$2\mathrm{.}147$ The system shown in Fig. $2\mathrm{.}120$ has a natural frequency of $5$ Hz for the following data: $m=10kg,J_0=5kg-m^2,r=10cm,r=25cm.$ When the system is disturbed by giving it an initial displacement, the amplitude of free vibration is reduced by $80\%$ in $10$ cycles. Determine the values of $k$ and $c.$

FIGURE $2\mathrm{.}120$ System with pulley, dashpot and spring.