fluid_mechanics

70 Chapter 2 ■ Fluid Statics
V2.8 Hydrometer
These same results apply to floating bodies which are only partially submerged, as illustrated
in Fig. 2.24d, if the specific weight of the fluid above the liquid surface is very small compared
with the liquid in which the body floats. Since the fluid above the surface is usually air, for practical
purposes this condition is satisfied.
In the derivations presented above, the fluid is assumed to have a constant specific weight,
g. If a body is immersed in a fluid in which g varies with depth, such as in a layered fluid, the
magnitude of the buoyant force remains equal to the weight of the displaced fluid. However, the
buoyant force does not pass through the centroid of the displaced volume, but rather, it passes
through the center of gravity of the displaced volume.
E XAMPLE 2.10
GIVEN The 0.4-lb lead fish sinker shown in Fig. E2.10a is attached
to a fishing line as shown in Fig. E2.10b. The specific
gravity of the sinker is SG sinker 11.3.
Buoyant Force on a Submerged Object
FIND Determine the difference between the tension in the line
above and below the sinker.
T A
T A
Pressure
envelope
F B
T B
T B
(a)
F I G U R E E2.10
(b)
(c)
SOLUTION
A free body diagram of the sinker is shown in Fig. E.10b, where
w is the weight of the sinker, F B is the buoyant force acting on the
sinker, and T A and T B are the tensions in the line above and below
the sinker, respectively. For equilibrium it follows that
Also,
T A T B w F B
w g sinker V gSG sinker V
where g is the specific weight of water and V is the volume of the
sinker. From Eq. 2.22,
F B gV
By combining Eqs. 2 and 3 we obtain
F B wSG sinker
(1)
(2)
(3)
(4)
Hence, from Eqs. 1 and 4 the difference in the tensions is
T A T B w wSG sinker w31 11SG sinker 24
0.4 lb 31 1111.324 0.365 lb
(5)
(Ans)
COMMENTS Note that if the sinker were raised out of the
water, the difference in tension would equal the entire weight of
the sinker (T A T B 0.4 lb) rather than the 0.365 lb when it is
in the water. Thus, since the sinker material is significantly heavier
than water, the buoyant force is relatively unimportant. As
seen from Eq. 5, as SG sinker becomes very large, the buoyant force
becomes insignificant, and the tension difference becomes nearly
equal to the weight of the sinker. On the other hand, if SG sinker 1,
then T A T B 0 and the sinker is no longer a “sinker.” It is neutrally
buoyant and no external force from the line is required to
hold it in place.