Pascal's Principle, Hydraulic Systems and Torricelli's Theorem

Pascal's principle

A change in pressure applied to an enclosed, incompressible fluid is transmitted undiminished to every point in the fluid and to the container walls. Two pistons connected by fluid therefore feel the same pressure, so

This is the hydraulic lift: a $200$ N push on a piston of area $0{,}01$ m² supports a $10\,000$ N car on a piston of area $0{,}5$ m², a force multiplication of $50$. Note that it is the ratio of areas, so doubling the large piston's radius quadruples the force.

No free energy

The fluid is incompressible, so the volume pushed out of the small cylinder equals the volume entering the large one: $A_1d_1=A_2d_2$. The small piston must move farther by the same factor that the force is multiplied, and the work on each side is equal:

To raise the car by $1$ cm, the $200$ N piston travels $50$ cm. A hydraulic system is a lever made of liquid — it trades distance for force, never creating energy.

Torricelli's theorem

Bernoulli applied to a tank open to the air, with a small hole a depth $h$ below the surface, gives the exit speed of the jet. At both the surface and the hole the pressure is atmospheric, and the surface drops so slowly (large area, continuity) that its speed is taken as zero:

This is exactly the speed of an object falling freely through a height $h$ — energy conservation per unit volume, with the fluid's density cancelling out. Once the jet leaves the hole, it is a projectile: the horizontal distance it reaches is $v$ multiplied by the free-fall time from the hole to the ground. Water $2$ m below the surface leaves at $\sqrt{2(10)(2)}\approx6{,}3$ m/s regardless of whether the tank holds water or oil.