A body when immersed in a liquid, 0.25 of its volume remains outside the liquid. It is taken 60m below the liquid surface and released. What height it will reach above the liquid surface?
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ARCHIMEDES’ PRINCIPLE
According to this principle the buoyant force on an object equals the weight of the fluid it displaces. In equation form, Archimedes’ principle is
FB = wfl,
where FB is the buoyant force and wfl is the weight of the fluid displaced by the object.
Density plays a crucial role in Archimedes’ principle. The average density of an object is what ultimately determines whether it floats. If its average density is less than that of the surrounding fluid, it will float. This is because the fluid, having a higher density, contains more mass and hence more weight in the same volume. The buoyant force, which equals the weight of the fluid displaced, is thus greater than the weight of the object. Likewise, an object denser than the fluid will sink. The extent to which a floating object is submerged depends on how the object’s density is related to that of the fluid. In Figure 4, for example, the unloaded ship has a lower density and less of it is submerged compared with the same ship loaded. We can derive a quantitative expression for the fraction submerged by considering density. The fraction submerged is the ratio of the volume submerged to the volume of the object, or
fraction submerged =
V
sub
V
obj
=
V
fl
V
obj
.
The volume submerged equals the volume of fluid displaced, which we call Vfl. Now we can obtain the relationship between the densities by substituting
ρ
=
m
V
into the expression. This gives
\displaystyle \frac{{V}_{\text{fl}}}{{V}_{\text{obj}}}=\frac{{m}_{\text{fl}}/{\rho }_{\text{fl}}}{{m}_{\text{obj}}/{\overline{\rho }}_{\text{obj}}}
V
obj
V
fl
=
m
obj
/
ρ
obj
m
fl
/ρ
fl
,
where
¯¯¯
ρ
obj
is the average density of the object and ρfl is the density of the fluid. Since the object floats, its mass and that of the displaced fluid are equal, and so they cancel from the equation, leaving
fraction submerged
=
¯¯¯
ρ
obj
ρ
fl
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Answer :
Volume of solid = V
Volume of water displaced = 3V/4
Weight of solid = Buoyant Force
mg = F
Vd’g = (3V/4)dg
d’ = 3d/4
d’ = (3 × 1 g/cm³) / 4
d’ = 0.75 g/cm³
Density of solid is 0.75 g/cm³
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