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Exam 3 is here: exam3.pdf The solutions for the first two problems are: p1.pdf and p2.pdf. Note that for problem 1 it is not necessary to plot the force as is done in the solution.

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Problem 3:

Then the mass is at the center of the ring the force on it is zero. Along an axis perpendicular to the plane of the ring and through the center at distance z above the ring the net force is:

Note that at some z all points along the ring are equidistant from the mass m. Only the component of force along the axis survives - the components perpendicular sum to zero. The sign is negative - displacing along +z or -z gives rise to a forcce pointing to the center of the circle. For z << R:

This is a linear resoting force like F = -kz where:

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Problem 4:

There are three half-wavlengths in the string. The velocity is given by:

And the wavelength and frequency:

Note that if the tension is doubled the frequency increases by sqrt(2)