If the amplitude of a lightly damped oscillator decreases by $1.5 \%$,then the mechanical energy of the oscillator lost in each cycle is: (in $\%$)

$A$ $3 \ kg$ sphere dropped through air has a terminal speed of $25 \ m/s$. (Assume that the drag force is $F_d = -bv$). Now suppose the sphere is attached to a spring of force constant $k = 300 \ N/m$,and that it oscillates with an initial amplitude of $20 \ cm$. What is the angular frequency of its damped $SHM$? ..... $rad/s$

If the amplitude of a damped harmonic oscillator becomes half of its initial amplitude in a time of $10 \ s$,then the time taken for the mechanical energy of the oscillator to become half of its initial mechanical energy is (in $s$)

The amplitudes of a damped harmonic oscillator after $2 \ s$ and $4 \ s$ are $A_1$ and $A_2$ respectively. If the initial amplitude of the oscillator is $A_0$,then

Derive the differential equation for damped oscillations and write its solution.

The amplitude of a damped oscillator becomes one third in $2 \, s$. If its amplitude after $6 \, s$ is $1/n$ times the original amplitude,then the value of $n$ is