Can an extremely small force lift a mountain in a hydraulic system? If not, then why not?

As you see in the picture, a car can be lifted with a small force F1. Here it is shown that 'F1 . A1 = F2 . A2'. What will happen if A1 is of the size of a pin's cross-sectional area. Can a child lift a truck in such a way, using a safety pin? If not, and I feel not, then what is it that will prohibit this from happening? Will the metal structure break? Assume that the child does not hurts itself with the pin as it pushes it...

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Just a tip: When I hear "If not, then why not?" I go uh-oh - sounds like a homework question. – Mike Dunlavey Mar 19 '14 at 15:53

What will happen if A1 is of the size of a pin's cross-sectional area. Can a child lift a truck in such a way, using a safety pin?

Assuming an incompressible fluid is used and that nothing structurally breaks, the answer is yes, a child can lift a truck.

This may seem unintuitive at first glance. However, if you've ever worked in a car shop, you know that a person can lift up an entire truck by pushing a small lever up and down on a car jack; this notion of force amplification by using different areas is the guiding principle behind how hydraulic jacks and related technology operates.

There is a trade-off, however. When you push down the pin a distance $d_1$, by conservation of fluid volume you will also have $$d_1A_1=d_2A_2\rightarrow d_2=d_1\frac{A_1}{A_2}.$$ For example, with a pin of diameter 1mm and a truck on a platform of diameter 2m, if the child pushes the pin down by 6 inches, the truck will lift $d_2=38\text{nm}$. This is so small that you couldn't even see the truck lift up. So in essence, you're trading more force for less lift distance.

You can also compute that this process obeys conservation of energy in a similar matter, using the forces and distances instead of forces and areas.

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But considering that it is a very long pin, and we go on displacing water with it. As you said, the amount by which the heavy object lifts is proportional to the volume of the pin. Increasing the length of the pin would do that, would it not? – Raja Mar 19 '14 at 4:24
@Raja: Yes, if you pushed the pin down a long distance you could theoretically lift it. A pin would actually be overkill, as in reality you really only need a factor of ~200 multiplication of force to have a human lift a truck, so a 5 inch diameter for $A_1$ would actually be fine. – DumpsterDoofus Mar 19 '14 at 12:27

"Give me a place to stand and with a lever I will move the whole world"-Archimedes.

Yes, in principle the mountain could be lifted. However, the distance it would be lifted would be extremely small compare to the distance the pin moves. The ratio of the distances moved is the same as the ratio of the cross-sectional areas.

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But considering that it is a very long pin, and we go on displacing water with it. As you said, the amount by which the heavy object lifts is proportional to the volume of the pin. Increasing the length of the pin would do that, would it not? – Raja Mar 19 '14 at 4:24
The volume of fluid displaced on one side will equal the volume of fluid displaced on the other side. Just lengthening the pin is not enough, the column of fluid also needs lengthened. In a practical jacking situation, you would devise some kind of ratchet mechanism, so that you could lift iteratively. For example, in the fluid case, perhaps a check valve, so you could force the pin down, seal off the fluid, retract the pin, add more fluid above the seal, and force the find down agai in a cycle. – DavePhD Mar 19 '14 at 12:33