A compact object with a mass of $5\mathrm{.}60$ kg oscillates at the end of a vertical spring with a spring constant of $2\mathrm{.}40{10}^4\frac{N}{m}.$ The motion is damped by air resistance, and the damping coefficient is $b=3\mathrm{.}00N*\frac{s}{m}.$

$($a$)$ What is the frequency $($in Hz $)$ of the damped oscillation?

Hz

$($b$)$ By what percentage does the amplitude of the oscillation decrease in each cycle?

$\%$

$($c$)$ Over what time interval $($in s$)$ does the energy of the system drop to $4\mathrm{.}00\%$ of its initial value?

s

$($d$)$ What If$?$ The atmosphere of Venus is $50$ times thicker than that on Earth. If the effect of air resistance on Venus is represented by $b=150N*\frac{s}{m},$ recalculate the answers for parts $($a$)$ to $($c$)$ for this system if it is set in motion in the atmosphere of Venus.

What is the frequency $($in Hz $)$ of the damped oscillations?

Hz

What is the percentage decrease in amplitude in each cycle?

$\%$

What is the time interval $($in s$)$ for the energy to drop to $4\mathrm{.}00\%$ of its initial value?

$]$s