Question #43068

A block of mass $m = 0.8 \text{ kg}$ is attached to a spring of force constant $k = 80 \text{ N/m}$ and undergoes damped oscillations on a smooth horizontal surface. The damping force is non-linear and given by $F_d = -c v |v|$, where $v$ is the velocity of the block and $c = 0.01 \text{ kg/m}$. At $t = 0$, the amplitude of oscillation is $A_0 = 0.3 \text{ m}$. Assuming the damping is weak enough that the motion remains approximately sinusoidal over any single cycle, find the approximate number of complete oscillations $N$ after which the amplitude reduces to $A_N = 0.15 \text{ m}$.