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Lecture 7 - 2 - Conservation of Momentum | Physics I | YouTubeToText
YouTube Transcript: Lecture 7 - 2 - Conservation of Momentum
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This content explains how the total momentum of a system of particles changes based on external forces, leading to the fundamental principle of conservation of momentum for isolated systems.
so i mentioned in the previous video how
momentum is a fundamental quantity and
in this video
we're going to find out one of the
reasons why that's the case to start out
with though
we need to start talking about systems
of particles in other words
how do we handle more than one particle
more than one
object how do we handle the momentum of
that situation
so i'm going to draw out a really simple system
system
let's suppose we've got three particles
say one two
and three so just three particles of
system it doesn't matter how many
it could be 15 it could be millions uh
but we'll start it nice and simple with three
three
and so this is going to define my system
okay i'm going to put a little box
around there i'll write the word system
right here
that is our system of three particles
now i need to worry about
the forces between these particles and so
so
there could be for example interaction
forces between them so maybe
for example object two is attracting
object one right so we could call this
f2 on one but then if that's the case
newton's third law says that object one
must be attracting object two as well
f21 and f12
they must be equal and opposite right so
remember that for later
and then there could be some forces
between you know these two
these two and they don't have to
necessarily be attractive they could be repulsive
repulsive
it doesn't really matter the point is
that there are interaction forces
happening between
these three objects maybe it's gravity
maybe it's something else
there are interaction forces those are
what we call internal forces right so
all of these forces that i've drawn so
far are internal
well maybe i'll do this here we go there
those are
internal internal i have to spell it right
right
internal forces there can also be
external forces as well so maybe for
example something outside the system
is pushing on on object three on
particle three
that would be an external force right so
we'll write that maybe just
f3 and then maybe something is pushing
on object two
something like this we'll call that f2
that's an external force as well
those two forces are different from the
others because they're coming from
something outside the system
all right so that's our basic idea here
we've got a system of three particles
and we have two really different kinds
of forces forces that are internal
that are interaction forces and forces
that are external coming from outside
the system pushing
in okay so that said
what are we trying to do here we want to
talk about the momentum
not just of one particle but of all
three of our particles
in the system and so i want to define
for you a
total momentum and this total momentum
is going to have a symbol of capital p
so this is my capital p and of course
it's a vector since
it's going to be a momentum and it's defined
defined
really simply you just take the momentum
of object one
and you add it to the momentum of object
two and you add it to the momentum of
object three
and if there's more than three particles
you keep going
the only thing you have to be careful
with is that it's a vector sum right so
you have to take the directions into
account as well as the
uh as well as the magnitudes so that is
our total momentum of the entire system
right you just add up the momentum
of each particle okay that's fine
what i'm really interested though is how
the total momentum
changes with time and to figure that out
i'm going to take the time derivative of
the total momentum
right so there is dp dt and that's how p
changes with time and uh and so to do this
this
i'm going to make a little tiny shortcut
here because i might have mentioned
before but physicists are kind of lazy
and so in order to um write things down
a little bit more compactly
i'm going to use a summation notation so
this says that the sum
over k of pk is the total momentum
and that summation symbol there just
stands for adding up all of the
different momentum
so if i do that i can write down this
derivative a little bit more simply in
fact i can take the derivative
because taking the derivative is
something that you can do to each
individual term and so i can write down
the derivative like that and so the
derivative of the total
is equal to the sum of the derivative of
the individual
momentum okay but dp dt
that's something special we just learned
about in the last video
that is actually equal to the force
on particle k right so remember k here
is standing for each of the particles so
particle one two or three
that says to add up all of the forces on
particle one
add it to all of the forces on particle
two and add it to all of the forces
on particle three and that's what this
says here
so let me actually make that really
clear right so fk
in this case is the net force
on uh on particle k okay
so that's nice but remember we have two
different kinds of particles
sorry two different kinds of forces not
particles two different kinds of forces
we said that there were internal
interaction forces
and then external forces coming from
something outside the system
i want to make that really clear by
writing this down like this i'm going to
write it as a sum
over k of f k
internal plus fk
external uh spell that right there we go
so that says
x that's supposed to be x extra um and
so that again makes it clear we have two different
different
classes of forces two different reasons
for the forces
and then i can just split up this sum
one more time so we'll write this down again
again
but i'll write this as the sum over fk internal
internal
plus the sum over k of
fk external okay so again i'm
getting a little bit complicated over
here we're starting from how the
momentum changes with time
but there's something interesting in our
last result here this first term right here
here
is the sum of all of the forces but
just the internal ones and remember
newton's third law says that for every
force there's an equal and opposite force
force
in that action reaction pair and so for
example we've got
f12 and f21 and if we add those two
forces together well we get
zero right they're exactly the same
length they're pointing in opposite directions
directions
and so adding those two together gives
you zero and likewise adding this one
and this one
should also give you zero and adding
this one and
this one should also give you zero so
hopefully you know where i'm going with this
this
what i'm suggesting is that all of the
internal forces in a particle
are going to sum to zero and so that
entire term
goes away thanks to newton's third law
and so that means that we have the
change in momentum
of the entire system due only to the forces
forces
acting outside of that system on the system
system
so i'm going to write that down finally
over here
as being just f net
and so that f net there means the net
okay and so i'm going to write this over
here just so we have it a little bit
more cleanly
what i'm saying is that f net
which again is all of the forces acting
on the entire system not those internal forces
forces
is equal to dp dt
and that's the capital p there that's
the total momentum of the system
this if you like is newton's second law but
but
restate it for a system of particles and
it's a really important result
conceptually and as we'll see
practically in a second
because it's going to lead to our first
conservation law but i want you to imagine
imagine
the following situation right here's a
tennis ball and we've already talked
about tennis balls in the last video
this tennis ball as i throw in the air
right it goes through a nice projectile
motion kind of shape there right a nice
parabola and it does that because of
course the only force acting on it is gravity
gravity
right pulling it down as usual but wait
a second
this tennis ball is not really a particle
particle
it's actually made up of a bunch of
molecules and atoms and those molecules
and atoms have forces between them
so how come we don't have to worry about
all of those internal forces between the
atoms and molecules that make up this
tennis ball
the answer comes from this and this
derivation this is why i went through
this derivation in such detail
it allows us to take a complicated
object something where there might be a
huge amount of very complicated internal forces
forces
and allow us to completely ignore those
internal forces
and focus only on the motion of the
entire object itself
under the influence only of external forces
forces
like the force of gravity so it's a big
deal can't be overstated but
but think about that carefully but we're
going to see how to use it practically
in just a second
something interesting happens to our
newton's second law over here
in a very special case and we're going
to call this case an isolated system
okay for an isolated system what that
means basically is that there are no
external forces acting on the system
from outside
or i should say should clarify a little
bit not that there are no external forces
forces
but the net external force is zero so
something might be pushing on it
something might be pushing another way
but if those forces balance we're still
calling that an isolated system
so in an isolated system the net force
acting on the system externally is equal
to zero i put a little vector sign above
that zero just to be clear that it's the
zero vector
not a zero magnitude doesn't really
matter too much the key thing
is that f net is equal to zero what that means
means
according to our newton's second law
down there is that
the derivative of the total momentum
with respect to time must be zero
but that means something very special
about the momentum
p the total momentum not the momentum of
any one of these particles but the total
of all of them
it must be constant right the only thing
that you take a derivative of and get zero
zero
is a constant and so what we're saying
here is that for an isolated system
the momentum whatever happens to be at
one point in time
stays that amount for all of the rest of
the time right as long as that system is isolated
isolated
the momentum is totally a constant we
give that kind of a special name
in physics because this is a very
special situation
so i'm going to call that the law we
don't use the word
allah in physics right newton's law is
the only other place where we really see
them but this is just as important as
newton's laws
of momentum and what it says
is that for an isolated system we have
p i equal to p f i'm going to put vector
signs above those because the vector nature
nature
of the momentum is really important
there are two things to be careful of here
here
first the vector nature that i already
mentioned and the fact that it has to be
an isolated system but
second this is the total momentum only
it doesn't say that an
individual particle's momentum is
conserved it says that when you add up
all of the momentum of all of the
particles in your system that number is
always the same value whatever it is initially
initially
is the same at some final point as well
so that
is a really important statement the law
put a big box around that that is probably
probably
so far you know maybe after f equals m a
the most important thing that we've
actually discussed because
it's actually a really fundamental thing
in physics
um there are situations where f equals m
a might not be the best to apply
situations like relativity for example
we have to modify the ideas
behind it but conservation of momentum
is always
true in every situation that we've ever seen
seen
whether we're talking about very tiny
objects under quantum mechanics or very
fast objects in relativity
this law of conservation of momentum
under this situation where we have an
isolated system
if that's the case this has never ever
been found to be false
and so it's a fundamental rule of our universe
universe
that momentum is conserved as long as
the net force on a system is zero
we'll do some practical examples in a
minute to
explore this law of conservation of
momentum in more detail but before we do that
that
i want to emphasize one aspect of it and
that's the idea of a system
and so think about this tennis ball
again right here's our good old tennis
ball that we've used a bunch of times already
already
and i would imagine this really simple
situation it's just going to drop
all right we're going to drop the tennis
ball and it's going to move downwards of
course towards the surface of the earth
because gravity is pulling on it
so i'm going to try to sketch this out
here here's the tennis ball
and the tennis ball is moving downwards
with some velocity
that's the that's the situation
so the question is is momentum conserved
in this tennis ball dropping to the
ground example
and the answer is that it depends right
and this is often the case in physics
because we have a choice to make we need
to know
what is the system and so there are two
different ways that we can do this and so
so
the first choice that i'll make is that
we will say
this is going to be our system here
right so this is
uh choice number one and in choice
number one we'll suppose that our system is
is
just the ball in that case we can
definitively as soon as we answer this
rate it's no longer going to be vague
we can definitively say that momentum is
not conserved and the reason is is very simple
simple
there is an external force right the
force of gravity is pulling
down on that tennis ball and that's an
external force that's coming from
outside the system since it's coming
from the entire earth
and so momentum is not conserved that
makes sense if you think about the
momentum of the tennis ball
as it drops to the ground right it goes
faster and faster and faster
its momentum is changing and so it
starts out you know with very small
momentum ends up with a lot
momentum is not conserved on the other
hand right we can have a situation like
this here's our tennis ball again
and drawing it like it's a trying it
more like it's a baseball it kind of
looks like a baseball to me it's a
tennis ball
um and so in this case again we're
dropping it right it's got some velocity
but now this is a not to scale
but this is the entire earth right uh and
and
and so we're dropping that tennis ball
but i want to make my system this time
i want to make it both of these things i
want to make it the tennis ball
and the entire earth and if that's the case
case
then we actually end up go back to that
same marker
we end up with a situation where we've
got a force of gravity pulling down on
the ball
but from newton's third law an equal
force pulling up
on the earth from the tennis ball and
this again plays into this weird sort of idea
idea
that the forces are equal but the
reactions are not what i'm suggesting is
that because these are now
internal forces there is no external
force here
momentum is conserved right and so in
this case
we can say that it is conserved but that
seems strange to me
what i'm suggesting is that the tennis
ball as it drops and gains momentum
has to do so at the expense of the earth
losing momentum because the total
has to be the same but how can the earth
lose momentum how can we
notice that how come when we drop the
tennis ball we notice the tennis ball
moving towards the earth
but not the earth coming up to meet the
tennis ball and the answer of course
is that the earth's mass is huge
compared to the tennis ball
and if that's the case remember momentum
is m times v
and if the mass is really really big in
order to have a
momentum that's equal to what the tennis
ball gains
the velocity is vanishingly small right
and so the physics says that yes the
earth is going to gain momentum
when you drop a ball towards it and
that's because the ball
drops and the earth comes up to meet it
but it does so at a
ridiculously tiny velocity because the
mass of the earth is so so big so we're
never going to be able to notice that
but it's a similar situation we talked
about with the mosquito in the truck
or dropping a tennis ball to the earth
and thinking about forces
the forces are the same the momenta are
the same but because the masses are
different the reactions are very different
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