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# Introduction to tension

An introduction to tension. Solving for the tension(s) in a set of wires when a weight is hanging from them. Created by Sal Khan.

Video transcript

I will now introduce you to
the concept of tension. So tension is really just the
force that exists either within or applied by
a string or wire. It's usually lifting something
or pulling on something. So let's say I had a weight. Let's say I have
a weight here. And let's say it's
100 Newtons. And it's suspended from this
wire, which is right here. Let's say it's attached to
the ceiling right there. Well we already know that the
force-- if we're on this planet that this weight is being
pull down by gravity. So we already know that there's
a downward force on this weight, which is
a force of gravity. And that equals 100 Newtons. But we also know that this
weight isn't accelerating, it's actually stationary. It also has no velocity. But the important thing is
it's not accelerating. But given that, we know that the
net force on it must be 0 by Newton's laws. So what is the counteracting
force? You didn't have to know about
tension to say well, the string's pulling on it. The string is what's keeping
the weight from falling. So the force that the string or
this wire applies on this weight you can view as
the force of tension. Another way to think about it
is that's also the force that's within the wire. And that is going to exactly
offset the force of gravity on this weight. And that's what keeps this point
right here stationery and keeps it from
accelerating. That's pretty straightforward. Tension, it's just the
force of a string. And just so you can
conceptualize it, on a guitar, the more you pull on some of
those higher-- what was it? The really thin strings that
sound higher pitched. The more you pull on it,
the higher the tension. It actually creates a
higher pitched note. So you've dealt with
tension a lot. I think actually when they sell
wires or strings they'll probably tell you the tension
that that wire or string can support, which is important if
you're going to build a bridge or a swing or something. So tension is something that
should be hopefully, a little bit intuitive to you. So let's, with that fairly
simple example done, let's create a slightly more
complicated example. So let's take the same weight. Instead of making the
ceiling here, let's add two more strings. Let's add this green string. Green string there. And it's attached to the
ceiling up here. That's the ceiling now. And let's see. This is the wall. And let's say there's another
string right here attached to the wall. So my question to you is, what
is the tension in these two strings So let's call
this T1 and T2. Well like the first problem,
this point right here, this red point, is stationary. It's not accelerating in
either the left/right directions and it's not
accelerating in the up/down directions. So we know that the net forces
in both the x and y dimensions must be 0. My second question to
you is, what is going to be the offset? Because we know already that
at this point right here, there's going to be a downward
force, which is the force of gravity again. The weight of this
whole thing. We can assume that the wires
have no weight for simplicity. So we know that there's going
to be a downward force here, this is the force of
gravity, right? The whole weight of this entire
object of weight plus wire is pulling down. So what is going to be the
upward force here? Well let's look at each
of the wires. This second wire, T2, or we
could call it w2, I guess. The second wire is just
pulling to the left. It has no y components. It's not lifting up at all. So it's just pulling
to the left. So all of the upward lifting,
all of that's going to occur from this first wire, from T1. So we know that the y component
of T1, so let's call-- so if we say that
this vector here. Let me do it in a
different color. Because I know when I draw these
diagrams it starts to get confusing. Let me actually use
the line tool. So I have this. Let me make a thicker line. So we have this vector
here, which is T1. And we would need to figure
out what that is. And then we have the other
vector, which is its y component, and I'll draw
that like here. This is its y component. We could call this T1 sub y. And then of course, it has an
x component too, and I'll do that in-- let's see. I'll do that in red. Once again, this is just
breaking up a force into its component vectors like we've--
a vector force into its x and y components like we've been
doing in the last several problems. And these are just
trigonometry problems, right? We could actually now, visually
see that this is T sub 1 x and this is
T sub 1 sub y. Oh, and I forgot to give you an
important property of this problem that you needed to
know before solving it. Is that the angle that the
first wire forms with the ceiling, this is 30 degrees. So if that is 30 degrees, we
also know that this is a parallel line to this. So if this is 30 degrees,
this is also going to be 30 degrees. So this angle right here is also
going to be 30 degrees. And that's from our-- you know,
we know about parallel lines and alternate
interior angles. We could have done
it the other way. We could have said that if this
angle is 30 degrees, this angle is 60 degrees. This is a right angle,
so this is also 30. But that's just review
of geometry that you already know. But anyway, we know that this
angle is 30 degrees, so what's its y component? Well the y component,
let's see. What involves the hypotenuse
and the opposite side? Let me write soh cah toa at the
top because this is really just trigonometry. soh cah toa in blood red. So what involves the opposite
and the hypotenuse? So opposite over hypotenuse. So that we know the sine-- let
me switch to the sine of 30 degrees is equal to T1 sub y
over the tension in the string going in this direction. So if we solve for T1 sub y we
get T1 sine of 30 degrees is equal to T1 sub y. And what did we just say
before we kind of dived into the math? We said all of the lifting on
this point is being done by the y component of T1. Because T2 is not doing any
lifting up or down, it's only pulling to the left. So the entire component that's
keeping this object up, keeping it from falling
is the y component of this tension vector. So that has to equal the force
of gravity pulling down. This has to equal the
force of gravity. That has to equal this
or this point. So that's 100 Newtons. And I really want to hit this
point home because it might be a little confusing to you. We just said, this point
is stationery. It's not moving up or down. It's not accelerating
up or down. And so we know that there's a
downward force of 100 Newtons, so there must be an upward force
that's being provided by these two wires. This wire is providing
no upward force. So all of the upward force must
be the y component or the upward component of this force
vector on the first wire. So given that, we can now solve
for the tension in this first wire because we have
T1-- what's sine of 30? Sine of 30 degrees, in case you
haven't memorized it, sine of 30 degrees is 1/2. So T1 times 1/2 is equal
to 100 Newtons. Divide both sides by 1/2
and you get T1 is equal to 200 Newtons. So now we've got to figure out
what the tension in this second wire is. And we also, there's
another clue here. This point isn't moving left
or right, it's stationary. So we know that whatever the
tension in this wire must be, it must be being offset by a
tension or some other force in the opposite direction. And that force in the opposite
direction is the x component of the first wire's tension. So it's this. So T2 is equal to the
x component of the first wire's tension. And what's the x component? Well, it's going to be the
tension in the first wire, 200 Newtons times the cosine
of 30 degrees. It's adjacent over hypotenuse. And that's square root
of 3 over 2. So it's 200 times the square
root of 3 over 2, which equals 100 square root of 3. So the tension in this wire is
100 square root of 3, which completely offsets to the left
and the x component of this wire is 100 square root of
3 Newtons to the right. Hopefully I didn't
confuse you. See you in the next video.