HYDROSTATICS: Fluids at Rest

Properties of Fluids

Fluids flow easily and do not preserve their shapes.

Density=mass/volume r
º m/V [kg/m^{3}]

Pressure=force/area Pº F/A
[Pascal=N/m^{2}]

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- P_{0}A=0 (N’s II Law)

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

\ __P= P _{0}+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. \
P_{s}=P_{l}.

f=P_{s}*a*; (*a*=p
r^{2}) F=P_{l}A Þ F/A=f/*a
Þ *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.

F_{B}=W_{FlDp}=m_{FlDp}g=r
_{Fl}V_{Dp}g

*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´ 10^{2} kg/m^{3}? (r
_{w}=1.00´ 10^{3} kg/m^{3})

*Hint*: An object floats when h<H.

*Solution*: W_{0}=m_{0}g=r
_{0}V_{0}g=r _{0}Hwlg.

F_{B}=r _{w}V_{Dp}g;
V_{Dp}=hwl.

F_{B}= W_{0Þ
r 0}Hwlg=r _{w}hwlgÞ

__h=H(r _{0}/r
_{w})__=0.72H.

\ The objects float when
r _{0}<r _{w}!!!