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## Circular motion and centripetal acceleration

# Loop de loop answer part 1

## Video transcript

What I want to do
now is figure out, what's the minimum
speed that the car has to be at the top of this
loop de loop in order to stay on the track? In order to stay in
a circular motion. In order to not
fall down like this. And I think we
can all appreciate that is the most difficult part
of the loop de loop, at least in the bottom half
right over here. The track itself
is actually what's providing the centripetal force
to keep it going in a circle. But when you get
to the top, you now have gravity that is
pulling down on the car, almost completely. And the car will have to
maintain some minimum speed in order to stay in
this circular path. So let's figure out what
that minimum speed is. And to help figure
that out, we have to figure out what the radius of
this loop de loop actually is. And it actually does not
look like a perfect circle, based on this little screen
shot that I got here. It looks a little
bit elliptical. But it looks like the radius
of curvature right over here is actually smaller
than the radius of the curvature of the
entire loop de loop. That if you made
this into a circle, it would actually be maybe
even a slightly smaller circle. But let's just assume, for
the sake of our arguments right over here, that this
thing is a perfect circle. And it was a perfect
circle, let's think about what that
minimum velocity would have to be up here at the
top of the loop de loop. So we know that the magnitude
of your centripetal acceleration is going to be equal
to your speed squared divided by the
radius of the circle that you are going around. Now at this point
right over here, at the top, which is
going to be the hardest point, the magnitude
of our acceleration, this is going to be 9.81
meters per second squared. And the radius,
we can estimate-- I copied and pasted
the car, and it looks like I can get it to stack
on itself four times to get the radius of this
circle right over here. And I looked it up on the
web, and a car about this size is going to be about
1.5 meters high from the bottom of the
tires to the top of the car. And so it looks like--
just eyeballing it based on these copying
and pasting of the cars, that the radius of this
loop de loop right over here is 6 meters. So this right over
here is 6 meters. So you multiply both
sides by 6 meters. Or actually, we could keep
it just in the variables. So let me just rewrite
it-- just to manipulate it so we can solve for v. We have
v squared over r is equal to a. And then you multiply
both sides by r. You get v squared is
equal to a times r. And then you take the principal
square root of both sides. You get v is equal to
the principal square root of a times r. And then if we plug
in these numbers, this velocity that we
have to have in order to stay in the circle is going
to be the square root of 9.81 meters per second
squared, times 6 meters. And you can verify that
these units work out. Meters times meters is meter
squared, per second squared. You take the square
root of that, you're going to get
meters per second. But let's get our calculator
out to actually calculate this. So we are going to get
the principal square root of 9.81 times 6 meters. It gives us-- now here's
our drum roll-- 7.67. I'll just round to three
significant digits, 7.67 meters per second squared. And significant digits
is a whole conversation, because this is just a very,
very rough approximation. I'm not able to measure
this that accurately at all. But I get roughly 7
point-- I'll just round, 7.7 meters per second. So this is approximately
7.7 meters per second. And just to give a sense
of how that translates into units that we're used
to when we're driving cars, we can convert 7.7
meters per second. If we want to say how many
meters we go to an hour, well, there's 3,600
seconds in an hour. And then if you want to
convert that into kilometers-- this will be in meters--
you divide by 1,000. One kilometer is
equal to 1,000 meters. And you see here,
the units cancel out. You have meters, meters,
seconds, seconds. You're left with
kilometers per hour. So let's actually
calculate this. And so we get our
previous answer. We want to multiply
it times 3,600 to figure out how many
meters in an hour. And then you divide
by 1,000 to convert it to kilometers per hour. So you divide by 1,000. And we get 27.6
kilometers per hour. So this is equal
to 27.6 kilometers per hour, which is
surprisingly slow. I would have thought
it would have to be much, much, much faster. But it turns out, it does not
have to be much, much faster. Only 27.6 kilometers per hour. Now the important
thing to keep in mind is this is just fast
enough, at this point, to maintain the circular motion. But if this were a perfect
circle right over here, and you were going at exactly
27.6 kilometers per hour, you would not have much
traction with the road. And if you don't have much
traction with the road, the car might slip
and might not be able to actually
maintain its speed. So you definitely
want your speed to be a good bit larger
than this in order to keep a nice
margin of safety-- in order to especially
have traction with the actual loop
de loop, and to be able to maintain your speed. Now what I want to
do in the next video is actually time the car to
figure out how long does it take it to do this loop de loop. And we're going to assume
that it's a circle. And we're going
to figure it out. And we're going to
figure out how fast it's actual average velocity
was over the course of this loop de loop.