity we need to separate
into the tangential force of gravity component and the radial force of gravity component.
Billy, please tell me what you see happening to the radial component of the force of gravity
as the ball moves through the vertical circle. Absolutely. Okay, the radial component of the
force of gravity. Well, starting at the top of the path, as the ball moves down, the inward
radial component of the force of gravity decreases in magnitude until it reaches zero when the string
is horizontal. Then the radial component of the force of gravity increases in magnitude and is
directed radially outward. The radial component of the force of gravity reaches its maximum magnitude
at the bottom of the arc and then decreases in magnitude until it reaches zero when the
string is horizontal. Then the radial component of the force of gravity increases in magnitude,
is directed radially inward, and reaches its maximum magnitude at the top of the arc.
Very nice Billy. Bobby, please do the same thing with the tangential component
of the force of gravity. Alright, the tangential component
of the force of gravity. Well, at the top, there is no tangential component of the force of gravity
and as the ball moves downward the tangential component of the force of gravity is in the
direction of motion of the ball and increases in magnitude until the string is horizontal.
Then the tangential component of the force of gravity decreases in magnitude until the ball
reaches the bottom of the arc where there is no tangential component of the force of gravity.
As the ball moves upward the tangential component of the force of gravity is opposite the direction
of motion of the ball and increases in magnitude until the string is horizontal.
Then the tangential component of the force of gravity decreases in magnitude
until the ball reaches the top of the arc. Bobby, that was also very nice. Now,
because net force equals mass times acceleration, we know net force is proportional to acceleration
and we can describe the acceleration in the tangential direction and the acceleration in
the radial direction. And class, the acceleration acting inward in the radial direction is called …?
Centripetal acceleration. The center seeking acceleration.
Yeah. Let’s start with tangential acceleration,
because the only force acting in the tangential direction is the tangential component of the force
of gravity, the tangential acceleration will be proportional to the tangential component of
the force of gravity. Bobby, the description of the tangential acceleration of the ball
is very similar to the description of the tangential component of the force of gravity.
You just described the tangential component of the force of gravity, so please also describe
the tangential acceleration of the ball and how that affects the tangential speed of the ball.
At the top, there is no tangential acceleration and as the ball moves downward the tangential
acceleration is in the direction of motion of the ball and increases in magnitude until the string
is horizontal. That means the tangential speed of the ball is increasing. Then the tangential
acceleration decreases in magnitude until the ball reaches the bottom of the arc where there
is no tangential acceleration. Ah, the tangential speed of the ball is increasing all the way from
the top to the bottom. And the tangential speed of the ball is at a maximum at the bottom of the
circle. As the ball moves upward the tangential acceleration is opposite the direction of motion
of the ball and increases in magnitude until the string is horizontal. Then the tangential
acceleration decreases in magnitude until the ball reaches the top of the arc. The tangential speed
of the ball is decreasing all the way from the bottom to the top. And the tangential speed of
the ball is at a minimum at the top of the circle. Again, very well done Bobby. And I just want to be
absolutely clear that because the tangential acceleration of the ball is not zero,
the tangential speed of the ball is not constant, and this must be Non-uniform Circular Motion.
I still think it looks like a constant angular velocity.
Uh… Eh…
We’re almost there Bo. After you describe
what happens to the centripetal acceleration of the ball as it moves around the circle,
we will prove this is Non-uniform Circular Motion. Bo, centripetal acceleration please?
Sure. Well, because centripetal acceleration equals tangential speed squared over radius,
as the tangential speed of the ball increases, the centripetal acceleration of the ball increases.
In other words, from the top to the bottom, the centripetal acceleration of the ball increases.
And the centripetal acceleration of the ball has its maximum magnitude at the bottom of the circle.
And as the tangential speed of the ball decreases, the centripetal acceleration of the
ball decreases. In other words, from the bottom to the top, the centripetal acceleration of the ball
decreases. And the centripetal acceleration of the ball has its minimum magnitude at the top of
the circle. Oh, and at the top, the centripetal acceleration of the ball is not zero because the
ball does have a tangential speed. Now can we prove it is Non-uniform Circular Motion?
Yes Bo, Let’s do that. We have already established that the ball has its maximum
tangential speed at the bottom of the circle and its minimum tangential speed at the top of
the circle. Right class? (Yes. Yeah. Yep.)
Therefore we would expect the time for the ball to travel through the bottom semicircle
to be less than the time for the ball to travel through the top semicircle. Right class?
(Yes. Yeah. Yep.) And, as you can see the
time for the ball to travel through the bottom semicircle is 0.350 seconds and the time for
the ball to travel through the top semicircle is 0.480 seconds. So, it does take less time
to travel through the bottom semicircle because it is moving, on average, faster while moving
through the bottom semicircle. And that means the ball is moving in Non-uniform Circular Motion.
The physics works! The physics works! Uh huh, uh huh! The physics works! The physics
works! Uh huh, uh huh! The physics works! Yes it does. Now, I do want to point out
(Uh, mr.p?) Yes, Billy?
We could also use conservation of mechanical energy to show that this must be Non-uniform
Circular Motion, right? (That is a good point
Billy. Could you do that please?) Certainly! Because mechanical energy is
conserved as the ball moves through the circle, as the ball moves down, the gravitational potential
energy of the ball decreases, so the kinetic energy of the ball must increase, therefore,
as the ball moves down, the speed of the ball must increase and therefore, the ball
must be moving in Non-uniform Circular Motion. Thank you Billy, that was remarkably easy.
Okay, so I do want to point out that, because the motion of my hand moving in
a small circle is what causes this ball to move in Non-uniform Circular Motion, the ball does not
quite move through a perfect circle, the string does not quite represent the radius of the motion,
and all of the magnitudes of the vectors shown today are only estimates of the real values.
Does that mean the physics did not work? Nah, the physics still works.
Uh huh. Good!
Thank you very much for learning with me today, I enjoy learning with you.