Bukhovtsev-et-al-Problems-in-Elementary-Physics

284 ANSWERS AND SOLUTIONS
Fig. 403
mg+4aa. ~ 2 3
x a 2pg -. em
Equilibrium is possible only if Fall
is greater than Fcd by an amount
equal to the weight of the section
of the film being considered.
The difference between the forces
of surface tension can be explained
by the difference in the concentration
of the soap in the surface
layers of the film.
354. The force of expulsion balances
the weight of the cube mg
and the force of surface tension 4aa.
i.e., a 2 xpg = mg+ 4aa , where x is
the sought distance. Therefore,
The forces of surface tension introduce a correction of about 0.1 em.
355. The water rises to a height h = 2a . The potential energy of the wapgr
ter colurnn is
E = mgh = 2na;2
p 2 pg
4n(£2
The forces of surface tension perform the work W=2nrah=--. One half
Pi
of this work goes to increase the potential energy, and the other half to
evolve heat. Hence,
2na
Q=pg
2
356. The pressure inside the liquid at a point that is at a height h above
a certain level is less than the pressure at this level by pgh. The pressure
is zero at the level of the liquid in the vessel. Therefore, the pressure at
the height h is negative (the liquid is stretched) and is equal to p= -pgh.
357. The forces of attraction acting on a molecule in the surface layer
from all the other molecules produce a resultant directed downward. The
closest neighbours, however, exert a force of repulsion on the molecule which
is therefore in equi librium.
Owing to the forces of attraction and repulsion, the density of the liquid
is smaller in the surface layer than inside. Indeed, molecule / (Fig. 404) is
acted upon by the force of repulsion from molecule 2 and the forces of
attract ion from all the other molecules (3, 4, ...). Molecule 2 is acted upon
by the forces of repulsion from 3 and / and the forces of attraction from the
molecules in the deep layers. As a result, distance /-2 should be greater
than 2-3, etc.
This course of reasoning is quite approximate (thermal motion, etc., is
disregarded), but nevertheless it gives a qualitatively correct result.
An increase in the surface of the liquid causes new sections of the rarefied
surface layer to appear. Here work should be performed against the forces
of attraction between the molecules. It is this work that constitutes the
surface energy.