Homework problems
1  Find an equation for the frequency of simple harmonic motion in terms
of k and m.
 
2  Many singlecelled organisms propel themselves through water with
long tails, which they wiggle back and forth. (The most obvious example
is the sperm cell.) The frequency of the tail's vibration is typically about
1015 Hz. To what range of periods does this range of frequencies correspond?
 
3  (a) Pendulum 2 has a string twice as long as pendulum 1. If we define x
as the distance traveled by the bob along a circle away from the bottom,
how does the k of pendulum 2 compare with the k of pendulum 1? Give a
numerical ratio. [Hint: the total force on the bob is the same if the angles
away from the bottom are the same, but equal angles do not correspond to
equal values of x.]
(b) Based on your answer from part (a), how does the period of pendulum
2 compare with the period of pendulum 1? Give a numerical ratio.
 
4  A pneumatic spring consists of a piston riding on top of the air in a
cylinder. The upward force of the air on the piston is given by F_{air}=ax ^{1.4},
where a is a constant with funny units of N.m ^{1.4}. For simplicity, assume
the air only supports the weight, F_{W}, of the piston itself, although in
practice this device is used to support some other object. The equilibrium
position, x_{0}, is where F_{W} equals F_{air}. (Note that in the main text I have
assumed the equilibrium position to be at x=0, but that is not the natural
choice here.) Assume friction is negligible, and consider a case where the
amplitude of the vibrations is very small. Let a=1 N.m ^{1.4}, x_{0}=1.00 m, and
F_{W}=1.00 N. The piston is released from x=1.01 m. Draw a neat,
accurate graph of the total force, F, as a function of x, on graph paper,
covering the range from x=0.98 m to 1.02 m. Over this small range, you
will find that the force is very nearly proportional to xx_{0}. Approximate
the curve with a straight line, find its slope, and derive the approximate
period of oscillation.
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5  Consider the same pneumatic piston described in the previous problem,
but now imagine that the oscillations are not small. Sketch a graph of the
total force on the piston as it would appear over this wider range of
motion. For a wider range of motion, explain why the vibration of the
piston about equilibrium is not simple harmonic motion, and sketch a
graph of x vs t, showing roughly how the curve is different from a sine
wave. [Hint: Acceleration corresponds to the curvature of the xt graph, so
if the force is greater, the graph should curve around more quickly.]
 
6  Archimedes' principle states that an object partly or wholly immersed
in fluid experiences a buoyant force equal to the weight of the fluid it
displaces. For instance, if a boat is floating in water, the upward pressure of
the water (vector sum of all the forces of the water pressing inward and
upward on every square inch of its hull) must be equal to the weight of the
water displaced, because if the boat was instantly removed and the hole in
the water filled back in, the force of the surrounding water would be just
the right amount to hold up this new "chunk" of water. (a) Show that a
cube of mass m with edges of length b floating upright (not tilted) in a
fluid of density r will have a draft (depth to which it sinks below the
waterline) h given at equilibrium by h _{o} = m / b ^{2}ρ . (b) Find the total force
on the cube when its draft is h, and verify that plugging in h = h _{o} gives a
total force of zero. (c) Find the cube's period of oscillation as it bobs up
and down in the water, and show that can be expressed in terms of h _{o} and
g only.
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7 
The figure shows a seesaw with two springs at Codornices Park in
Berkeley, California. Each spring has spring constant k, and a kid of mass
m sits on each seat. (a) Find the period of vibration in terms of the
variables k, m, a, and b. (b) Discuss the special case where a=b, rather than
a>b as in the real seesaw. (c) Show that your answer to part a also makes
sense in the case of b=0.
 * √ 
8  Show that the equation
has units that make sense.  
