prove that liquid pressure P is equal to dhg
Answers
more sophisticated derivation which works for any shape of container uses the ideas of work and energy. Imagine a small cylinder fitted with a piston of area A, which is inserted in the liquid so the piston is at depth h, at any orientation we choose, and in contact with the liquid on one side (of the piston). If we push the piston out, a small distance x into the liquid, we'll displace a mass ρAx of liquid, and an equal mass of liquid will, effectively, be displaced from depth h to the surface, gaining gravitational potential energy ρAxhg. We can equate this gain in GPE to the work p dV = p A dx done by the piston against the liquid pressure p. This gives p = ρgh. This is delightful because it shows the result to be totally independent of the container shape and of the orientation of the surface against which the liquid pushes. The piston is assumed to move slowly enough for viscous (dissipative) forces to be negligible; after all we
p =dhg is that formula ....