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# Visual understanding of centripetal acceleration formula

Video transcript

let's have some object this traveling in
a circular path just like this and what I've drawn here is its velocity vector
at different points along that path and so this right over here is going to be
the one to vector wanted this is going to be velocity vector too and this right
over here is going to be velocity vector 300 going to assume in this video is
that the magnitude of these velocity vectors is constant or another way to
think that the speed is constant so I'll just say lower case of you without the
Erawan tops is going to be scalar quantity I'll call this the speed or you
could call this the magnitude of these vectors and this is going to be constant
so this is going to be equal to the magnitude of vector one which is equal
to the magnitude of actor to the direction is clearly changing but the
magnitude is going to be the same which is equal to the magnitude the magnitude
of the vector 3 I'm going to assume those travelling in a path in a circle
with radius r what I'm going to do is i'ma drop position vector at each point
so let's call this call are one actual student pink let's call our one that
right over there that's position vector one or one that is position vector are
too so the position is clearly changing its position vector are too and that is
position vector are three but the magnitude of our position vectors are
clearly the same and I'm going to call the magnitude of our position vectors
are that's just the radius of the circle is this distance right over here so are
as you go to the magnitude of our 10 to the magnitude of our to which is equal
to the magnitude magnitude of our three know what I want to do in this video is
proved to you visually that given this radius and given this speed that the
magnitude of the centripetal acceleration and I'll just write that is
a subsea I don't have an error on top of this is a scalar quantity so the
magnitude of the centripetal acceleration is going to be easy will to our speed squared are constant
speed squared divided by divided by the radius of the circle this is what I want
you I want you to feel good that this is indeed the case by the end of this video
and to understand that what I want to do is I want to reap lot these these
velocity vectors on another circle and just think about how the vectors
themselves are changing so let's copy and paste this so let me copy and paste
of you on so copy and paste so that is actually do it from the center so that
his view on that we do the same thing for you to let me copy and paste it that
is easy to let me do it also 433 let's get back to Parliament get the label so
copy and paste it and that right over there
is Victor the three let me clean this up a little bit too so that we don't so
that's clearly v2 to lay blame or we know that v2 is orange v2 is an orange
and what is the radius of the circle going to be right over here with the
radius of the circle is going to be the magnitude of the velocity vectors and we
are do you know the magnitude of the velocity vectors is this quantity V the
scalar quantity so the radius of the circle is the the radius of the circle
we are the party know is equal to arm and just as the velocity vectors what's
giving us the change in position over time the change in position vector
overtime what's the vector that's going to give us the change in our velocity
vector overtime well that's going to be our acceleration
vectors so you will have some acceleration will call this a one will
call this a 282 and I'll call this a 383 and I want to make sure that you get the
analogy that's going on here as we go around the circle displays the position
vectors first they point out to the left then in the upper kind of a kind of eleven o'clock
position I guess that the top of the top left then to the top it's pointing in these different
directions like a hand in a clock and what's moving it along there is the
change in position vector overtime which is are these velocities factors over
here the velocity vectors are moving around like the hands of a clock and
what it what is it doing the moving around is are these acceleration vectors
and over here the velocity vectors they are tangential to the radius or shine
their tangential to the path which is a circle their perpendicular to a radius
and you learn that in geometry that a that a line that is tangent to a circle
is perpendicular to a radius is also going to be the same thing right over
here and just going back to what we learned when we learn about the
intuition of centripetal acceleration if you look at a one writer year and you translate this vector it'll
be going just like that it is going towards the center a to once again is
going towards the centre 83 once again if you translate that that is going
towards the center so all of these are actually center seeking vectors and you
see that right over here these are all these are actually centripetal
acceleration vectors right over here here time I just the magnitude of it I
would assume that all of these have the same magnitude so we're going to assume
that our sincere but all they all have a magnitude of will call a subsea so
that's the magnitude it's equal to the magnitude of a one and that vector is
equal to the magnitude of a two and it's equal to the magnitude of a three now
what I want to think about is how long is it gonna take for this thing to get
from this point on the circle to that point on that circle right or they're so
the way to think about it is what's the length of the arc that it traveled the
length of this arc that it traveled right over there that's one-fourth or
around the circle it's going to be one fourth of the circumference the
circumference is too high are it is going to be one-fourth of that so that
is the length of the arc that is the length of the arc and then how long
would take it to go that well you would you divide the length of your path
divided by the actual speed actual thing that's not get along that path so you
want to divide that by your actual the magnitude of your velocity or your speed
this is a magnitude of a city not velocity is not a vector right over here
this is a scalar so this is going to be the time the time to travel along that
path now that the time of travel along this path is going to be the exact same
amount of time it takes to travel along this path for the velocity vector so
this is for the position vector travel like that this is for the velocity
vector of travel like that it's going to be the exact same tea and what is the
length of this path and now think of it in the purely geometrical sense we look
we're looking at the circle year the radius of the circle is v so the length of this path right over
here is going to be one fourth is going to be be doing that same color CCD analogy
it's equal to 14 times of the circumference of the circle the
circumference of the circle is to buy times the radius of the circle which is
now what is nudging it along this circle what is it not doing a long as pat what
is the analogy for speed right over here is what's module along the path over
here it is the magnitude of the velocity vector so it's a nudging along this arc
right over here is the magnitude of the acceleration vector so it is going to be
it is going to be a sun sea and these times are going to be the exact same
thing the amount of time it takes to go for this vector to go like that for the
position vector is the same amount of time it takes the velocity vector to go
like that so we can set these two things equal each other so we get on this side
we get one-fourth to buy our overview is equal to 1 4th to päivi to Ivy over
over the magnitude of our acceleration vector and now we can simplify little
bit we can divide both sides by one-fourth care that we can divide both
sides by Tupac I get rid of that let me rewrite it so that we get our over the
is equal to the over the centripetal acceleration and I could cross multiply
and so you get V times so I'm just multiplying I'm cross multiplying right
over here V time zev squared is equal to AC times are and cross multiplying
remember is really just the same thing is multiplying both sides by both
denominators by multiplying both sides times viene si times the AC so such a
magical thing if you multiply both sides viene see these views cancel out these
AC's cancel out you get these times easy squared is equal to a subsea times are
is equal do a subsea times are and how to solve for the magnitude of our
centripetal acceleration you just divide both sides by are you divide both sides
by our and you are left with and I i guess we we've earned a drum roll now you're left with the magnitude of our
centripetal acceleration is equal to the magnitude are constant magnitude of our
velocity so this right here is our speed divided by the radius of the circle and
we're done