Physics question for pocket protector types

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OK, here's the question:

The definition of one horsepower is the power to lift 33,000 Lbs, one foot off the ground in one minute.

I was explaining to a friend of mine that the reason for a heavy weight being lifted a short distance is to minumize the effects of inertia becase power increases as a square of velocity. For an example, I said that it would take much more than one horsepower to lift 1 pound 33,000 feet into the air in one minute because of the acceleration forces involved. He asked me how much more, and in my best Butt-Head voice, I said "Uuuuhh-huh I don't know Beavis"

So that's my question, How much horsepower does it take to lift 1 pound 33,000 feet off the ground in one minute?

Not UPWARDS.. just to move 33000 pounds one foot horizontally. Big difference. Think of it like this: the horse weighs 1000 pounds.. the most he could life in any single instance would be that 1000 points. Assuming no time is spent transfering loads or attaching the horse to another 1000 lbs weight, that horse would have to lift the equivalent of 15 cars.. you'd kill that horse.

Now on the other hand, a horse could pull several vehicles easily.

Its possible for 1 horse power to lift 33000 pounds in one go, but not a horse. It wouldnt have the traction, mass, nor the friction needed. Nor would most 1 horse power motors.
http://www.web-cars.com/math/horsepower.html

Sorry, Fuzzy. Upwards is right (but the web cars link you posted does have some useful info). Moving something horizontally will take different amounts of force depending on the friction between it and the substrate. The first definition (moving 33000 pounds 1 foot upward in 1 minute) is more technically correct, as one horsepower is equal to exerting 33000 foot-pounds of work over one minute of time. To lift 33000 pounds one foot requires exertion of 33000 foot-pounds of work (energy). The rate at which this work is done/energy is expended is defined as power. If this amount of work can be done in one minute, then you have 1 horsepower.

However you can still use a horizontal-movement definition if you assume a frictionless interface between the mass and the substrate, and instead of moving something a particular distance you accelerate it by a certain velocity. This sort of analysis would be equivalent, and I'll go back to it in a moment.

It takes the same amount of energy to lift 1 pound 33,000 feet as it does to lift 33,000 pounds 1 foot. Therefore by extension, if you can do either in the span of one minute, you have one horsepower.

A horse could easily lift 33,000 pounds by one foot in one minute, given a pulley or other simple machine that provides enough mechanical advantage. A simple machine does not create any energy; it just allows you to exchange force for distance, or vice versa, leaving the overall expenditure of energy (work) or production of power the same.

This means that if you had a frictionless, massless pulley and rope with a mechanical advantage of 33,000 (not technically feasible, but theoretically possible) and 33,000 feet of rope, you could lift a 33,000 pound weight one foot by reeling in all 33,000 feet of rope --- and it would only require you to exert one pound of force to lift. If you could accomplish this task in one minute, you will exert one horsepower.

It is still hard to imagine reeling in 6 miles of rope in one minute (you'd have to pull at about 360 miles per hour!) so let's change the mechanical advantage. First of all, nobody is so weak they can only exert 1 pound of force. Let's make it 100 pounds of force, which is imaginable. Now you only have to pull in 330 feet of rope - just over a football field. You now only have to reel it in at about 5 feet per second, or just over 3 mph. Can you exert 100 pounds on 330 feet of rope for one minute? It would be tough for even a strong guy, but a horse could do it!

Another way to imagine it is to simplify the quantities to 550 foot-pounds in one second. It's equivalent to the 33,000 foot-pounds in one minute. Can a horse lift 550 pounds one foot in one second? Yes.

How is pulling something horizontally different from pulling it upwards? Here's another way to imagine it. Power (of which horsepower is a unit, another one is the watt which is about 1/746 hp), is defined by how rapidly you can change an object's energy state. If you have your 33,000 pound mass sitting on the ground and displace it sideways by one foot, you have not changed its energy state (though you will have expended energy in the form of heat as you overcame friction to move it sideways).

If, however, you lifted it upward by one foot, you have changed its potential energy. If you did this in one minute, again, bam, one horsepower. If you accelerated it sideways over a frictionless surface by applying one pound of force for one minute, you have changed its kinetic energy by the same amount. Again, one horsepower.

So why might a 1 horsepower motor connected to a pulley not necessarily be able to lift the requisite weight in the right amount of time, even given a frictionless environment? This is because of another quantity, torque. Torque is analogous to 'rotational force'. It uses the same dimensional units as energy (foot-pounds), which makes it easy to work with. Torque is formally defined as angular acceleration multiplied by rotational moment of inertia (analogous to force being equivalent to linear acceleration multiplied by mass).

So how do we bring all of this together? Do some unit dimensional analysis, and you'll see that you can relate torque, horsepower, and angular velocity (RPM)! For the sake of continuity, I'll stick with imperial units, though in practice it's much easier to use SI units.

Beginning with unitary units...
angular velocity units = 1 / s
torque units = ft * lb
power units = ft * lb / s

And making them more familar...

1 RPM = PI / (30 s) or appx. 0.1047 / s
1 hp = 550 ft * lb / s

If you bring that all together into one equation, you get

Horsepower = Torque * RPM / 5252

Er, no. The inherent inertia is the same, but you're working against gravity. In the former case you have 9.8 m/s^2 * 1 pound working against you for the duration of that 1 minute. In the latter case you have 9.8 m/s^2 * 33,000 pounds.



Apples and oranges.

Let's not forget that "potential energy" is just a bookkeeping convention. That doesn't mean it's false; it just means it's something humans have incorporated into our definition of "energy" to account for the fact that what what goes up must come down. The kinetic energy we spent lifting an object out of a gravity well will become kinetic energy again when the object falls back into the well, and in the mean time we call it "potential energy" which allows us to define conservation of energy in a useful way.

That's completely independent of inertia, which is the only thing that's relevant if we're moving something horizontally over a frictionless surface.

Did you read the sentence? I said energy, not force. Obviously it takes much more force to lift a 33 thousand pound object than a 1 pound object. But since in the case of the 1 pound object, the force is exerted over a much greater distance, the total energy expended is the same. (Also, whether or not [in 1 minute] is left in is irrelevant, as the amount of energy exerted is independent of time. Power is the measure of energy output over time).

That potential energy is a bookkeeping convention (agreed) does not discredit its usefulness here. Energy can be compared in all forms. We set the ground plane as the datum of zero potential energy and measure all potentials with respect to this datum. Similarly, we set zero velocity as the datum of zero kinetic energy and measure all kinetic energies with respect to this datum. With both data established, we can freely interchange kinetic energy for potential and vice versa in the same way that we can use the dynamics equations. And in this situation, where we are interested in power which again is a measurement of energy over time, the energy equations are particularly appropriate.

The same amount of energy exerted over the same amount of time by definition means the production of the same amount of power.



Sorry, incorrect again. Don't forget that an object on a frictionless surface does not require a continuous input of force to keep it moving. As such, power is only measurable when a force is being exerted.

The case of vertical lift and horizontal movement are certainly not apples and oranges. Comparing vertical displacement to horizontal displacement, which is what the last few other posts seemed to have been trying to do, is apples and oranges. But I am concerned with the comparison of vertical displacement to horizontal acceleration.

In the vertical scenario, we expend energy by lifting an object a specified distance above the reference datum. This works because, conveniently, there is a constant force (gravity) which must be overcome at all points along the path of lift. The confusion with the comparison to the horizontal scenario arises because in the vertical scenario, the mass begins and ends at rest.

In the horizontal scenario, the mass cannot begin and end at rest. If you have a 1 pound mass sitting on a frictionless surface, how much energy is expended in moving it 33000 feet? This problem is unsolvable as it is underconstrained. ANY force is necessary and sufficient to set the mass in motion, but once set in motion it will move forever. You must apply continuous force, which is necessary and sufficient to cause continuous acceleration of the object. Applying such a force over a given distance horizontally is operationally equivalent to applying the same force to the same amount of mass over the same distance vertically. The only difference is the end state of the masses. In each case we changed its energy state by the same amount (energy again being measured from our reference datum) and if each is done over the same amount of time, the same amount of power is evolved.

It takes expenditure of energy to lift an object, but once it is up there it takes no more expenditure of energy to keep it up there. Just like lifting a book from the floor and setting it on a table. The energy goes into the lift; there is no more expenditure as it sits on the table.

It takes expenditure of energy to accelerate an object over a frictionless surface, but once it is at the desired velocity, it takes no more expenditure of energy to keep it moving.



And now we understand why that statement is not quite right.

Displacing a mass vertically is NOT operationally equivalent to displacing a mass horizontally, especially not over a frictionless surface. I think this is where the confusion is arising.

Displacing a mass vertically is operationally equivalent to accelerating a mass horizontally in terms of force, energy, and power.

I've got a hundred bucks that says it takes the same amount of energy to lift one pound 33000 feet as it takes to lift 33000 pounds one foot (both masses begin and end at rest).

If there is still any confusion about this, I'll be happy to break out the equations.

Here is where it gets stupid.

The potential energy (vertical) scenario gets confusing because unless we are dealing with a quantum mass, we can't lift it from one point to another without necessarily moving it through all the points in between. Therefore, by necessity, for a certain period of time, the mass needs to have a velocity. Since it began at rest on the ground plane, it requires expenditure of energy to accelerate it, in order to start it in motion, and this does not seem to fit anywhere in the operational equivalency we discussed above. Well here is why it still makes sense if you look at a practical construct.

At the beginning of the lift, you need to exert a force sufficiently greater than the force of gravity on our mass (or by definition, its weight) to accelerate it to a speed that will allow it to transit the required distance in the required time. But once you have made this initial impulse, and the mass is moving, you now only need to exert a force equivalent to its weight and it will continue to move upward at constant velocity (since the net force is now zero). When it gets close to the top, all you need to do is reduce the force you are applying, and gravity will do the work of decelerating our mass back to rest, as it reaches the desired energy height. This work done by gravity is exactly equal and opposite to the extra work we did when we started lifting the mass, and they cancel out.

Of course, to keep the mass up there, you will need to apply force again, equivalent to its weight. But since the mass is now at rest, you are expending no more energy keeping it up there, only exerting force. Again, just like a book sitting at rest on a table. The table exerts upward force on the book exactly equivalent to the downward force exerted by gravity, but since nothing is moving, there is no energy exchange.

So this reminds me that there is yet a third set of equations that are useful in understanding this. We already have the kinematics equations of time, displacement, velocity, and acceleration; the dynamics equations of force, mass, and acceleration; the energy equations of force and displacement or mass and velocity; now, to tie it all back together, we need the momentum and impulse equations of mass, velocity, force, and time.

No, this isn't correct because Power=Mass x Velocity². The one pound moving 33,000 ft is going 34,090 times faster than the 33,000 pounds moving 1 ft (375mph vs .011mph respectively) This number of 34,090 gets squared before it's multiplied by mass to get power.This is why it takes 4 times the power to go twice as fast.

I ran the numbers and it takes quite a bit more horsepower than 1 to get one pound up to 33,000 feet. I just want to see if someone else gets the same answer I do.

Sice we're talking about horsepower (torque over time) The amout of time is totaly relavent, in fact crucial to the discussion, becaus now you have to take into consideration, the acceleration of mass. (Increased G forces)

I hate to get squabbly, but no it isn't! Power is not the same as energy! That is not even the right equation for energy, anyway.

Power = displacement * force / time.

Kinetic energy = 1/2 * mass * velocity^2.

It takes 4 times the energy to make something go twice as fast, not four times the power.

Power is only a measure of how quickly the energy is applied. It may take 4 times the energy to accelerate something to twice the speed, but if a greater amount of time is required to make this acceleration, then lower power has been produced.

Side note. If we're going to start putting equations on here, bear in mind that the conventional notion of pounds(mass) is not the same as pounds(force). When we refer to the weight of something in common parlance, we are actually speaking of pounds(mass), whereas the horsepower equation is derived from pounds(force). Formally, pounds are a unit of force, and the imperial unit of mass is a slug. To convert conventional units of pounds(mass) to pounds(force), multiply by 32.2 ft/s^2. This is analogous to multiplying a kilogram by 9.81 m/s^2 to produce one newton.

This is high school physics, folks.

Power is not torque over time, though the units look the same.

Reread my paragraph that you quoted. The point I was making is that in both cases, the same amount of energy was expended. Since both cases took the same amount of time, the same amount of power was produced in each case.

In the vertical-lift scenario, the acceleration of the masses is irrelevant because they begin and end at rest. This means that the extra force required to accelerate the mass off the ground and get it moving is exactly counterbalanced by the force that you don't need to expend at the top of the lift, because the inertia of the mass will carry it the last bit of the way.

Here's a better way to look at it. Let's say we have the one-pound brick A (ugh, I hate these units but let's press anyway) being lifted upward at a constant velocity by motor+pulley A. There is no acceleration of mass going on, as the motor is spinning at a constant rate. Thus the mass is being lifted at 33000 feet per minute, or 550 feet per second, or 375 mph. I will reiterate that the speed is constant, so there is no acceleration.

Meanwhile, we have the 33,000 pound brick B being lifted upward by motor+pulley B, at a rate of 1 foot per minute or 0.01136 mph. Again, a constant velocity. No acceleration is going on.

So motor+pulley A is exerting a constant 1 pound force, no more, no less. Since Brick A is already in motion, the only force required is to counteract the force of gravity. This makes the net force zero, and Brick A continues upward traversing 550 feet each second.

Meanwhile, motor+pulley B is exerting a constant 33000 pounds force. Again, Brick B is already in motion, so all the motor is doing is counteracting the force of gravity, allowing Brick B to continue upward at one foot per minute.

And both motors are producing one horsepower! You can interchange the motors (while leaving the pulleys where they are) and everything will work exactly the same.

No.

First of all, I don't know how you're calculating the power produced by these two motors, but if the velocities of 550 feet/sec and 1 foot/min enter the equations at all, you've taken a wrong turn. As you pointed out, the bricks were already in motion and not accelerated to those velocities by the motors, so their velocities and distance traveled aren't relevant in any measure of the motors' work or power.

Second of all, all discussions of power aside, you've recognized that motor B is exerting 33,000 times as much force as motor A. That alone should tell you that swapping them and expecting no consequences would be absurd, and any power calculations to the contrary must either be wrong or measurements of the wrong thing.