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
