The Viscous Pendulum Clock · AI Generator #GEN-7008B5CE
Physics · fluid-mechanics
Problem
A brilliant inventor designs a unique clock using a steel sphere that oscillates vertically in a tall cylinder filled with glycerin. The sphere is attached to a spring and oscillates between two equilibrium positions due to the interplay of spring force, buoyancy, and viscous drag.
The steel sphere has radius R=2.0cm and density ρs=7800kg/m3. The glycerin has density ρg=1260kg/m3 and dynamic viscosity η=1.5Pa⋅s. The spring has spring constant k=12.0N/m.
When the sphere is displaced from its equilibrium position and released, it undergoes damped oscillations. The inventor notices that the sphere reaches a terminal velocity of vt=0.15m/s when falling freely in the glycerin (without the spring).
For small oscillations about equilibrium, determine the period of oscillation of this viscous pendulum clock. You may assume the oscillations are underdamped and use g=9.8m/s2.