Now let's get to a few mechanics before we finish this section
on our mechanical part of this course.
Let's imagine for ourselves a lever.
So, you have a lever here, you have a mass on one end
and you're going to push down, and as we know from Archimedes' great statement
that with any lever, you can pick up an object as big as you'd like
as long as you make the lever long enough.
But when I push down on the side of this lever, notice what happens.
I have to move my hand a great distance
while the object that I'm lifting only goes up by a small distance.
The energy, however, is the same.
Energy is always conserved and I cannot use a mechanical engine
like this very simple one, to cheat energy.
I can use it to help me mitigate how much force I have to apply,
but the total amount of energy that I need to contribute to this object
to lift it up by a distance d, in this case d2,
will always be the same no matter what sort of engine or mechanism that I use.
So the fact that the forces are different,
the force that I have to apply being much smaller
than the force that the box is going to feel is offset by the fact
that I have to move my hand in much greater distance.
So the force times the distance, which gives us the work done,
will still be the same because I still have to lift it to that same height.
I still have to give it the same potential energy if I want it to be up in the air.
So the force times the distance, in summary, is the same for both sides because again the work,
the amount of energy I have to do, is always the same.
But because I'm having my hand go a much longer distance,
the force that I have to exert is much less throughout that entire time.
So for a given energy input,
we always have a trade-off and that's the key idea here,
is that we can input the same amount of energy,
it's still going to be the gravitational energy, but if I want to mitigate the force,
I'm going to have to trade off with distance.
In this case, we can actually see specifically what the equation for conservation of energy would be like.
It would be the force times the distance for the left hand side
would be equal to the force times the distance that I'm applying,
again because the amount of work that I'm doing is just going straight into lifting this box right into the air.
Let's see the same sort of mechanical principle applied to something like a pulley system
and why pulleys are so useful.
We can see here as I'm lifting a box that has some gravitational weight
that's pulling it down, some gravitational force,
and I'm going to use a system of pulleys to lift it up
rather than trying to lift it up directly myself.
So let's look at the forces that are at play here
and see why they're so useful in this pulley setup.
If I'm applying a tension force with my hand,
we apply the same tension throughout the entirety of the rope
as we've discussed when we talked about tension,
and so the entire rope will have a force of tension in it
which means that we're going to lift up our pulley with two forces of tension,
one from either rope on each side.
If that seems a little counterintuitive it looks like
well, maybe it should only be one force of tension because the rope's going all the way around.
There's another way to think about the exact same thing
just conceptually which is that you could sort of imagine these two ropes
going right to your box or even imagine the two ropes
being pinned right to the sides of the pulley.
If we just rewrite it really, like that,
you can see very clearly that these must be considered as two different forces,
two forces of tension from your rope.
So, in equilibrium, let's just look at our box system as we have it here,
just looking at the box and pulley system,
there are forces downwards and forces upwards.
So to put this box in equilibrium, we know that the total number of forces upwards,
one from each tension force here,
and then the force downwards acting against it must come to equal zero if we're in equilibrium.
This tells us that twice the tension force
has to be equal to the gravitational force pulling it downwards.
And here's where we see the utility of this and why this is so useful.
The tension force, which is coming from the force that I'm applying,
only has to be half the weight of the box, only half the weight of the object.
Now just because of one pulley, like we said we have a rope going around,
we get twice the tension force and therefore we only have to apply half the weight of the box
in order to lift the box or in order to keep it in equilibrium.
We can define for ourselves, using these sorts of tools,
a mechanical advantage which is just a way to quantify exactly how much advantage
we're gaining by using these kinds of tools or levers or pulleys,
and it's a long sort of explanation here
but this is just exactly what you would intuitively think it would be,
which is that your advantage is whatever force would be required
without using the machine divided by the force required using the machine.
So let's look at some examples to make this more concrete.
Another way to think about it is that it's just the amount of output force that you get,
the amount of force, for example, the weight of the box
divided by how much force you had to put in to bring the system into equilibrium or to move your box.
So that would be F in, how much force you are applying.
In our box example, what we saw was that the output force,
the amount of force that we're getting on the box that we're trying to lift was Fg,
the weight of the box, because we're trying to lift it up,
that's sort of the result, that's what we get out of the system,
but all we had to do, as we saw, was apply the force of tension with our hand.
Using the analysis that we just showed,
you have that the gravitational force is twice the tension force
because we only had to apply half the gravitational force.
Cancelling the tension forces from both the numerator and dominator here,
you can see that we have a mechanical advantage of 2.
In other words, what we've done is written how much force we got out of our system,
which was the gravitational force, to lift the box up,
how much force we had to put into the system by pulling
and creating a tension force and then writing one of them in terms of the other.
In this case, we wrote the gravitational force in terms of the tension force.
By doing that, we could see what the ratio was, the mechanical advantage,
the ratio between the forces was, just in terms of a pure number.
So in this case, we have a mechanical advantage of 2.
We can look even more closely at an example of mechanical advantage.
So if I have a box and it's going to be moved up a slope,
instead of lifting the box directly, we now move it somewhat easier.
You've maybe seen in many situations instead of climbing the stairs,
which is a direct lifting, you have to go just straight down to up,
you can instead climb a slope and that sort of makes the force you have to apply more gradual.
And so let's see what the mechanical advantage of a slope is, like this one,
instead of lifting this box directly upwards,
where there's a force of gravity acting directly downwards through the entire lift
which can be very difficult for a very heavy object,
we can instead slide it up a slope, in which case,
the force from gravity acting down the slope directly is just Fg sine of theta,
so it's less than the total gravitational force if you were just lifting the thing up by yourself.
The amount of force that we have to apply is, in that case,
only the amount of force to fight against the gravity going up the slope.
We haven't included friction here.
It's considered to be a frictionless slope
because in this kind of example you would try to use as smooth a slope as you could.
So if this is the case, then the amount of force that you have to use
to push this box up the slope will just be the force to combat this Fg sine of theta term,
which is the only force fighting against you.
So what we're saying and what we?re seeing here
is that the amount of force I get out of this system,
in other words sort of the weight I?m able to lift, is Fg, the weight of the box,
whereas the only force I have to put in
to actually do this work to push the box is only Fg sine of theta.
Fortunately, these are already written in terms of each other
so we can quickly cancel the Fg and see that our mechanical advantage is 1 over the sine of theta.
We can apply this to a particular example to see what a particular number is.
So for example, suppose we used a 30-degree slope at an angle of 30 degrees from the ground,
that we know the sine of 30 degrees is 1/2,
so our mechanical advantage, which is 1 over the sine of theta
is 1 over 1/2 or 1 over 0.5 which is again a mechanical advantage of 2.