Rotational Motion Video Lecture - NEET

hi it's mr. Andersen and this is ap
physics essentials video 53 it's on
rotational motion which is turning
motion and that would be dangerous if
you're a slackliner like this person in
Yosemite as he applies a force or a net
torque on one side you're going to get
rotation which he has to counteract with
the net torque in the other direction
and so by moving your arms out you can
increase your inertia but it still is
difficult that's why he's attached on
here with a rope and so if we take an
object and apply a net torque to it what
we get is rotational emotion we measure
that in physics according to its angular
displacement so the angle at which it's
turning its angular velocity so the
speed at which it's turning and then the
angular acceleration which is how much
that angular velocity is increasing or
decreasing over time now the whole
object has the same for example angular
velocity but if you look at different
distances out from the axis of rotation
obviously out here it's going to be
going faster than it is in here and so
all we do for example is take the
angular velocity times the radius or the
distance from the center and that's
going to give us its linear velocity or
its speed in that one direction now
another term that you should understand
is rotational inertia and so we know in
regular physics that force is equal to
mass times acceleration if we apply a
force times a mass we're going to get an
acceleration now in rotational motion
the equation looks a little bit
different if we apply a net torque so
for plying a force remember that is
perpendicular to the lever arm if we
apply a net torque that's going to be
equal to the rotational inertia in other
words that's how much this object
resists change or resists that change in
motion times the angular acceleration
and so a good understanding of
rotational inertia is important when
you're trying to solve problems like
this before we get to that we have to
understand the importance of the Radian
because that's the base unit when we're
measuring angular velocity for example
or angular acceleration so if I draw a
radius like this and then I take that
distance of the radius and wrap it
around the circle I now have the angle
of one Radian and so this would be one
Radian
to radians three radians this would be
PI radians 180 degrees or half a
revolution or if we go all the way
around that's two pi radians and so you
should understand what a Radian is it's
about 57 degrees but also it's the base
unit as we're measuring this rotational
motion and in the back your mind you
should know that one revolution equals
360 degrees equals two pi radians and so
for measuring angular displacement we
measure that using the angle theta so
it's going to be how much it's turning
or the distance by which it's turning
and so again we would measure that in
radians and this is one Radian right
here it's about 57 degrees
what's angular velocity then angular
velocity and we use the symbol Omega to
represent that is simply the change in
that angle over the change in time
so let me animate this for a second
we're going to move that one Radian in
one second and so what would our angular
velocity be it would be one Radian per
second and so it's how fast this thing
is turning so if it took one second to
go all the way around then it's going to
PI R or two pi radians rather per second
now let's do an angular acceleration so
what is that it's the change in angular
velocity over time or the change in
omega over time so let me start the
spinning at the beginning it has an
angular velocity of zero but after five
seconds let's say it's going at one
Radian per second so how would I figure
out my angular acceleration it's going
to be the change in Omega which it went
from zero to one Radian per second so
that's going to be one Radian per second
how long did it take to do that it took
five seconds and so my angular
acceleration would be one-fifth or 0.2
radians per second squared now let's do
some simple problems and I'm using an
album of mine of course it's Pink
Floyd's Dark Side of the Moon you can
see I have to clean up the album a
little bit but let's say that I started
spinning like this and it takes about
2.8 seconds for this to get up to speed
and then it spins at thirty three point
three revolutions per minute well you
can see right away that this revolutions
per minute is how much it's rotating in
a given period of time and so we're
already halfway there when we're trying
you're out the angular velocity angular
velocity remember is equal to the change
in theta so the change are in angle over
the change in time and so if I start
solving this what's my change in angle
well I said it's thirty three point
three revolutions per one minute now
these units are bad we have to get this
two radians and we have to get this two
seconds and so we can just do some
factor-label I know that one revolution
equals two pi radians and so right now
my revolutions cancel and I know that
one minute is equal to 60 seconds so my
minutes cancel and so if I calculate
that out it's going at about 3.49
radians per second which makes sense if
it takes a minute to go around 30 times
it makes sense that it's going a little
more than you know pi radians or 3.49
radians per second so that's my angular
velocity and it's the same for any point
along that album let's say I want to
figure out my angular acceleration now
how do I do that what do you need to do
well I'm giving you the time at which
it's changing and so it's changed or my
change in omega is going to be three
point five four nine radians per second
and it took 2.8 seconds to get up to
that it started at zero and it got up to
that speed after two point eight zero
seconds and so I'm going to get an
acceleration of 1.25 Ratings per second
squared so that's how I calculate that
now one thing you should understand is
that even though this whole album has an
angular velocity of three point four
nine seconds not every point along that
album is going at that same linear speed
and so we have an equation for that as
well it's called linear velocity and
that's simply equal to Omega or angular
velocity times the radius and so if we
do a point right here in the middle
let's say that's zero centimeters from
the center so it's right at the center
well there's one problem here this
should be converted to meters but I
don't have to because it's just zero so
if I take my angular velocity of three
point four nine radians per second times
zero what's my velocity zero and so it's
linear velocity zero at the center what
would it be as we move farther out so
let's say out here since this is a
regular album it's six inches out here
fifteen point three centimeters so if I
want to calculate that I'm going to say
my velocity equals to Omega times radius
what's my
megha 3.49 radians per second what's my
radius
it's 0.15 three meters I've converted
that from centimeters to meters and so
what's my linear velocity 0.534 meters
per second so it's going you know a
little over a mile an hour if we're
going around the outside so it seems
like it's going fast but not that fast
and now we're going to start talking
about applying forces to an object and
creating these motions and so there's a
nice phe T simulation you can see that
if I move it in this direction at a
constant speed I've got linear velocity
if I move it in the other direction I've
got linear velocity let me freeze that
for just a second and so now it's not
accelerating it's just moving with a
constant angular velocity but what you
can see is the linear velocities greater
on the outside that it is as we move
farther in but also this is accelerating
even though the velocities are constant
their direction is changing and so what
we have is called a centripetal
acceleration to the inside but what I'm
now going to do is start applying a net
torque to it and so I'm going to start
pushing and pulling on it and what
you'll see is that we'll get huge
accelerations in one direction and then
as it stops it goes back in the other
direction so as you apply a net torque
to an object we're going to get that
acceleration and so just like force is
equal to mass times acceleration in
rotational motion our torque or that
force that we're applying is equal to
not mass but rotational inertia times
its angular acceleration and so what is
rotational inertia again it's the
resistance to motion of an object in
rotation so to demonstrate that I have
two pieces of PVC pipe they're exact
same mass exact same length and I'm
going to drop them at the exact same
time you can see that they land at the
exact same time now what I did is I
taped a weight to one of the PVC pipes
and then I'm going to drop them at the
same time so make a prediction for a
second which one do you think is going
to land first the heavier one the
lighter one are they both going to land
at the same time let's let that go and
so you can see that the lighter one hit
first why is that it's because it has
less rotational inertia and so as this
force force of gravity acts on it it's
going to turn with a lower acceleration
and so another way to look at this is
the demonstration here what I have are
four objects that all have the same mass
this one right here is a solid cylinder
and this is a cylinder where all the
masses on the outside this is a solid
sphere and this is a hollow sphere and
so which one do you think is going to
make it to the bottom first if we race
them and they all have the same mass
well this is kind of counterintuitive as
well if we let this run you'll see that
the solid cylinder gets there first and
so if we kind of pause that for a second
so this solid rather sphere is going to
make it to the bottom first why is that
it's because it's going to have a lower
rotational inertia more of its mass is
going to be towards the center of that
object whereas up here all of that mass
is around the outside of the object now
they're going to be equations to
calculate the rotational inertia of each
of these but what you should understand
is that the bigger I is the bigger that
rotational inertia is the more torque we
have to apply to have a similar
acceleration and since the force is the
same on all of these the ones that have
the greatest eye or rotational inertia
are going to have the slowest
acceleration and so did you learn to
make predictions about velocity around
an object again we've got
angular velocity displacement and
acceleration if we want to figure out
linear we have to multiply it times the
radius and then finally could you
collect data to see how a force or a
torque applied to a rotating object
affects its acceleration there's a
wonderful Paet simulation I'll put a
link down below but I hope that was
helpful