Homework problems

1	Find an equation for the frequency of simple harmonic motion in terms of k and m.
2	Many single-celled 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 10-15 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 Fair=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, FW, of the piston itself, although in practice this device is used to support some other object. The equilibrium position, x0, is where FW equals -Fair. (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, x0=1.00 m, and FW=-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 x-x0. Approximate the curve with a straight line, find its slope, and derive the approximate period of oscillation.	√
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 x-t 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 2ρ . (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.	√
7	The figure shows a see-saw 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 see-saw. (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.