We can also look at this from the point of view of work.
So we know still that we have to apply an input work,
the total amount of work we have to do to raise a box of height, h
is still going to be mg times h, the force times the distance.
So in this particular example,
we have a force times a distance and the entire distance that we're lifting
is the entire length of our slope.
But the hypotenuse of this slope, just by trigonometry,
if you see that the height vertically is h, the hypotenuse of the slope,
the distance that we actually had to push the box up the slope,
will be the height divided by the sine of theta.
And using the trigonometry that we have up in the top right corner,
you can see more clearly why that is that the opposite side of this triangle,
which would be the vertical side on the top right here,
would just be h which is always the hypotenuse times the sine of theta.
So, on the other hand, you can see just by dividing that the hypotenuse
would then be the opposite side h divided by the sine of theta.
And so, as I was saying,
the entire distance that you have to push this box is h
divided by the sine of theta, which is a little bit longer than h.
So you see the trade-off here, the sine of theta cancelling
because what we've done by applying this mechanical advantage
is decrease the force that we have to apply.
The force we have to apply is now just F times the sine of theta.
But we still have to apply the same amount of work
because we can see that the total work we have to do
is force times distance, and the distance increased.
It went from just being h,
if I were lifting it directly, to being this longer,
slanted distance which you see is h over sine of theta.
So multiplying the force by the distance,
we still get the same number even though we've dropped our force
from Fg to Fg sine of theta,
and we've increased our distance from just being h,
if we were lifting it directly, to being h divided by the sine of theta
by having to move it up this entire slope.
So again, the entire work that we have to do
is still just equal to the gravitational potential energy
that we're giving in the box.
So we still don't cheat energy.
We could include friction here though, and what happens if we do that?
If we include friction in our slope, we actually do lose out on energy
because the force that we have to apply
is different while the distance over which
we have to apply it's still the same.
So, in other words, friction is still combatting our pushing motion,
so we have to apply a greater force than we would normally have had to apply.
So, that force that we have to apply now is Fg sine of theta
plus the force of the friction because we have to overcome both of these forces.
However, the distance that we're applying
at the entire hypotenuse of this triangle
is still the same, we still have to go that whole distance.
And so the force times the distance is just going to be greater
because of this friction term. In this case, we do lose out on energy.
Here's why. If I move the thing to the entire length
all the way to the top of this triangle,
I still have a potential energy of mgh.
But with friction included, we saw I had to apply more work than mgh.
So in other words, what's happening is I'm having to give this box energy
and some of it is being lost to heat.
So I don't get to use all the energy I give it to just be the mgh,
that a gravitational potential energy of the box. Some of it got lost.
But we still do this sometimes because the mechanical advantage we gain,
the decrease in the amount of force we have to apply,
is so great that it becomes useful regardless of the friction of our slope.
The last thing we could do here is define an efficiency.
So we saw that when there was no friction,
all the energy that I put in to this system, I got back out of this system.
I pushed the box all the way up the slope without friction.
I applied an energy,
a work to the system of mgh and the amount of energy I got out,
in other words the energy the box had,
the gravitational potential energy at the end of the day was also mgh.
So, the efficiency would then be the output work, how much energy I got out,
divided by the input work, how much work I had to do.
In a frictionless case where no energy is lost to your system,
you get the same input or sorry,
the same output work as the work that you put in. So you get everything.
You don't lose anything to any other sources and in that idealized case
which is never going to happen in real life, you get an efficiency of 100%.
But as soon as you start adding in any nonconservative forces,
forces that will take your energy,
then your efficiency will still be whatever output work you got,
mgh in this case, but it might be divided by a bigger number,
a bigger amount of work that you had to do
because as we saw in this case, we had to apply some amount of extra work
to overcome the frictional force that was combatting and fighting against us.
So in these cases, the numerator is the same
but the denominator of your efficiency equation got bigger
because we had to apply more force to fight friction and in this situation,
your efficiency becomes less than 100%,
and it could become lower and lower as the friction
or any other resisting force
or nonconservative force that you're fighting becomes greater and greater.
So with this,
we covered a few mechanical ideas and how to think about machines,
conservation of energy while it's also a trade-off between force and distance.
So we're ready with our mechanics to move to our final mechanics section
which is momentum.
Thanks for watching.