Using the kinetic energy approach show that the real vertica

Using the kinetic energy approach show that the real vertical spring ( with mass m linearly distributed along its length) may be replaced by the ideal (massless) spring of the same length, and stretched by the effective mass m/3 attached to its end. b) Consider such system with the a block of mass M is connected to the end of a vertical spring of mass ms and force constant k,. Show that when this system is set into simple harmonic motion, the period of its motion is given by Use the opposite site of this page to present the full solution T = 2 M + (ms /3)

Solution

Assume the lowest end of the spring is moving with velocity v, the topmost end will be at rest. Now consider an element of lenght dx and mass dm at a distance x from top end.

This element will move with velocity vx/L.

K.E. of element will be 1/2 dmv2x2/l2

dm= Mdx/L

Hence K.E = 1/2 Mv2x2dx/L3.

Integrating from x=0 to x= L

K.E = 1/2 (M/3) v2. Hence effective mass will be M/3.

When system will be in SHM effective mass of the spring will also be added to the system mass. Hence time period will be as given.