Q)
(chatquestion)...1.12 A mass M oscillates at the end of a spring that has spring constant k and finite mass
m. (a) Show that the total energy E of the system for oscillations of small amplitude
is given by
1.
E = (M + m/3)v² +kx?
where v and x are the velocity and displacement of the mass M, respectively. (Hint:
To find the kinetic energy of the spring, consider it to be divided into infinitesimal
elements of length d! and find the total kinetic energy of these elements, assuming
that the mass of the spring is evenly distributed along its length. The total energy E
of the system is the sum of the kinetic energies of the spring and the mass M and
the potential energy of the extended spring.) (b) Hence show that the frequency of the
oscillations is equal to k/(M + m/3).
...
