10 Example 25 on page 344 suggests analyzing the resonance
of a violin at 300 Hz as a Helmholtz resonance. However, we might
expect the equation for the frequency of a Helmholtz resonator to
be a rather crude approximation here, since the f-holes are not long
tubes, but slits cut through the face of the instrument, which is only
about 2.5 mm thick. (a) Estimate the frequency that way anyway,
for a violin with a volume of about 1.6 liters, and f-holes with a total
area of 10 cm^2. (b) A common rule of thumb is that at an open end
of an air column, such as the neck of a real Helmholtz resonator,
some air beyond the mouth also vibrates as if it was inside the
tube, and that this effect can be taken into account by adding 0.4
times the diameter of the tube for each open end (i.e., 0.8 times the
diameter when both ends are open). Applying this to the violin’s
f-holes results in a huge change inL, since the∼7 mm width of the
f-hole is considerably greater than the thickness of the wood. Try
it, and see if the result is a better approximation to the observed
frequency of the resonance. .Answer, p. 1064
11 (a) Atmospheric pressure at sea level is 101 kPa. The deepest
spot in the world’s oceans is a valley called the Challenger Deep, in
the Marianas Trench, with a depth of 11.0 km. Find the pressure
at this depth, in units of atmospheres. Although water under this
amount of pressure does compress by a few percent, assume for the
purposes of this problem that it is incompressible.
(b) Suppose that an air bubble is formed at this depth and then
rises to the surface. Estimate the change in its volume and radius.
.Solution, p. 1040
12 Our sun is powered by nuclear fusion reactions, and as a first
step in these reactions, one proton must approach another proton to
within a short enough ranger. This is difficult to achieve, because
the protons have electric charge +eand therefore repel one another
electrically. (It’s a good thing that it’s so difficult, because other-
wise the sun would use up all of its fuel very rapidly and explode.)
To make fusion possible, the protons must be moving fast enough
to come within the required range. Even at the high temperatures
present in the core of our sun, almost none of the protons are mov-
ing fast enough.
(a) For comparison, the early universe, soon after the Big Bang,
had extremely high temperatures. Estimate the temperatureT
that would have been required so that protons with average en-
ergies could fuse. State your result in terms ofr, the massmof the
proton, and universal constants.
(b) Show that the units of your answer to part a make sense.
(c) Evaluate your result from part a numerically, usingr= 10−^15 m
andm= 1.7× 10 −^27 kg. As a check, you should find that this is
much hotter than the sun’s core temperature of∼ 107 K.
.Solution, p. 1041
Problems 349