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# Centripetal force problem solving

In this video David gives some problem solving strategies for centripetal force problems and explains many common misconceptions people have about centripetal forces. Created by David SantoPietro.

## Video transcript

- [Instructor] There are
unfortunately quite a few common misconceptions
that many people have when they deal with
centripetal force problems, so in this video, we're
gonna go over some examples to give you some problem
solving strategies that you can use as well as going over a lot of the common
misconceptions that people have when they deal with these
centripetal motion problems. So, to start with, imagine this example, let's say a string is causing
a ball to rotate in a circle. And to make it simple, let's say this ball is tracing out a perfect circle, and let's say it's sitting on
a perfectly frictionless table so this would be the bird's eye view. This is the view from above. What it would look like from the side would be something like this. You'd have the ball tied to the rope and then you nail some sort of stake in the middle of the table. You tie the rope to the stake, and then you give the ball a push. And the ball's gonna take this
circular path on the table when we view it from the side. But when we view it from above, you see this path traced out. So this is a bird's eye view that you would see if
you were looking down from above the table, and this would be the side view. So let me ask you this question. What force is causing this
ball to go in a circle? Now, a lot of people want
to answer that question with the centripetal force. They'd say that it's the centripetal force that points inward that causes
this ball to go in a circle, and that's not wrong. It's the truth, but it's not the whole truth. And the reason is that when
we say centripetal force, all we really mean is
a force that's directed toward the center of the circle. So saying the force that causes
this ball to go in a circle is the centripetal force is a little unsatisfying. It'd be like answering the question, what force balances the force of gravity while the ball's on the table with the answer, the upward force. I mean, yeah, we knew it
had to be an upward force, but that really doesn't
tell us what force it is. Similarly, just saying
the centripetal force just tells us what
direction the force points. It doesn't really tell us
what type of force this is, so to answer this question
over here in a better way, if someone asked you what
force counteracts gravity that keeps the ball from
falling through the table, instead of saying upward force, it'd be better to just say
that's the normal force. And we can do better over here as well. Instead of just saying
the centripetal force, we could say what kind of force this is. It's gotta be one of the forces
that we already know about. I mean, it's gotta be
either the force of friction or normal force or tension
or the force of gravity. The centripetal force
isn't a new type of force. It's just one of the
forces we already know that happens to be pointing
toward the center of the circle. And that's important
because this is our first, big common misconception. People think the centripetal
force is a new kind of force, but it's not. It's just one of the
forces we already know that happen to be pointing
toward the center of the circle and that happen to be causing
an object to move in a circle. So in this case over
here, what force is it? Well, there's a rope tied to this mass, and that rope's gonna pull on it. And when a rope pulls, we call
that the force of tension, so I'm gonna call this the tension. So that's a little better. Now we know what kind of force is acting as the centripetal force. Now, be careful out there. Sometimes, people want to do this, they're like, oh yeah,
there's a force of tension, and there's also a centripetal force. But that's just crazy because this tension is the centripetal force. I wouldn't draw it twice anymore than I'd come over here and say, yeah, there's a normal force, there's also upward force. The upward force is the normal force. I wouldn't draw it again. Similarly, over here, I'm not gonna draw the centripetal force twice. The tension was the centripetal force. I mean, it's possible you
could have two forces inward. Maybe there's two ropes and you had a second tension
over here pulling inward, but you'd better be able to
identify what force it is before you draw it. Don't just call it F centripetal, so you might be like,
yeah, yeah, I get it. The centripetal force is
just an extra title we give to a force that happens to point toward the center of the circle, but how would I ever
solve a problem like this? What strategy do I use? I've got forces that are up,
that are down, that are in. So let me show you how
to solve some problems and some things to keep in mind. So let me add some numbers in here. So let's say I told you this. Let's say the mass of the
ball was two kilograms, the rope's length was 0.5 meters, and the ball is traveling
around the circle at a constant speed of
five meters per second. So what kind of question
might you be asked if given a problem like this? A possible question would be, well, what's the force
of tension in the rope? And so, now's a good time for me to let you in on a little secret. The secret to solving
centripetal force problems is that you solve them
the same way you solve any force problem. In other words, first, you
draw a quality force diagram. And then you use Newton's second law for one of the directions at a time. And if the direction you chose to analyze Newton's second law for didn't get you to where you needed to be, just do it again. Use Newton's second law
again for another direction, and that'll get you to
where you need to be. So in other words, let's
draw a quality force diagram. We've got forces, but they're
kind of all over here. This side view's gonna better illustrate all the forces involved. So we've already got
the normal force upward and the force of gravity downward. Now, I'm gonna draw this
tension pointing inward, that's the force that's acting
as the centripetal force. Now, we're gonna use Newton's second law for one of the directions. Which direction should we pick? Well, which force do we want to find? We want to find this force of tension, so even though I could if I wanted to use Newton's second law for
this vertical direction, the tension doesn't even point that way, so I'm not gonna bother
with that direction first. I'm gonna see if I can get
by doing this in one step, so I'm gonna use this horizontal direction and that's gonna be the
centripetal direction, i.e., into the circle. And when we're dealing
with the centripetal force, we're gonna be dealing with
the centripetal acceleration, so over here, when I use
a and set that equal to the net force over mass, if I'm gonna use the centripetal force, I'm gonna have to use the
centripetal acceleration. In other words, I'm gonna only plug forces that go into, radially
into the circle here, and I'm gonna have the radial
centripetal acceleration right here. And we know the formula for
centripetal acceleration, that's v squared over r, so I'm gonna plug v squared over r into the left hand side. That's the thing that's new. When we used Newton's second
law for just regular forces, we just left it as a over here, but now, when you're using this law for the particular direction that is the centripetal direction, you're gonna replace a
with v squared over r and then I set it equal to the net force in the centripetal
direction over the mass. So what am I gonna plug in up here? What forces do I put up here? I mean, I've got normal
force, tension, gravity. A common misconception is that
people try to put them all into here. People put the gravitational
force, the normal force, the tension, why not? But remember, if we've selected
the centripetal direction, centripetal just means pointing toward the center of the circle, so I'm only going to plug in forces that are directed in toward
the center of the circle, and that's not the normal force
or the gravitational force. These forces do not point inward toward the center of the circle. The only force in this case that points toward the
center of the circle is the tension force, and like we already said, that is the centripetal force. So over here, I'd have v squared over r, and that would equal the only force acting
as the centripetal force is the tension. Now, should that be positive or negative? Well, we're gonna treat
inward as positive, so any forces that point
inward are gonna be positive. Is it possible for a centripetal
force to be negative? It is. If there was some force
that pointed outward, if for some reason
there was another string pulling on the ball outward, we would include that
force in this calculation, and we would include it
with a negative sign, so forces that are
directed out of the circle, we're gonna count as negative and forces that are
directed into the circle, we're gonna count as positive in here. And if they're not directed into or out, we're not gonna include them
in this calculation at all. Now, you might object. You might say, wait a minute. There is a force out of the circle. This ball wants to go out of the circle. There should be a force this way. This is often referred to
as the centrifugal force, and that doesn't really exist. So when people say that
there's an outward force trying to direct this
ball out of the circle, they're usually referring
to this centrifugal force, but this doesn't exist. It turns out this is not a real thing if you're using a good reference frame. There is no natural outward force for something going in a circle. You might object. You might be like, wait a minute. If I let go of this ball, it flies out of the circle. Won't it go flying off this way? And no, it won't. If you let go of the string right now, for some reason the string broke, at this moment this ball
would not veer off that way. There's no force pushing it to the right. The ball, if the string broke, would just follow Newton's first law. It says it would just
travel in a straight line with constant velocity, and
it would roll off the table. So the reason you have to pull on the rope to get the ball to go in a circle is not because there's an outward force but because this ball wants
to maintain its velocity. It has inertia, it wants to
keep moving in a straight line, but you have to keep pulling on it to keep changing the
direction of this velocity. So even though many people think there's an outward centrifugal force that's just naturally occurring on an object going in a circle, there is not. So finally, we can come back over to here. I can put my mass here. I can finally solve for
my force of tension. If I do this, I'll multiply
both sides by mass, and I just get that
the force of tension is mass times the speed squared
over the radius of the circle, and if I plug in my values, the mass was two, the speed was five, and you can't forget to square it. You divide by the radius which was 0.5, and you get that the force of
tension had to be 100 Newtons. So in this case, the force of tension, which is the centripetal
force, is equal to 100 Newtons. Now, some of you might be thinking, hey, this was way too much work for what ended up being
a really simple problem. Why did we have to go
through all the trouble of stating all of this
problem solving strategy? And I agree. This one was easy, but other problems won't be easy. And if you don't have some sort of problem solving framework to fall back on, you'll be shooting blind and that's a lonely, lonely place to be. So let's use this same procedure, but let's look at a new problem. Let's say, you have this. Let's say you were riding your bike over a circular hill. So this gray line represents the pavement, and it starts off flat. But then the pavement veers upward and it creates this concrete hill that you ride over and then down and you ride over to this side. And all this purple circle is representing is the fact that if you were to continue this crest of the hill
around into a circle, it would form this shape, so that gives us a way to
define what the radius is of this top part of the hill. So, let's put some numbers in here. Let's say the radius of
this hill was eight meters. Let's say the mass of you
and your bike together are about 100 kilograms. And let's say you're riding over this hill at six meters per second. And let's say I asked you, what's the size of the normal force exerted on you and your bike as you ride over the crest of this hill at six meters per second? Now, let me show you what you can't do because most people would try to do this. They really want to say
that the normal force is just gonna be equal
to the force of gravity. Therefore, since the
force of gravity is mg, the normal force should just be mg, but that can't be right. If the forces on an object
are balanced and they cancel, the object is just gonna
maintain its velocity, size, and direction, so this object, since
it's going to the right, this bike would just
continue going to the right and it would just hover
straight off this hill. That'd be awesome, but
that doesn't happen. This bike moves downward. It accelerates downward after this moment since it rides down the hill, so the downward force
has got to be bigger. The force of gravity's gonna
be bigger than the normal force 'cause if it wasn't, this bike would just hover off into space. So how do you solve this problem? We use the same strategy we used before. We're gonna draw a force diagram, but we already did that. We're gonna use Newton's second law for one of the directions, and the direction we're gonna pick is the vertical direction. Now, is that vertical direction
the centripetal direction? Yeah, it is because look at into the circle is downward. Because this bike is at
the crest of the hill, down corresponds to pointing
toward the center of the circle and upward corresponds to pointing away, radially away from the
center of the circle. So, since I'm dealing with
the centripetal direction, we plug in the formula for
the centripetal acceleration, and the part where you
have to be most careful is what you plug into
the centripetal forces. Remember that into the
circle counts as positive and out of the circle counts as negative. So both of these forces,
normal and gravity, are gonna be included,
but only one of them are gonna be included
with a positive sign. Think about which one. Can you figure out which force would be included in here
with a positive sign? If you said the force of gravity, you're right, which is weird. Usually, we treat the force
of gravity as negative because it points down, but for centripetal forces,
what we care about is into or out of the circle. So, I'm gonna treat gravity as
a positive centripetal force. Gravity is the force pointing toward the center of the circle, and the normal force in this case is gonna be a negative centripetal force since it's directed out of
the center of the circle. And then, we divide by our mass. And so, if we solve this
for the normal force, if you do some algebra, we'll multiply both sides by m, we move over the F N and
then move the m v squared to the other side and what
we end up getting is that mg minus m times v squared over r is equal to the normal force, which if we plug in numbers, gives us 100 kilograms times 9.8 minus 100 kilograms
times the speed squared, that's gonna be six
meters per second squared, divided by the radius of the
circle we're traveling in which is eight meters, and you end up getting 530 Newtons. So the normal force on you and your bike as you ride over this hill is 530 Newtons. That is not equal to your weight. This is less than your weight. The force of gravity on you is gonna be m times g, that would be about 980 Newtons. So you experience less normal force, and this is natural. This is what happens when
you ride over a hill fast. You feel slightly weightless
as you go over that hill. If you've ever gone with a car a little too fast over a hill, you feel that whoa in your stomach, and you're like, hey, that was cool. That was the weightlessness
you felt for a moment. If you go too fast, if you go too fast, this normal force will become zero. You'll subtract so much
m v squared over r here, the normal force becomes zero. When that happens, you do become airborne, so be careful driving over those hills. If you drive too fast,
you'll become airborne since your normal force
is gonna become zero. So, recapping, when you solve
centripetal force problems, be sure to draw a quality force diagram. Then use Newton's second law for one of the directions at a time. If you use the centripetal direction, the direction pointed
radially into the circle, you can say that the
acceleration in that direction is v squared over r, but be sure to only plug in forces that are directed radially, that is to say, forces
that are pointed into or out of the circle. If they point into the circle, they're gonna be positive forces, and if they point out of the circle, they're gonna be negative forces.