Treating systems (the easy way) | Forces and Newton's laws of motion | Physics | Khan Academy

- [Voiceover] 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 even wanna know the hard way.

"Just show me the easy way please."

Well, that's what we're
gonna 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 29.4 divided by eight kilograms.

But when you do enough of these,
you might start realizing,

"Wait a minute. 29.4 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.

Or at least the only external
force propelling it forward.

And then eight kilograms down here.

You're gonna 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 it's not a coincidence.

Turns out you'll always get this.

That what you'll end up with
after solving this hard way,

you'll get 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 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 gonna 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 this 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
trying to make it go,

you've got another tension over here

resisting the motion on this mass,

trying to make it stop.

That's what internal forces do.

There's always equal and opposite

on one part of the object than 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 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 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.

I'm 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 gonna do all
of the external forces,

all of the external forces on our system,

treat it as if it were a single object,

divide it 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 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 these two objects,

between objects inside of our system.

But the force of gravity
on the three kilogram mass,

that's an external force
'cause 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 cancelled
by the gravitational force.

So even though those are external,

they're not gonna make it in here.

I mean, you can put 'em in there

but they're just gonna cancel anyway.

We only look at forces in
the direction of motion

and if it's a force that causes motion,

we're gonna 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?"

Well, 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 gonna accelerate

if there's forces that make it go

and we're gonna plug in negative forces,

the forces that make the system stop

or resist the motion of system.

So maybe I should say,
"Resist 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 resist the motion.

I would plug in that
external force as a negative

'cause it resist the motion.

So even though this might sound weird,

it makes sense if you think about it.

The acceleration of our system
treated as a single object

is only gonna 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 it by the total mass

which gives the total measure
of the inertia of our system,

we'll get the acceleration of our system.

It makes sense and it works.

Turns out it 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.

And I would say that the
acceleration of this system

which is composed of
this five kilogram mass

and our three kilogram mass

is just gonna be equal to---

I'd ask myself, "What
force makes this system go?

"What force drives this 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's gonna happen here.

This is the force making the system go

so I'd put it in my three
kilograms times nine point eight.

And at this point,

you might be like, "Well,
okay, that gravity made it go.

"Should I include this gravity, too?"

But no, that gravity is
perpendicular to the motion for one

so this gravity isn't making the system,

that's just causing this
mass to sit on the table

and for two, it's cancelled
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

'cause 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'll
save you a ton of time.

This is supposed to be a three here.

And to show you how useful it is,

let say there was friction,

let's say there was a
coefficient of friction

of zero point three.

Well, now I'd have a frictional force

so there'd be an external
frictional force here.

It'd be applied this five kilogram mass.

I'd have to subtract it up here.

So if I get rid of this---

So it's not gonna be three
point six eight anymore.

I'm gonna 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 gonna be equal to---

Well, I know three times
nine point eight is---

Let me just write this in here,

29.4 Newtons minus the force
of friction it's given by.

So there's a formula
for force of friction.

The force of friction is always mu K FN.

So the force of friction
on this five kilogram mass

is gonna be mu K which is point three.

So it's gonna be zero point
three times the normal force,

not the normal force on our entire system.

I don't include this three kilogram mass.

It's only the normal force
on this five kilogram mass

that's contributing to this
force of friction here.

So even though we're treating
this system as a whole,

we still have to find individual forces

on this individual boxes correctly.

So it won't be the entire
mass that goes here.

The normal force on the five kilogram mass

is just gonna be five kilograms
times nine point eight

meters per second squared.

I divide by my total mass down here

because the entire mass is
resisting motion through inertia.

And if I solve this from my
acceleration of the system,

I get one point eight four
meters per seconds 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,

tryna 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 mass
accelerating to the right,

gonna 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 seconds squared.