expect I desire to carry some logs from the ground come the roof that a tower. Initially I deserve to use a lift, or part cables, or also move the logs upwards manually; climate the energy is converted to the potential energy of logs.

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Now, if I build a high water tank from ground to the roof of the tower, and also fill it with water, and also suppose the I can push the logs native the bottom that the tank with a fine shaped door, climate let the logs float to the roof of the tower. What is the energy source that supplied to transform to the potential power of logs?


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edited Dec 15 "15 in ~ 16:13
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Drarp
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asked Dec 14 "15 at 6:38
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GstestsoGstestso
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The occupational you should do (to insert the log) versus the push of the fluid at the depth is equal to the job-related done by the fluid to obtain the log approximately the height you desire.

If you take into consideration a log of volume $V$ and also a tank the depth $h$, the push at that depth would be $ ho gh$, wherein $ ho$ is the thickness of the fluid, and $g$ the acceleration because of gravity.

The work you should do come insert the log right into the fluid at that depth is $ ho ghV$. (the pressure times the volume)

The buoyant force on the log because of the liquid is $ ho Vg$, for this reason the work-related done by the buoyant pressure to elevator the log in up by a height $h$ is $ ho Vgh$. (the pressure multiplied by the displacement, because the pressure is consistent here.)

Both these amounts are equivalent, for this reason the energy source here is you.


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edited Jul 4 "16 in ~ 12:03
reply Dec 14 "15 in ~ 6:56
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Hritik NarayanHritik Narayan
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Just to increase on Hritik"s an excellent answer with a little more physical insight:

Energy together a "stuff", friction in this picture.

Very frequently we can specify a scalar field (a smoothly-varying collection of numbers, one because that each suggest of the space) which defines everything about a sample of forces. We contact this the "potential energy" field. That is useful due to the fact that a quantity dubbed the total energy is conserved, this has the kinetic and also potential energies of every one of the corpuscle in a system.

Now we need to talk around friction: the is the large situation where energy is not conserved. Yet this is really basic to know too, because friction leads to heat-energy, and also heat-energy is recognized to just be kinetic-energy for points that we don"t treatment about.

Example image of the people in terms of energies

So if girlfriend think around dropping a bouncy round from the height of a building, it"s the very same as if you have 2 cups of part liquid "stuff" (total energy) in some basin (energy of all the things you don"t care about). Your cup labeling "kinetic power of ball" has small holes drilled with the sides (air drag). Your potential power cup is bigger, and is full. Nature pours energy from the potential into the kinetic cup in ~ first, v droplets running out of the sides together the feet leak the end some energy to waiting drag. Climate the liquid level might concerned some stable level (terminal velocity) whereby the fluid poured in is the exact same as the liquid going out v those holes. Ultimately the sphere hits the ground, which usually shakes this kinetic energy cup, sending out a spray that liquid right into the basin, prior to Nature starts putting the kinetic energy ago into the potential power cup. As soon as it"s north (particle is at its maximum), Nature starts putting the potential energy back in everywhere again.

The process continues till both cups are empty -- or, more precisely, when they have actually both concerned the very same level as the power level in the basin (thermalization), which might not be 0. (Consider waiting molecules when they hit the ground: hope the planet thermally-bounces them earlier with enough oomph to acquire up to the upper atmosphere, otherwise our whole atmosphere is around to collapse!)

Minimizing potential energy, flotation

Now we deserve to use this together a general preeminence for wherein things ultimately trend: a system eventually finds itself thermally jittering roughly a minimum that potential energy. If we neglect the heat jitter for large systems, we have actually that Nature simply in general directs things to minimize their potential energy. Call this the "minimum power principle," the world "wants" things to it is in at your lowest (potential) power state, in the feeling that energy will frequently gradually leak out into the surrounding people until this happens.

(Notice the it likewise depends top top the sort of friction: "static" friction overcomes this tendency for quick timescales, i m sorry is why you have the right to park a auto on a hill. Basically, in this snapshot of the universe, static friction creates small potential energy wells whenever 2 surfaces of sliding solids room at rest family member to every other, as atoms between these 2 "bond".)

We deserve to use this to number out as soon as something floats: simply ask what the full energies are. Let"s store the object totally submerged therefore that us don"t have to ask challenging questions about how the water level lowers as it comes the end of the water and so forth.

You have actually an object, like a crate of stuff, through some volume $V$ and also density $ ho_o$ which have the right to either be in ~ the bottom of the fluid $(h=0)$ or the peak of the liquid $(h=H)$, and also the fluid has density $ ho_ell.$ We know that there will be every sorts of complicated drag forces but the system will at some point minimize the full potential energy, and every one of the other energy will be heat in the liquid and currents of liquid and also the like.

Now calculate the total potential power when it is at the top: $ ho_o V g H.$ easy peasy. (Add a continuous term $L$ because that the potential energy of every one of the liquid.) now what happens as soon as it"s at the bottom? 0? No.

Well, it"s a tiny complicated: there is no place for the object at $h=0$ simply yet, it"s all full of liquid. Once the object is down there, the displaces that very same volume $V$ that liquid, moving it the end of the an are it occupies. The liquid ultimately displaces various other liquid and also so on till the "hole" at the optimal of package is filled! We deserve to actually abstract every one of this away by imagining that us freeze time and just cut a box-sized feet of volume $V$ in the fluid at height $h=0$ and take the "box that liquid" and also swap it through the object. The energy is as such $ ho_ell V g H$ together that box of fluid was moved to the top. To compare this through the earlier expression we deserve to see the $V$, $g$ and also $H$ space the same. Our minimum-energy rule says thus that things float as soon as $ ho_o ho_ell.$

Furthermore, we discover that if miscellaneous naturally has actually a density less than the liquid, so that it desires to float, a full energy $( ho_ell - ho_o) V g h$ will certainly be liberated by permitting it to float upwards it is in a height $h$.

(We can also now recognize why logs favor to float lengthwise -- water desires to be lower more than the log does therefore the log in tries to concentration its fixed up in ~ the surface ar of the water. And we can understand the an ext general case of ingredient floating "out of" water: the just thing that the water notices is the "submerged volume" $V_s$ listed below the water line, i m sorry as far as the water is pertained to bears the entirety mass $m$ the the ship/log/whatever. Therefore it has actually an effective thickness $ ho_e = m / V_s$ -- but due to the fact that it"s no floating no one sinking, $ ho_e = ho_ell$ and also we can therefore calculate $V_s = m / ho_ell$ to figure out just how much volume it s okay submerged.)

How this uses to her case

When you support the log into the water, you must therefore additionally displace the water, essentially pushing a "box" that water from the bottom to the top of the water column. This expenses you one up-front energy $ ho_ell~V~g~H$ just to support the log into the water.

Now mean that you tied the log to some really thin but strong fishing line, i beg your pardon goes the end the bottom the the rig: it pulls a flywheel via part gearing mechanism. Therefore every tiny bit of power that this system gets by floating upwards is now going to it is in harvested into this flywheel. (Maybe we then want to use that energy to shove other logs right into the water column?)

Well, we discover that we invested an energy $ ho_ell~V~g~H$ but in right circumstances we recover an energy $( ho_ell - ho_o)~V~g~h,$ leaving us through the net deficit $- ho_o~V~g~H$ which we have to supply from some outside means. But... That"s simply the power that you"d need to supply via any device to lift the log.

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A similar situation happens, for example, if you try to slurp increase a volume $V$ of fluid at the bottom together the log goes in (zero price to get in, yay!) and then let it float come the top. You gain lifting because that free, hooray! Except... Now you"ve lost a volume $V$ from her reservoir, and also if you ever want to placed it earlier in, you"re walking to have to pay that cost to background a volume $V$ the water earlier to elevation $H.$ You just used the water pillar itself as a "waterfall" of species to power the lifting that the log.