In forced oscillations,a particle oscillates simple harmonically with a frequency equal to

Explain the behaviour of the oscillator when the driving frequency is close to the natural frequency in small damped oscillation and define resonance.

The angular frequency of the damped oscillator is given by $\omega = \sqrt{\frac{k}{m} - \frac{r^2}{4m^2}}$,where $k$ is the spring constant,$m$ is the mass of the oscillator,and $r$ is the damping constant. If the ratio $\frac{r^2}{mk}$ is $8\%$,the change in time period compared to the undamped oscillator is approximately as follows:

$A$ damped harmonic oscillator has a frequency of $5$ oscillations per second. The amplitude drops to half its value for every $10$ oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to .... $s$

You are riding in an automobile of mass $3000\; kg$. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags $15\; cm$ when the entire automobile is placed on it. Also,the amplitude of oscillation decreases by $50 \%$ during one complete oscillation. Estimate the values of
$(a)$ the spring constant $k$ and
$(b)$ the damping constant $b$ for the spring and shock absorber system of one wheel,assuming that each wheel supports $750 \;kg$.

When an oscillator completes $100$ oscillations,its amplitude is reduced to $\frac{1}{3}$ of its initial value. What will be its amplitude after it completes $200$ oscillations?