Treating Systems (the easy way) Video Lecture | Study JEE Foundation - Class 9 | Best Video for Class 9

so in the previous video we solved this
problem the hard way maybe you watched
it maybe you didn't maybe you just
skipped right to here and you're like I
don't you want another hard way just
show me the easy way please well as
we're going to talk about now turns out
there's a trick and the trick is after
you solve this problem the hard way with
a five kilogram mass and a three
kilogram mass when you find the
acceleration what you get is this that
the acceleration of the five kilogram
mass is just 29 point four divided by
eight kilograms but when you do enough
of these you might start realizing wait
a minute twenty-nine point four Newtons
that was just the force of gravity
pulling on this three kilogram mass in
other words the only force that was
really propelling this whole entire
system forward release the only external
force propelling it forward and eight
kilograms down here you might be like
wait eight kilograms that's just five
kilograms plus three kilograms is that
just a coincidence or is this telling us
something deep and fundamental and if
not a coincidence turns out you'll
always get this that what you'll end up
with after solving this the hard way
you'll get that in the very end you'll
get all the external forces added up
here where forces that make it go like
this force of gravity end up being
positive and forces that try to like
resist the motion so if there was
friction that would be an external force
that tries to resist motion would be up
top and then you get the total mass on
the bottom and this makes sense the
acceleration of this entire system if we
think about it as a single object so if
you imagine this was just one single
object and you asked yourself what's the
total acceleration of this entire system
well it's only going to depend on the
external forces and in this case the
only external force making it go was
this force of gravity right here you
might object you might be like wait what
about this tension right here isn't the
tension pulling on this five kilogram
mass making the system go it is but
since it's an internal force now if
we're treating this entire system as our
one object since this tension is pulling
trying to make it go you've got another
tension over here resisting the motion
on this mass trying to make
stop that's what internal forces do
there's always equal and opposite on one
part of the object on the other so you
can't propel yourself forward with an
internal force so these end up
cancelling out essentially the only
force you have in this case was the
force of gravity on top only external
forces and the total mass on the bottom
and that's the trick that's the trick to
quickly find the acceleration of some
system that might be complicated if you
had to do it in multiple equations and
multiple unknowns but much much easier
once you realize this so the trick
sometimes it's called just treating
systems as a single object let me just
show you really quick if that made no
sense let me just show you what this
means if we just get rid of this so what
I'm claiming is this if you ever have a
system where multiple objects are
required to move with the exact same
magnitude of acceleration right because
maybe they're tied together by a rope or
maybe they're pushing on each other
maybe there's many boxes in a row and
these boxes all have to be pushed at the
same acceleration because they can't get
pushed through each other right if
there's some condition where multiple
objects must have the same magnitude of
acceleration then you can simply find
the acceleration of that system as if it
were a single object and writing sys for
system by just using Newton's second law
but instead of looking at an individual
object for a given direction we're just
going to do all of the external forces
all the external forces on our system
treat it as if it were a single object
divided by the total mass of our system
and so when you plug in these external
forces these are forces that are
external so external means not internal
to the system so if I if I think of this
five kilogram box and this three
kilogram box as a single mass tension
would be an internal force because it's
applied internally between between these
two objects between objects inside of
our system but the force of gravity on
the three kilogram mask that's an
external force because that's the earth
pulling down on the three kilogram mass
and the earth is not part of our system
similarly the normal force is an
external force but it's exactly canceled
by the gravitational force so even
though those are external they're not
going to make it in here I mean you can
put them in there but they're just going
to cancel anyway we only look at forces
in the direction of motion and if it's a
force that causes motion we're going to
make that a positive force if it's a
force in the direction of motion like
this force of gravity is we make those
positive forces so forces will be
plugged in positive into here if they
make the system go and that might seem
weird you might be like wait how do I
decide if it makes the system go we'll
just ask yourself is that force directed
in the same direction as the motion of
the system so we're just saying the
system is going to accelerate if there's
forces that make it go and we're going
to make negative we're going to plug in
negative forces the forces that make the
system stop or resist the motion of the
system so maybe I should say resists
motion of the system in this case for
this one down here I don't have any of
those so resist motion of the system I
don't have any of those I could have if
I had a force of friction then there'd
be a external force that resists the
motion I would plug in that external
force as a negative because it resists
the motion too even though this might
sound weird it it makes sense if you
think about it the acceleration of our
system treated as a single object is
only going to depend on the forces that
try to make the system go and the forces
that try to make the system stop or
resist the motion so if we add those
accordingly with positives and negatives
we divide by the total mass which gives
a total measure of the inertia of our
system we'll get the acceleration of our
system it makes sense and it works
turns out this always works and it saves
a ridiculous amount of time for instance
if we wanted to do this problem if you
just gave me this problem straight away
and you were told do this however you
want I would use this trick I would say
that the acceleration of this system
which is composed of this five kilogram
mass and our three kilogram mass which
is going to be equal to I'd ask myself
what force makes this system go what
force drives the system and it's this
force of gravity on the three
kilogram mass that's driving this system
right if you took this force away if you
eliminated that force nothing is going
to happen here this is the force making
the system go so I'd put in my three
kilograms times 9.8 and at this point
you might be like whoa okay that gravity
made it go should I include this gravity
too but now that gravity is
perpendicular to the motion for one so
this grip this gravity isn't making the
system go that's just causing this mass
to sit on the table and for two it's
canceled by that normal force so those
cancel anyway even though they're
external forces this is it this is the
only one that drives the system so I put
that in here and I divide by my total
mass because that tells me how much my
system resists through inertia changes
in velocity and this is what I get I get
the same thing I got before I get back
my three point six eight meters per
second squared and I get in one line I
mean this trick is amazing and it works
and it works in every example where two
masses or more masses are forced to move
with the same acceleration so this is
great this will save you a ton of time
this is supposed to be a three here and
to show you how how useful it is let's
say there was friction let's say there
was a coefficient of friction of 0.3
well now I'd have a frictional force so
there'd be an external frictional force
here it'd be applied to this five
kilogram mass I'd have to subtract it up
here so if I get rid of this so it's not
going to be three point six eight
anymore I'm going to have a force of
friction that I have to subtract so
minus mu K so the force of friction
I'll just put force of friction and so
to solve for the force of friction the
force of friction is going to be equal
to well I know three times 9.8 is let me
just write this in here twenty nine
point four Newtons minus the force of
friction is given by so there's a
formula for force of friction the force
of friction is always mu K F n so the
force friction on this five kilogram
mass is going to be mu K which is 0.3 so
it's going to be 0.3 times the normal
force not the normal force on our entire
system
I don't include this three kilogram mask
it's only the normal
on this five kilogram mass that's
contributing to this force of friction
here so even though we're treating the
system as a whole we still have to find
individual forces on these individual
boxes correctly so it won't be the
entire mass that goes here the normal
force on the five kilogram mass is just
going to be 5mm kilograms times 9.8
meters per second squared I divide by my
total mass down here because the entire
mass is resisting motion through inertia
now if I solve this for my acceleration
of the system I get one point eight four
meters per second squared so this is
less less than our three point six eight
and that makes sense now there's a
resistive force a resistive external
force trying to prevent the system from
moving but you have to be careful what
I'm really finding here I'm really
finding the magnitude of the
acceleration this is just giving me the
magnitude if I'm playing this game where
positive forces are ones that make it go
and negative forces are ones that resist
motion external forces that is I'm just
getting the magnitude of the
acceleration individual boxes will have
that magnitude of the acceleration but
they may have positive or negative
accelerations in other words this five
kilogram mask accelerating to the right
going to have a positive acceleration in
other words the acceleration of the five
kilogram mass will be positive one point
eight four and the acceleration of the
three kilogram mass since it's
accelerating downward will be negative
one point eight four meters per second
squared