Problem 4.85

A ball of mass $m$ when suspended by a spring stretches the latter by $\Delta l$ . Due to external vertical force varying according to a harmonic law with amplitude $F_{0}$ the ball performs forced oscillations. The logarithmic damping decrement is equal to $\lambda .$ Neglecting the mass of the spring, find the angular frequency of the external force at which the displacement amplitude of the ball is maximum. What is the magnitude of that amplitude?

\omega_{r e s}=\sqrt{\frac{1-(\lambda / 2 \pi)^{2}}{1+(\lambda / 2 \pi)^{2}} \frac{g}{\Delta l}}, a_{r e s}=\frac{\lambda F_{0} \Delta l}{4 \pi m g}\left(1+\frac{4 \pi^{2}}{\lambda^{2}}\right)

Mechanical Oscillations