Problem #$2$

Let's take a look at how your friend with the snowboarding accident from last homework is doing. Unfortunately, the first rescue helicopter had a rotor issue and needed to return your friend to the ground. A second helicopter arrives to transport your friend. Upon realizing that they have forgotten their standard rescue line, they fashion a makeshift line from a bungee cord, which behaves like a spring with no damping.

$g$

Figure $1$: The second helicopter, hovering at a constant altitude with your friend attached via a bungee cord.

Consider the first stage of the second rescue attempt, when the helicopter reaches and maintains a constant altitude. Assume that your friend does not sway back and forth $)={}^{}=(0.$

a$)$ Derive the equations of motion for the first stage of the rescue attempt, when the helicopter has lifted your friend and is hovering at a constant altitude. Discuss the difference between the unstretched spring length $L_0$ of a horizontal spring and the static equilibrium length $L_s$ between your friend and the helicopter $(L_s$ is the length of the spring at equilibrium$).$

b$)$ What is the natural frequency of the system? How could you change the system in order to double the natural frequency? What would that mean for the period?

c$)$ Assume your friend has a mass of $65kg.$ Your friend's initial conditions are $y(0)=0($referenced from the equilibrium point$)$ and $y^{}(0)=-0\mathrm{.}2[c\frac{m}{s}].$ What spring constant would result in your friend oscillating $5$ meters up and $5$ meters down from the equilibrium point?

d$)$ Find $y(t)$ for the initial conditions given above. Make a hand$-$drawn sketch of your solution and label important features. Indicate the period, and what form of the solution you chose to use when making the sketch. Compare your sketch with a MATLAB$/$PYTHON plot of $y(t).$

e$)$ Now assume the bungee has some damping properties and can be modeled as a spring$-$damper system with spring constant $k$ and damping coefficient $c.$ Given your friend's mass $m$ and the spring constant $k,$ what range of $c$ will result in an underdamped system? Give your answer both symbolically and numerically $($using the value for $k$ you found in part c$).$ Pick three different values in this range and plot the resulting motion. Compare these results with part d$).$

Be sure to follow all the proper steps for deriving the EOM.