- Good morning.
This object, which has a mass of 155 grams,
and I will call the hanging mass,
when I place it right here on the string,
it will cause the right string to be completely horizontal
and the left string to be at an angle
of 28 degrees with the horizontal.
Given those two pieces of information,
we can determine the tension in each of these two strings.
♫ Flipping Physics ♫
In order to solve this problem,
we need to draw a free body diagram.
That is step one. Class, what is step two?
- [All] Break forces into components.
- Step three.
- [All] Redraw the free body diagram.
- Step four.
- [All] Sum the forces.
- And step five.
- [All] Sum the forces.
- Again.
- [mr.p] Very nice.
Step one.
Billy, could you please tell me what forces are
in the free body diagram?
- The force of gravity is straight down.
The force normal is up,
and there are two forces of tension.
Each tension force is in the direction of the string.
- In an effort not to muddle our picture,
I have drawn the free body diagram down here
where this point right here corresponds
to that point right there in our picture.
And, also, I have identified this as force of tension one
and this one as force of tension two.
- [Bo] Mr. P.
- Yes, Bo.
- Why is there a force normal in our free body diagram?
- Because Billy said there was a force normal.
- I thought a force normal was caused by
the push of a surface.
What surface is the mass hanging touching?
- The mass hanging isn't touching a surface,
so there is no force normal.
So, sorry.
I'm just so used to having a force normal
in our free body diagrams that I assumed it was there.
- It's okay.
- That's absolutely fine, Billy.
It's actually very common for students to add
a force normal to every free body diagram
simply because it seems like every free body diagram
we've done so far had a force normal in it.
However, when you draw your free body diagrams,
you have to think through the reasons for every force
and not just follow patterns.
Step two, break forces into components.
Bo, which force or forces do we need to
break into components and why?
- The force of gravity is in the y direction,
and the force of tension two is in the x direction,
so we don't need to break either of those forces
into components.
The force of tension one is neither directly
in the x or y direction, so we need to break
the force of tension one into components.
- Bobby, could you please break force of tension one
into its components?
- But, um, but we don't know tension force one,
so how can we break it into components?
- Yeah. What?
- Okay, now, we do not have a number
for force of tension one, but that does not mean
we can't break it into its components.
Now, generally, many of you have
what I like to call a numbers dependency.
You have a really hard time moving forward
if you don't have numbers for your variables,
and I will help you let go of your numbers dependency.
It is very important that you do so.
Here's what I'd like you to do.
Bobby, please solve for the force of tension one
in the y direction even though we do not have
a number for the force of tension one.
- Uh, okay.
- Um, so...
- [Mr. P] SOH is correct, Bobby.
You can use that.
- [Bobby] So sine theta - [Billy] That funny.
equals opposite over hypotenuse.
Opposite theta is the force of tension one
in the y direction over the hypotenuse,
which is the force of tension one.
Multiplying by force of tension one gives us
force of tension one in the y direction
equals force of tension ones times the sine of theta.
Oh, and we know theta, so we can plug in 28 degrees for--
- [Mr. P] Oh, hold on.
Please, don't plug in numbers at this point.
That would just stoke your numbers dependency,
which is something I am unwilling to do.
Now, rather than having a number for the force
of tension one in the y direction, we have an equation,
and we can't do anything more with this equation,
so we need to put it in our equation holster
just in case we need to pull it out
at some point and use it.
(laughs)
- What did he just do?
- He put that equation in our equation holster.
- [Bobby] Yeah, that doesn't really make sense to me.
- Equation holster. Oh, okay.
We save the equation for later when we need to use it.
Okay, I got it.
That makes sense.
- We can do the same thing with cosine theta.
The cosine of theta equals adjacent over hypotenuse
or the force of tension one in the x direction
divided by the force of tension one.
Therefore, the force of tension one in the x direction
equals the force of tension one times the cosine of theta.
And, class, we're going to put this equation in where?
- [All] Our equation holster.
- That's right, in our equation holster.
- We have completed step number two,.
We have broken forces into components.
Bobby, could you please tell me what step number three is
and do it?
- We need to redraw the free body diagram.
We still have the force of gravity down
and the force of tension two along the string to the right.
Instead of force of tension one, we now have
force of tension one in the x direction
and force of tension one in the y direction.
- Bo, could you please do step number four?
- We need to sum the forces.
Let's start with the y direction, so the net force
in the y direction equals the force of tension one
in the y direction minus the force of gravity,
and the net force is, according to Newton's second law,
always equal to mass times acceleration, in this case,
the acceleration in the y direction.
The hanging mass isn't moving in the y direction,
so its acceleration in the y direction is zero.
Therefore, the tension force one in the y direction
minus the force of gravity equals zero.
Add the force of gravity to both sides,
and we get the force of tension one in the y direction
equals the force of gravity.
- Bo, please keep going.
- For the force of gravity, we can substitute
mass times the acceleration due to gravity and, uh,...
- Oh, oh, the equation holster.
- Right.
From the equation holster, we know the force of tension one
in the y direction equals the force of tension one
times the sine of theta.
Dividing by sine theta gives us the force of tension one
equals mass times the acceleration due to gravity
divided by the sine of theta.
Can we plug in numbers now?
- Absolutely.
Now that we know everything in the equation,
we can substitute in numbers.
- Wait a second.
First we need to convert the mass hanging to kilograms
because force is in Newtons, and a Newton is a kilogram
times meters per second squared.
So 155 grams times one kilogram
over 1,000 grams, and that means
the mass hanging is 0.155 kilograms.
- Therefore, the force of tension one equals
0.155 times 9.81
divide by the sine of 28 degrees,
which is 3.238854
or with two sig figs, 3.2 Newtons.
- We have solved for force of tension one.
Billy, could you please do step five?
- The last step is to sum the forces in a direction
perpendicular to our previous step.
So the net force in the x direction equals tension force two
minus the tension force one in the x direction,
which then is equal to mass times the acceleration
in the x direction, and the acceleration in x direction
is equal to zero because the mass hanging isn't moving.
- Hey.
The mass hanging is an object in equilibrium
because the net force acting on it in all directions
is equal to zero, right, Mr. P?
- Yeah, it's in equilibrium.
- Bobby, that is correct.
Billy, could you please continue?
- Sure.
Therefore, tension force two equals tension force one
in the x direction.
From our equation holster, we know tension force one
in the x direction equals tension force one
times the cosine of theta.
Therefore, tension force two equals
3.238854
times the cosine of 28 degrees,
which is 2.85974,
with two sig figs, 2.9 Newtons.
- We have solved for the force of tension
in each of the two strings.
And you may not have noticed, but these two blue items
right here are force sensors, and let's go back
to the very beginning when I added the mass to the string.
And you could see the force that we measured
using those two force sensors matches exactly
what we got from solving the problem, which is nice.
- [Billy] That's pretty cool.
- [Bo] Absolutely.
- Thank you very much for learning with me today.
I enjoyed learning with you.