Lastly, before we finish this lecture.
We're briefly going to consider the idea of the center of gravity.
We're going to do this so briefly
because we will see how the center of gravity
is related to the center of mass which we've already discussed.
We'll introduce this topic as brief topic by a way of a quick example
which asks us to find the center of gravity for a system of two masses,
10 kg and 5 kg, located respectively at 1 and 4 meters away from the origin.
The way the center of gravity is defined
is very similar to the center of mass.
Say, the position X of the center of gravity is simply going to be again,
the forces of gravity times the position of the object.
In other words, what I'm saying here is that we have force of gravity
on object one times its position
plus the force of gravity on object two times its position
plus any other object you have on your system.
And just like we did with the center of mass, we do want a position,
a position coordinate at the end of the day
and so we need to divide by the units of force.
We'll add up all of our forces, all of our gravitational forces acting on our object,
however many there are.
And so in this problem we only have two masses,
so let's just write down with the gravitational forces for these two objects.
We again have a gravitational force, we have mass 1 times gravity, is its force,
times its position, plus mass two times gravity times its position divided by
and again the sum of the gravitational forces.
And now we get to see why this is such a simple idea,
the center of gravity relative to the center of mass,
because look what happens to this factor of G in the numerator and denominator.
We can factor this out. And then I'll factor it out from the denominator as well.
And then we see immediately what happen is that these factors of G cancel.
The expression that we have left here after we've cancel our factors of G,
looks exactly like the equation that we have for the center of mass.
We have masses times their positions,
all added up divided by the total mass of our system
and so the center of gravity will be in fact exactly the same
as the center of mass when and only when
these factors of G where able to be factored out from the numerator and denominator
and be the same for each object in our system.
So the only time that the center of gravity might be different from the center of mass
is if you have some more exotic questions or a strange system or maybe have a big object
or maybe object in space that are being experience by different amounts of gravity.
But these are more atypical situations are not something
we're going to have to worry about.
And so for our case again, we can cancel these factors of G
and actually plug in our values and see what we get.
So here we have mass 1 is equal to 10 kg
is multiplied by its distance which is 1 meter,
plus our second mass which is 5 kg times its position which is 4 meters
and then again we'll divided by the total sum of the mass which is 10 kg plus 5 kg.
And so all we have to do is simplify here. We have 10 plus 20, divided by 15.
This is 30 divided by 15, which is simply 2 meters.
And so our center of gravity as at a position of 2 meters
which is located between the two masses that we've introduced.
This finishes our quick discussion of the center of gravity
and this lecture covering torque, equilibrium and systems and the center of gravity.
Thanks for watching.