If the mass of a spring, m, is not negligible but small compared to the mass M suspended from it, show that the period of oscillations is given by T= 2pisqrt((M+m/3)//k), where k is the force constant of the spring. [Hint: If x and v be the instaneous displacement and velocity of the lower end, then the displacement of an element at a distance z from the fixed end is xz/l and velocity zv/l. The kinetic energy of the spring = (int_(0)^(t))1/2(mdz/l)z^(2)v^(2)/l^(2) = 1/6 mv^(2) Total energy of the system = 1/6mv^(2)+1/2Mv6(2) + 1/2kx^(2) = a constant. Differentiate anmd find T.]

If the mass of a spring, m, is not negligible but small compared to th

1.	If the mass of a spring, m, is not negligible but small compared to the mass M suspended from it, show that the period of oscillations is given by T= 2pisqrt((M+m/3)//k), where k is the force constant of the spring. [Hint: If x and v be the instaneous displacement and velocity of the lower end, then the displacement of an element at a distance z from the fixed end is xz/l and velocity zv/l. The kinetic energy of the spring = (int_(0)^(t))1/2(mdz/l)z^(2)v^(2)/l^(2) = 1/6 mv^(2) Total energy of the system = 1/6mv^(2)+1/2Mv6(2) + 1/2kx^(2) = a constant. Differentiate anmd find T.]
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