- [Instructor] We're told that a metal ball
lies on a flat, horizontal surface.
It is attracted by two magnets placed around it.
We're told that the first magnets force on the ball is 5 N.
We're then told the second magnets force on the ball
is 3 N in a direction
that is 100 degree rotation from the first magnets force.
And we can see that drawn here.
This is the first magnets force, it's 5 N.
And then the second magnets force is 3 N
at 100 degree angle, 100 degree rotation
from the first magnets force.
Now they're asking us a few interesting questions.
What is the combined strength of the magnets' pulls?
And then they also say,
what is the direction of the magnets' combined pulls,
relative to the direction of the first magnets pull?
So I encourage you, pause this video
and have a go at this on your own
before we work through this together.
All right, now let's work through this together.
So they're really saying,
if I take the sum of these two vectors,
what is gonna be the resultant force vector?
What is going to be the magnitude
of that result in force vector?
And what is its direction going to be?
There's two ways we could approach this.
We could break it down each of these vectors
into their respective components,
and then add the respect of components,
and then from that figure out what the magnitude
and direction is.
And we do that in other videos,
or we could take the geometric approach.
So that's what we're gonna do here.
And to help us with that,
we're gonna use what we've called the Parallelogram rule
which is really the same idea
as the head to tail addition of vectors.
I can take the 3 N vector,
I can shift it over,
so its tails at the head of the 5 N vector.
It would look something like this.
So this is 3 N right over there.
And then I could also go the other way around.
I could take the 3 N vector first,
and take the tail of the 5 N vector
at the head of the 3 N vector and shift it like this.
You can add an either direction
and either way you look at it,
when you start at the tails and you get to the head
of the second vector,
you're going to have a resultant force
that looks like this,
which is the diagonal of this parallelogram.
So there we go.
And let me just call that our force vector right over there.
So if we can figure out the length of this line
of this diagonal right over here,
that would be the magnitude of this force vector.
Now, how can we do that?
Well, let's just think geometrically
what else we can figure out about what's going on over here?
This is a parallelogram.
So if this is 100 degree angle right over here,
this angle right over here is also going to be 100 degree.
And we also know that these two opposite angles
are also gonna have the same measure right over here.
And we also know that the sum of all of the angles
in a quadrilateral are going to be 360 degrees.
So these two make up 200 degrees,
we have 160 degrees left that have to be split
between that one and that one.
So we know that this is 80 degrees,
and we know that this is 80 degrees.
Well, how does that help us?
So we know the length of this brown side,
we know the length of this side right over here.
We know the angle between them,
and what we're trying to do,
is figure out the length of the side opposite this angle,
opposite this 80 degree angle.
And some of you might remember the Law of Cosines here.
And the Law of Cosines I always imagine it
as an adaptation of the Pythagorean theorem,
so that we can deal with non-right triangles.
And the Law of Cosines will tell us
that the magnitude, I'll just write it over here,
the magnitude of this vector,
which is the length of this diagonal,
is going to be equal to the square root of,
we're going to have this side squared,
so let me write 3 squared,
plus this side squared,
plus 5 squared minus
2 times this side.
So times 3 times that side.
So times 5 times the cosine of 80 degrees.
And so let's get our calculator out to calculate that.
I'll start with taking the cosine of 80 degrees,
then I'm just gonna multiply that times looks like 30.
So times 30 is equal to that.
Let's put it a little negative there.
And then to that, I'm going to add 25 and 9, which is 34.
So plus 34 is equal to that.
And now I just take the square root of all of that.
And they tell us to round our answer to the nearest tenths.
So I can round this to approximately 5.4.
So this is approximately 5.4 N.
Now they say, what is the direction
of the magnets combined poles
relative to the direction of the first magnets pole?
So really what we want to do,
is figure out this angle right over here,
let's call that data.
Well, we know what the length of the side opposite is.
So maybe we could use the Law of Sines.
The Law of Sines would tell us
that the sine of theta
over the length of the side opposite to it
is going to be equal to, let's pick another angle we know,
sine of this angle.
Sine of 80 degrees
over the length of the side opposite to it.
And so this is approximately 5.4.
And so if we wanna solve a for theta,
we can multiply both sides by 3.
So we're going to get sine of theta,
I'll just stay in this purple color for simplicity,
is equal to 3 times sine of 80 degrees
divided by 5.4.
And then we could say that theta
is equal to the inverse sine of all of this business.
Three sine of 80 degrees over 5.4.
So we're going to take 80 degrees, take the sine of it,
we're going to multiply that by 3,
divide that by 5.4, that equals that,
and then I'm going to take the inverse sine of all of that.
And they want us to round to the nearest integer.
So that's approximately 33 degrees.
When you do the Law of Sines,
it's possible that you're also dealing with an obtuse angle.
And when you do all of this, you get the acute one,
and then you would have to make an adjustment.
But that's not what we're dealing here,
so we know that this data is approximately
equal to 33 degrees.
So we know the magnitude of the force
and we know that it forms an angle
of approximately 33 degrees
with the direction of the force of that first magnet.