A pile of sugar and a pile of salt look pretty similar, but which weighs more? If the volumes of both piles are the same, any difference in mass is due to their different densities (including the air space between crystals). Which do you think has the greater density? What values did you find? What method did you use to determine these values?
Calculating the mass of a reservoir from its volume
A reservoir has a surface area of
and an average depth of 40.0 m. What mass of water is held behind the dam? (See
[link] for a view of a large reservoir—the Three Gorges Dam site on the Yangtze River in central China.)
Strategy
We can calculate the volume
of the reservoir from its dimensions, and find the density of water
in
[link] . Then the mass
can be found from the definition of density
Solution
Solving equation
for
gives
.
The volume
of the reservoir is its surface area
times its average depth
:
The density of water
from
[link] is
. Substituting
and
into the expression for mass gives
Discussion
A large reservoir contains a very large mass of water. In this example, the weight of the water in the reservoir is
, where
is the acceleration due to the Earth's gravity (about
). It is reasonable to ask whether the dam must supply a force equal to this tremendous weight. The answer is no. As we shall see in the following sections, the force the dam must supply can be much smaller than the weight of the water it holds back.
An under-inflated volleyball is pumped full of air so that its radius increases by 10%. Ignoring the mass of the air inserted into the ball, what will happen to the volleyball's density?
The density of the volleyball will increase by approximately 25%.
The density of the volleyball will increase by approximately 10%.
The density of the volleyball will decrease by approximately 10%.
The density of the volleyball will decrease by approximately 17%.
The density of the volleyball will decrease by approximately 25%.
A piece of aluminum foil has a known surface density of 15 g/cm
2 . If a 100-gram hollow cube were constructed using this foil, determine the approximate side length of this cube.
A cube of polystyrene measuring 10 cm per side lies partially submerged in a large container of water.
If 90% of the polystyrene floats above the surface of the water, what is the density of the polystyrene? (Note: The density of water is 1000 kg/m
3 .)
A 0.5 kg mass is placed on the block of polystyrene. What percentage of the block now remains above water?
The water is poured out of the container and replaced with ethyl alcohol (density = 790 kg/m
3 ).
Will the block be able to remain partially submerged in this new fluid? Explain.
Will the block be able to remain partially submerged in this new fluid with the 0.5 kg mass placed on top? Explain.
Without using a container of water, explain how you could determine the density of the polystyrene mentioned above if the material instead were spherical.
(a) 100 kg/m
3 (b) 60% (c) yes; yes (76% will be submerged) (d) answers vary