Hydraulic Lift

Classical Mechanics Level 2

The picture above is that of a hydraulic lift. There is a piston A A of cross-sectional area A 1 A_{1} that is pushed down with a force F F while the valve V V is in an open state. This exerts a pressure on piston B B with the larger cross-sectional area A 2 A_{2} that lifts the car or any other heavy load up. Assume the liquid to be incompressible and nonviscous.

Consider the following questions whose answers have to be given in a Yes/No format:

  1. Is there any gain in terms of work?

  2. Does Piston A A have to traverse a larger downward distance as compared to the height raised by Piston B B ?

  3. Does the pressure in the liquid increase by an amount F A 1 \frac{F}{A_{1}} everywhere?

No, No, Yes No, Yes, No Yes, Yes, Yes No, Yes, Yes Yes, Yes, No

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1 solution

  1. Energy is conserved, therefore there can be no gain in terms of work.

  2. That is true. As we saw in 1, there can be no gain in terms of work, therefore the product F → ⋅ d → \overrightarrow{F} \cdot \overrightarrow{d} , where F → \overrightarrow{F} stands for the force on the side that has a piston displacement d → \overrightarrow{d} must be constant. By Pascal's principle, if the piston A A area is smaller, the force on it is also smaller; if the force is smaller and the product F → ⋅ d → \overrightarrow{F} \cdot \overrightarrow{d} has to be constant, the displacement must be greater.

  3. This is simply the statement of Pascal's principle.

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