Answer to Question #112480 in Mechanical Engineering for mwamba

Question

A heavy cylindrical mass is supported at the end of a close-coiled helical spring and is
located concentrically in a cylinder of oil as shown, with a small annular clearance
between the cylinder wall and mass. The mass has a value of 6 kg. When released
from the spring, the mass is observed to sink to the bottom of the cylinder at a steady
speed of 1.4 m/s. When secured to the spring, the spring is observed to extend 16 mm
when the mass was being immersed in the oil. The system is now caused to vibrate.
Determine the frequency of free response and the ratio of two successive amplitudes.
The specific gravity of the oil is 0.86 and that of the mass is 7.35.

Answer 1

mass of cylindrical weight= 6 kg, mass going down with steady state condition with speed of 1.4 m/s, mass of oil 7.35 kg and its specific gravity = 0.86 and spring extension= 16 mm= 0.016 m

Volume of cylinder= $\frac{mass}{density}= \frac{6}{7350}=8.163 \times10^{-4} m^3$

Now net downward force = weight of cylinder- upthrust

$F_d=$ $(6\times9.8)-($ $8.163 \times10^{-4}\times860 \times 9.8)$

$F_d=51.92$

this is responsible for extension of spring

$F_d= kx$

$k= \frac{51.92}{0.016}=3245$

$\nu= \frac{1}{2\pi}(\sqrt\frac{k}{m}= 3.70 Hz$

And ratio of two succesive amplitudes =2.32