Question

A mass of 2 kg stretches a spring 0.1 m. At time t= 0 the mass is released from its equilibrium position with a downward velocity of 1 m/s. Air resistance adds a damping force equal to one fifth of the velocity of the mass. Set up an initial value problem for the position x(t) of the mass at time t. Determine whether the system is overdamped, underdamped or critically damped. You can assume the acceleration of gravity g equals 10 m/s^2. Explicitly state your choice of a coordinate system for the problem (axis, direction, origin)

Answer 1

I Toka V T I Suppose a mass of 2kg in hanging throug the sponing in vertical dinection a. The movement of the mass is strictly restricted _x20 to ve Vertical ac dinection (Downward sis taken as positive) -x=0lm Due to 2kg mass the spring stretches through olm. - mg zk8 » 29=kXol zl kz 0.29 = 1,962 Nim. Damping due to ain is onelifth of velocity. gkapcă 1241 box C= 115 » Equation of motion- 20+ lit 1962 x 20 » 10xtit 9.810=0 x(0) = 1 mis xozo

Damping ratio is given as, Ez e amon where was a 1 ha natural frequency - Whzq 1.962 = 0.999.8. ez is ag Z 0.051 2x 2x0.99 since ę 20051 <1 the system is undercamped.