An introduction to physics in the context of everyday objects.
Why is it Easier to Pull a Wagon Up a Ramp Than to Lift it Up a Ladder?
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Из курса от партнера University of Virginia
How Things Work: An Introduction to Physics
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Ramps
Professor Bloomfield examines the physics concepts of Newton's third law including conserved quantities, support forces, work, energy, and mechanical advantage working with ramps.
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Louis A. BloomfieldProfessor of Physics
Why is it easier to pull a wagon up a ramp than it is to lift it up a ladder?
The answer to that question is that the ramp allows you to lift the wagon using a
smaller force exerted over a longer distance.
Regardless of how you lift the wagon you have to do work on it, to raise its
altitude from this to this. That increase in altitude comes with an
increase in gravitational potential energy in the wagon.
So you have to transfer a specific amount of energy to the wagon.
To pay for that increase in altitude and therefore increase in gravitational
potential energy. Well if you go up the ladder, you do that,
that work, you transfer the energy by exerting a large upward force on the wagon
as it moves a short distance, the minimum possible distance.
In the direction of your force. That involves a big upward force, which
may or may not be practical. Okay.
In going up the ramp, however, you increase its altitude, the wagon's
altitude gradually. It goes from the low level, to the high
level, while traveling quite a long distance.
Now, the distance it travels is in the direction of your force.
You are pulling the wagon uphill in order to cancel the downhill ramp force.
But that's a gentle uphill force and the wagon moves uphill.
So you're doing work on it. But it takes a much longer travel along
that, that ramp to do the work involved in lifting the wagon from low to high.
So while you do the same amount of work whether you go up the ladder or up the
ramp, the ramp version of this trip involves a small force exerted over a long
distance, whereas the ladder version went, involved a large force exerted over a
short distance. Well, it's all very well to do this in my
laboratory with a dinky little wagon. But we can do better.
Let's head on outside and do the trick with a real wagon and a real handicap
access ramp. Suppose I want to lift this wagon from
ground level to building entry level. I have two options.
On the right I can go up to that ledge and lift the wagon straight up.
In a very short distance I can manage to, to elevate the wagon from ground level to
entry level. On the left I can go up the ramp.
It's a much longer trip. I'll be pulling the wagon up, up the ramp,
but I can still do the job of lifting the wagon from ground level to entry level.
Let's start with the first option, the ledge.
I'll roll the wagon out to the ledge. And now to lift the wagon up I have to
support its entire weight so that it can coast upward constant velocity and go from
ground level to building entry level. So big upward force as the wagon moves a
short upward distance in the direction of that force.
Job done. I did a certain amount of work on the
wagon. That, that amount was the weight of the
wagon times the distance upward the wagon moved.
Its increase in, in altitude. Okay.
Let's try the second option. Back to the starting point and this time
I'm going to go up the ramp. As I go up the ramp, I have to pull uphill
gently just enough to balance the downhill ramp force so the wagon can coast upward.
I am doing work on the wagon though this entire trip because I am pulling it uphill
as it moves uphill. That means I'm doing work, it's moving in
the direction of my force on it. I'm almost there, almost there, I'm there.
So I have managed to lift the wagon from ground level to building entry level, this
time using a small force exerted on the wagon.
As the wagon moved a long distance in the direction of my force.
I did a certain amount of work on the wagon in that process.
And lo and behold, it's the same amount of work that I did in lifting the wagon from
ground to ledge. All that mattered was the increase in
elevation or increase in altitude of the wagon.
That's the increase in the wagon's gravitational potential energy.
The only form of energy that, that increased.
It went from motionless to motionless so It has the same kinetic energy it had at
the beginning, zero. And the only other form of energy that it
has is gravitational potential, and that went from the value at ground level to the
higher value at building entry level. And whether I went up the ledge or whether
I went up the ramp, makes no difference. The wagon has no memory of which approach
I took. It doesn't matter at all.
But there's a difference to me. In going up the ledge I had to exert a
large force on the wagon as the wagon moved a short distance in the direction of
that force. In going up the ramp I had to exert a
gentle force on the wagon as the wagon moved a very long distance in the
direction of that force. The product of those two, force times
distance, was the same in each case. But if this wagon were a little more
heavily occupied that it is going up that ledge would have been very difficult if
this were full of, of bricks. There is no way I can lift it up the
ledge. So I could still pull it up the ramp.
So, the I get mechanical advantage. The mechanical advantage of a simple
machine known as a ramp, or inclined plane.
What that, what that ramp is doing for me is allowing me to do something.
With, to, to rearrange the force, and, and direction of force.
The amount of force and the direction of force that I need to use to do some
particular activity. Instead of using a very large.
Upward force to do the activity in this case.
I was able to use a small uphill force to do that same activity namely lifting the
wagon. Now that the wagon is full of energy I can
release that energy by riding it back downhill.
And here we go. The gravitational potential energy is
becoming kinetic energy. And off I go. In case you're jealous and
wanted to make the trip yourself, here's your opportunity.
You get to go down the ramp also. Ready?
Off we go. It's time for a question.
Suppose you have two ramps with the same height.
That is, both ramps lift the wagon that rides them vertically upward the same
amount, in this case, about that much height.
But one ramp is long, and the other ramp is short.
In fact, the long ramp is twice as long. Along its surface as the short ramp.
In that case those two ramps look like this, this is the long ramp and the short
ramp looks, looks like, ignore this part of the board like this, its steeper
obviously because its only half as long as.
The long ramp. The question then is this: If it takes 100
newtons of uphill force to pull the wagon up the longer ramp, how much uphill force
does it take to pull the wagon up The steeper, shorter ramp.
If it took 100 newtons of uphill force, to pull the wagon up the long ramp at
constant of velocity, when you have the length of the ramp While keeping its
height constant, you must do the same amount of work in half for distance.
So now you have to pull uphill with twice the uphill force.
The product of the two, force times distance traveled by the wagon Remains the
same. But now instead of a 100 Newton force
exerted over a long ramp, it's a 200 Newton force exerted over the short,
half-length ramp. It's clear that ramps make it easier to
lift and lower heavy objects. For example, when you're moving something,
or delivering something, or taking something back out; if you're using a ramp
that's gradual enough, you can lift a very heavy load using a relatively gentle
force. One of the most important uses for ramps
is for handicap access. And access ramp allows a disabled person
to change elevations, to move up or down with relative ease.
But in order to be usable as an access ramp, the ramp has to be very gradually
sloped. In fact, the United States regulates
access ramps and will only approve them If their slope is 1 unit of rise to 12 units
of run. Rise is defined as the vertical movement
that occurs in going up a ramp. And run is defined as the horizontal
movement that occurs in going up the ramp. Now, I've been talking about rise.
All along, although I haven't called it that, I've talked about the attitude
increase that occurs when going up a ramp; but I've been talking not about run, but
about the length along the surface of the ramp because that's the one that's most
useful to us in calculating work. That is, after all, the distance moved by
an object rolling along the ramp. Fortunately for these very shallow angle
ramps, very gradual ramps, there's almost no difference between the length of the
ramps surface and the run, the horizontal extent of the ramp.
This is a trigonometry problem that basically for very small angles the
difference between this hypotenuse and this component of horizontal.
Horizontal component. There's almost no difference between them.
It's a simple problem in trigonometry. You can do it as homework if you like.
So, the bottom line is that ramps, access ramps, can be no steeper than, ramps in
which you move up 1 unit for every 12 units you go along the surface of the
ramp. And, if you recall, the uphill force that
you, you have to exert on an object coming up this ramp at constant velocity is 1, 1
12th its weight. If it's a 1 to 12 ramp, 1 unit of rise to
12 units of length, then the force you have to exert in the uphill direction Is 1
12th of the weight of the object. That means that a disabled person coming
up one of these access ramps at the maximum slope, needs an uphill force of
approximately 1 12th their weight. And that was deemed to be All you can
expect of the disabled person themselves or of their helpers in going up the ramp,
1 12th their weight is enough. In addition to limiting the slope of an
access ramp, US regulations limit the extent of that ramp as well.
And the extent that they limit Is its vertical extent.
That is, what altitude change you go through in riding that ramp.
To be approved, an axis ramp may have no uninterrupted inclines that take you
through an altitude change of more than about this much.
There's a sound physics basis for that requirement.
Gravitational potential energy is related to altitude.
And in going from this height to this height, your gravitational potential
energy increases by your weight times that distance.
So, in going up an access ramp the, the person responsible for doing the work
necessary to increase your gravitational potential energy from this value to this
value has to do the work necessary to increase that gravitational potential
energy. And evidently, this much increase in
altitude is enough. The person who's doing the work, whether
it's the helper of the disabled person or the disabled person themselves needs a
break. So don't make that person go through an
altitude change of more than this without giving them a flat area to regather their
energy. On the way down similar problems are, are,
are avoided, by having periodic breaks. In going from this altitude, to this
altitude, the person descending releases gravitational potential energy and that
energy takes some other form. If the, if this occurs in an uncontrolled
roll in a wheel chair, it becomes kinetic energy.
If it's done in a controlled fashion, it becomes typically thermal energy in the
people involved and maybe the equipment if a break is, if, if, if wheelchair brakes
are involved. So, evidently, this much decrease in
altitude and the gravitational potential energy that's released in that process is
considered to be enough. And it's time for another pause.
So in, in descending what would otherwise be a very long, uninterrupted ramp, there
are periodic flat spots put in to allow the people descending that ramp to pause,
get ready for, for the next descent, and then continue on their way.
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