HYDROSTATICS: Fluids at Rest

Properties of Fluids

Fluids flow easily and do not preserve their shapes.

Density=mass/volume r º m/V [kg/m3]

Pressure=force/area Pº F/A [Pascal=N/m2]

Fluids at Rest

If fluid is at rest, all its portions are in static equilibrium.

All points at the same depth must be at the same pressure, independent of the direction. Otherwise fluid would flow.

Pascal’s Principle:

Pressure applied to an enclosed fluid at rest is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.

Proof: otherwise the fluid will flow.

Variation of Pressure with Depth

S F=PA- W- P0A=0 (N’s II Law)

W=Mg; M=r V; V=hA.

\ P= P0+r gh !!!

\ The level of fluid filling connected vessels must be the same in all chambers regardless of their size or shape.

Hydraulic Press

A hydraulic press consists of two cylindrical chambers of different diameters connected by a tube and filled up with a fluid.

Problem: The fluid in the smaller chamber is covered by a piston of radius r and the fluid in the larger chamber is covered by a piston of radius R. If a force f is applied to the small piston, what force F will occur at the large piston?

Solution: At any place in the connecting tube the pressure is the same. Otherwise the fluid will be flowing. \ Ps=Pl.

f=Psa; (a=p r2) F=PlA ̃ F/A=f/a ̃ F=f(R/r)2

\ F is greater than f because R>r.

Archimedes’ Principle:

A body that is partly or entirely submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body.

FB=WFlDp=mFlDpg=r FlVDpg

Problem: Why wood floats? A block of oak wood H(cm) high, w(cm)-wide and l(cm)-long is placed into a tub of water. How far will it sink before it floats, if the density of oak is r 0=7.20´ 102 kg/m3? (r w=1.00´ 103 kg/m3)

Hint: An object floats when h<H.

Solution: W0=m0g=r 0V0g=r 0Hwlg.

FB=r wVDpg; VDp=hwl.

FB= W0̃ r 0Hwlg=r whwlg̃

h=H(r 0/r w)=0.72H.

\ The objects float when r 0<r w!!!