Exercise:
The mass oscillating on a spring is immersed in a liquid. In this situation the drag force is given by Stokes' law: vecF_d - pi eta R vecv_yt where R is the radius of the spherical mass and eta is the viscosity of the liquid. abcliste abc Derive the differential for the damped oscillation and show that the damping coefficient is abc A mass-spring system consists of spring with spring constant kO and an aluminium ball with radius rO oscillating in glycerole etaetO. Calculate the damping coefficient and the angular frequency. Discuss the result. abc For the same spring as in b what would be the radius of the steel ball that leads to critical damping? abcliste

Solution:
abcliste abc The net force acting on the mass is sscFnet -k yt - pieta R v_yt -k yt - pieta R dot yt Using Newton's second law this yields m ddot yt -k yt - pieta R dot yt Longrightarrow ddot yt -frackm yt - fracpieta Rm dot yt This is the differential for a damped oscillation with omega_ sqrtfrackm and delta fracpieta Rm The damping coefficient is thus delta fracpieta Rm Remark: For a fully immersed mass buoyancy does not affect the oscillation but only the equilibrium position analogously to gravitation. abc The mass can be expressed as m V rho fracpi R^ rho It follows for the angular frequency of the undamped oscillator omega_ sqrtfrackm omF sqrtfrac times kpi times r^ times rh om approx resultomP The damping coefficient is given by delta fracpieta Rm pieta R fracpi R^ rho deF frac times et times r^ times rh de approx resultdeP Since delta omega_ the oscillation is weakly damped. The angular frequency of the damped oscillation is omega omdF sqrtom^-de^ omd approx omdP which is as expected in good agreement with the angular frequency of the undamped oscillation. center includegraphicswidthtextwidth#image_path:damped-mass-on-spring-# center abc The damping ratio is zeta fracdeltaomega_ deF sqrtfracpi R^ rho k zeF Solving for the radius yields R frac pi eta^ k rho zeta^ For critical damping zeta this corresponds to a radius R_c RcF frac pi times et^ times k times rh Rc approx resultRcP- For a ball with a radius RR_c the oscillation would be overdamped. abcliste