http://www.ck12.org Chapter 12. Fluid Mechanics
FIGURE 12.7
Consider the pressure at the bottom of each cube.
Using the definition of density, we can rewrite the massmin terms of the volumeVand densityρasm=ρV. The
volume of each cube isd^3 and the area of the any side of the cube isd^2. Substitutingm=ρV=ρd^3 andd^2 forAin
each pressure equation we haveP 1 =FA=ρd
(^3) g
d^2 ;P^2 =
F
A=
2 ρd^3 g
d^2 ;P^3 =
F
A=
3 ρd^3 g
d^2 →P^1 =ρgd;P^2 =^2 ρgd;P^3 =^3 ρgd.
In each case,d, 2 d, 3 drepresent the depth of the water as measured from the top of the column whereP 0 =0. We can
replaced, 2 d, 3 d with∆h, where∆hrepresents the height difference in any two positions along the column. We can
then express the corresponding difference in pressure asP 2 −P 1 =∆P=ρg∆h. If we arbitrarily setP 1 =0, we can
writeP=ρgh. The pressure depends on the density and height (or depth) of the column of fluid. We will always
assume thatgis constant. If the density is constant, the pressure varies only with the heighth. Follow the link below
for a demonstration.
http://demonstrations.wolfram.com/FluidPressure/
Check Your Understanding
- Lake Pontchartrain in Louisiana,Figure12.8, has an area of approximately 1600km^2 and an average depth of
4.0 m. The average density of the water in the lake is very close to the density of fresh water.
a. What is the average pressure at the bottom of the lake?
Answer:
P=ρgh=
(
(^1000) mkg 2
)(
- (^81) sm 2
)
( 4. 0 m) = 39 , 240 → 3. 9 × (^104) mN 2
b. A tall thin tank of water has dimensions of radius 30 cm and height 4.5 m. Compare the pressure at the bottom of