A metal sphere with a mass of $5\mathrm{.}60kg$ oscillates at the end of a vertical spring with a spring constant of $1\mathrm{.}60{10}^4\frac{N}{m}.$ 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?

How does the damped angular frequency depend on undamped angular frequency and damping coefficient? How is the undamped frequency related to the spring constant? How does frequency relate to angular frequency? $Hz$

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

The amplitude will decrease exponentially. How does it depend on the original amplitude and time? Given the frequency found in part $($a$),$ how much time elapses from one cycle to the next? Can you use this to find the amplitude one cycle after the original amplitude? How much does it differ from the original, and what percentage is this difference with respect to the original? $\%$

$($c$)$ Over what time interval $($in s$)$ does the energy of the system drop to $5\mathrm{.}00\%$ 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?

Follow the same steps as in part $($a$),$ only use the new value of $b.Hz$

What is the percentage decrease in amplitude in each cycle?

Follow the same steps as in part $($b$),$ only use the new value of $b.$ Remember the angular frequency has now changed as well. Be sure to express your answer as a percentage, and not a fraction. $\%$

What is the time interval $($in s$)$ for the energy to drop to $5\mathrm{.}00\%$ of its initial value?

$s$