Super hot tension | Forces and Newton's laws of motion | Physics | Khan Academy

- [Voiceover] Oh, it's time.

It's time for the super
hot tension problem.

We're about to do this right here.

We've got our super hot can of red peppers

hanging from these strings.

We want to know what the
tension is in these ropes.

This is for real now, this
is a real tension problem.

And here's the deal.

You might look at this,
you might get frightened.

You might think, I've gotta come up with

a completely new strategy to tackle this.

I've gotta throw away
everything I've learned

and just try something new.

And that's a lie.

You should not lie to yourself.

Use the same process.

We're gonna use the same process we used

for the easy tension problems,

because it's gonna lead
us to the answer again.

Be careful. Don't stray
from the strategy here.

The strategy works.

So we're gonna draw our
force diagram first.

That's what we always do.

We're gonna say that the forces are

force of gravity on
this can of red peppers,

which is MG, and if it's 3 kilograms,

we know 3 kilograms times about 10,

we're gonna say, let's
approximate G as 10 again

to make the numbers come out nice.

So instead of using 9.8,
we'll say G is about 10,

and so we'll say 3 kilograms

times 10 meters per second
squared is gonna be 30 Newtons.

And so the force of gravity
downward is 30 Newtons.

What other forces do we have?

We've got this T1, remember
tension does not push.

Ropes can't push, ropes can only pull,

so T1's gonna pull that way.

So I'm gonna draw T1 coming this way.

So here's our T1.

And then we're gonna have
T2 pointing this way,

so this is T2.

Again, T2 pulls, just like all tension.

Tension pulls, tension can't push.

So I've got tension 2 going this way.

That's it, that's our force diagram.

There's no other forces.

I don't draw a normal force,

'cause this can isn't in
contact with another surface.

So there's no normal force,
you've got these two tensions,

the force of gravity.

And now we do the same thing we always do.

After our force diagram,
we use Newton's Second Law

in one direction or another.

So let's do it.

Let's say that acceleration
is the net force

in a given direction, divided by the mass.

Which direction did we pick again?

It's hard to say, we've
got forces vertical,

we've got forces horizontal.

There's only two directions to
pick, X or Y in this problem.

We're gonna pick the vertical direction,

even though it doesn't
really matter too much.

But because we know one of
the forces in the vertical

direction, we know the force of gravity.

Force of gravity is 30 Newtons.

Usually that's a guod
strategy, pick the direction

that you know something about at least.

So we're gonna do that here.

We're gonna say that the
acceleration vertically

equals to the net force
vertically over the mass.

And so now we plug in.

If this can is just sitting here,

if there's no acceleration,
if this is in not an elevator

transporting these peppers up or down,

and it's not in a rocket,
if it's just sitting here

with no acceleration, our
acceleration will be zero.

That's gonna equal the net force
and the vertical direction.

So what are we gonna have?

So what are the forces in
the vertical direction here?

One force is this 30
Newton force of gravity.

This points down, we're gonna
assume upward is positive,

that means down in a negative.

So I'll just put -30 Newtons.

I could have written -MG,

but we already knew it was 30 Newtons,

so I'll write -30 Newtons.

Then we've got T1 and T2.

Both of those point up.

But they don't completely point up,

they partially point up.

So part of them points to the right,

part of them points upward.

Only this vertical
component, we'll call it T1Y,

is gonna get included
into this calculation,

'cause this calculation
only uses Y directed forces.

And the reason is only Y directed forces,

vertical forces, affect
the vertical acceleration.

So this T1Y points upward,

I'll do plus T1 in the Y direction.

And similarly, this T2.

It doesn't all point vertically,

only part of it points vertically.

So I'll write this as
T2 in the Y direction.

And that's also upward,

so since that's up, I'll count it

as plus T2 in the Y direction.

And that's it, that's all our forces.

Notice we can't plug in
the total amount T2 in this

formula, 'cause only part of it points up.

Similarly, we have to plug in
only the vertical component

of the T1 force because only
part of it points vertically.

And then we divide by the
mass, the mass is 3 kilograms.

But we're gonna multiply
both sides by 3 kilograms,

and we're gonna get zero
equals all of this right here,

so I'll just copy this right here.

We use this over again,
that comes down right there.

But now there's nothing
on the bottom here.

So what do we do at this point?

Now you might think we're stuck.

I mean, we've got two unknowns in here.

I can't solve for either one,

I don't know either one of these.

I know they have to add up to 30,

so I'd do fine, if I
added 30 to both sides,

I'd realize that these
two vertical components

of these tension forces added up

have to add up to 30,
and that makes sense.

They have to balance the force downward.

But I don't know either of
them, so how do I solve here?

Well, let's do this.

If you ever get stuck on
one of the force equations

for a single direction, just
go to the next equation.

Let's try A in the X direction.

So for A in the X direction,
we have the net force

in the X direction, over the mass,

again, the acceleration is gonna be zero

if these peppers are not
accelerating horizontally.

So unless this thing's in
a train car or something,

and the whole thing's accelerating,

then you might have
horizontal acceleration.

And if it did, it's
not that big of a deal,

you just plug it in there.

But assuming it's acceleration zero,

because the peppers
are just sitting there,

not changing their velocity,
we'll plug in zero.

We'll plug in the forces
in the X direction.

These are gonna be T1 in the X.

So part of this T1 points
in the X direction.

Similarly, part of T2
points in the X direction.

We'll call this T2X.

We use these as the magnitude.

Let's say T2X is the
magnitude of the force

that T2 pulls with to the left,

and T1 is the magnitude that
T1 pulls with to the right.

So to plug these in, we've got to decide

whether they should be
positive or negative.

So this T1X, since it pulls to the right,

T1X will be positive.

We're gonna consider rightward
to be the positive direction,

'cause that's the typical
convention that we're gonna adopt.

And T2X pulls to the left.

That's gonna be a negative contribution,

so minus T2 in the X direction.

'Cause leftward would be negative.

We divided by the mass,
the mass was 3 kilograms,

but again, we'll multiply both sides by 3,

we'll get zero equals,
and then we just get T,

the same thing up here,

so we'll just copy this
thing here, put it down here.

And again, you might be concerned.

I can't solve this either.

I mean, I can solve for
T1X, but look at what I get.

If I just multi, or if I
added T2X to both sides,

I'm just gonna get T1 in the X direction

has to equal T2 in the X direction.

And that makes sense.

These two forces have to
be equal and opposite,

because they have to cancel so
that you have no acceleration

in the X direction.

And this was not drawn
proportionately, sorry,

this should be the exact
same size as this force

because they have to cancel,

since there's no horizontal acceleration.

But what do we do?

We can't solve this equation
we got from X direction.

We can't solve this equation
we got from the Y direction.

Whenever this happens,
when you get two equations,

and you can't solve either

because there's too many unknowns,

you're gonna have to end up
plugging one into the other.

But I can't even do that yet.

I've got four different variables here.

T1X, T2X, T1Y, and T2Y,

these are all four different variables,

I've only got two equations,
I can't solve this.

So the trick, the trick we're gonna use

that a lot of people don't like doing

because it's a little more sophisticated,

now we've gotta put these
all in terms of T1 and T2

so that we can solve.

If I put T1Y in terms of the total T1,

and then sines of angles,
and cosines of angles,

and I put T2Y in terms of T2 and angles,

and I do the same thing for 1X and 2X,

I'll have two equations,
and the only two unknowns

will be T1 and T2, then
we can finally solve.

If that didn't make any
sense, here's what I'm saying.

I'm saying figure out what
T1Y is in terms of T1.

So I know this angle here,
let's figure out these angles.

So these angles here are, if this is 30,

this angle down here has
to be 30 because these

are alternate interior angles.

And if you don't believe me,

imagine this big triangle over here,

where this is a right angle.

So this triangle from here
to there, down to here,

up to here, if this is 30,
that's 90, this has gotta be 60,

'cause it all adds up
to 180 for a triangle.

And if this right angle
is 90, and this side's 60,

this side's gotta be 30.

Similarly, this side's a right angle.

Look at this triangle, 60, 90,

that means this would have to be 30.

And so if I come down here,
this angle would have to be 60.

Just like this one,

'cause it's an alternate
interior angle, so that's 60.

So this angle here is 60,
this angle here is 30,

we can figure out what
these components are

in terms of the total vectors.

Once we find those,

we're gonna plug those
expressions into here,

and that will let us solve.

In other words, T1Y is gonna be,

once you do this for awhile you realize,

this is the opposite side.

So this component here is going to be

total T1 times sine of 30.

Because it's the opposite side.

And if that didn't make sense,
we'll derive it right here.

So what we're saying is that sine of 30,

sine of 30 is opposite over hypotenuse,

and in this case, the
opposite side is T1Y.

So T1Y over the total T1
is equal to sine of 30.

And we can solve this for T1Y now,

we can get the T1Y if I
multiply both sides by T1.

I get that that's T1 times sine of 30.

So that's what I said down here.

T1 is just T, oh sorry, forgot the one.

T1 times sine of 30.

Similarly, if you do the
same thing with cosine 30,

you'll get that T1X is T1 cosine 30,

by the exact same process.

Similarly over here, T2 is going to be,

I'm sorry, T2X is gonna be 2.

So T2 cosine 60, because
this is the adjacent side.

And T2Y is gonna be T2 sine of 60.

And if any of that doesn't make sense,

just go back to the
definition of sine and cosine,

write what the opposite side is,

the total hypotenuse side,
solve for your expression,

you'll get these.

If you don't believe me on
those, try those out yourselves.

But those are what these components are,

in terms of T2 and the
angles T2, T1 and the angles.

And why are we doing this?

We're doing this so that
when plug in over here,

we'll only have two variables.

In other words, if I plug
T1Y, this expression here,

T1 sine 30 in for T1Y,
similarly if I plug in

T2Y is T2 sine 60 into
this expression right there

for T2Y, look at what I'll get.

I'll get zero equals.

So I'll get negative 30 Newtons,

and then I'll get plus
T1Y was T1 sine 30, so T1,

and then sine 30, we can
clean this up a little bit.

Sine 30 is just a half.

So I'll just write T1 over 2, and then

'cause sine 30 is just one half.

And then T2Y is gonna be T2 sine 60,

and sine 60 is just root 3 over 2.

So I'll write this as plus T2
over 2, and then times root 3.

And you might think this is no better.

I mean this is still a
horrible mess right here.

But, look at. This is
in terms of T1 and T2.

That's what I'm gonna do over here.

I'm gonna put these in terms of T1 and T2,

and then we can solve.

So T1X is T1 over cosine 30,

so I'm gonna write this
as T1 times cosine 30,

and cosine 30 is root 3 over 2,

so this is T1 over 2 times root 3.

And that should equal T2X is right here,

That's T2 cosine 60, cosine 60 is a half.

So T2X is gonna be T2 over 2.

So T2 over 2.

So what I'm doing is, if
this doesn't make sense,

I'm just substituting
what these components are

in terms of the total
magnitude in the angle.

And I do this, because
look at what I have now,

I have got one equation with T1 and T2.

I've got another equation with T1 and T2.

So what I'm gonna do to solve these,

when we have two equations
and two unknowns,

you have to solve for
one of these variables,

and then substitute it
into the other equation.

That way you'll get one
equation with one unknown.

And you try to get the math right,

and you'll get the problem.

So I'm gonna solve this one is easier,

so I'm gonna solve this
one for, let's just say T2.

So if we solve this for T2,

I get that T2 equals, well, I can multiply

both sides by 2, and
I'll get T1 times root 3.

So T1 times root 3, because
the 2 here cancels with this 2,

or when I multiply both
sides by 2 it cancels out.

So we get that T2 equals
T1 root 3. This is great.

I can substitute T2 as T1
root 3 into here for T2.

And the reason I do that,

is I'll get one equation with one unknown.

I'll only have T1 in that equation now.

So if I do this, I'll
get zero equals negative,

you know what, let's
just move the -30 over.

This is kind of annoying here.

Just add 30 to both sides,

then take this calculation here.

We get plus 30 equals,
and then we're gonna have

T1 over 2, from this T1, so T1 over 2

plus, I've got plus, T2 is T1 root 3.

So when I plug T1 root 3 in for T2,

what I'm gonna get is,
I'm gonna get T1 root 3,

and then times another route 3,

because T2 itself was T1 root 3.

So I'm taking this expression here,

plugging it in for T2, but
I still have to multiply

that T2 by a root 3 and divide by 2.

And so, what do we get?

Root 3 times root 3 is just 3.

So we have T2 times 3
halves, plus T1 over 2.

So I'll get 30 equals,

and then I get T1 over 2,
we're almost there, I promise.

T1 over 2, plus, and this is
gonna be T1 times 3 over 2,

so it's gonna be 3 T1 over
2, or what does that equal?

T1 over 2 plus 3 T1
over 2 is just 4 halves.

So that's just 2 T1. So
this cleaned up beautifully.

So this is just 2 times T1,
and now we can solve for T1.

We get that T1 is simply 30 divided by 2.

If I divide both sides,
this left hand side by 2,

and this side here, this right side by 2,

I get T1 is 30 over 2 Newtons,

which is just, these should be Newtons,

I should have units on these,
which is just 15 Newtons.

Whoo, I did it, 15 Newtons.

T1 is 15 Newtons.

We got T1. That's one of them.

How do we get the other?

You start back over at the very beginning.

No, not really, that would be terrible.

You actually just take this T1,

and you plug it right into
here, boop, there it goes.

So T2, we already got it.

T2 is just T1 root 3.

So all I have to do is
multiply root 3 by my T1,

which I know now.

And I get that T2 is just
15 times root 3 Newton.

So once you get one of the forces,

the next one is really easy.

This is just T2.

So T2 is 15 root 3, and T1 is just 15.

So in case you got lost in the details,

the big picture recap is this.

We drew a force diagram,
we used Newton's Second Law

in the vertical direction
we couldn't solve,

because there were too many unknowns.

We used Newton's Second Law
in the horizontal direction,

we couldn't solve because
there were two unknowns.

We put all four of these unknowns

in terms of only two unknowns, T1 and T2,

by writing how those components depended

on those total vectors.

We substituted these expressions
in for each component.

Once we did that, we had two equations,

with only T1, T2, and T1 and T2 in them.

We solved one of these
equations for T2 in terms of T1,

substituted that into the other equation.

We got a single equation
with only one unknown.

We were able to solve for that unknown.

Once we got that, which is our T1,

once we have that variable,

we plug it back into that first equation

that we had solved for T2.

We plug this 15 in, we get
what the second tension is.

So even when it seems
like Newton's Second Law

won't get you there, if you have faith,

and you persevere, you will make it.

Good job.