Down at the ice skating $($assume frictionless ice$)$ rink you watch a customer try out the

new exercise machine consisting of a large spring allowing the skater to oscillate back

and forth. The skater $($mass $=64kg)$ is stretched $0\mathrm{.}84m$ from equilibrium with a force $250N$

and remains at rest. At $0\mathrm{.}84m$ from equilibrium, the skater is then pushed toward the

equilibrium position. At a distance of $0\mathrm{.}22m($from equilibrium$),$ the skater has a speed of

$2\mathrm{.}4\frac{m}{s}.$ Please note that the $250N$ force and the push is not part of the oscillation

dynamics except for determining various constants!

a$)$ Starting with Newton's second law, show that this system obeys the simple harmonic

oscillator differential equation. Assume Hooke's Law is valid. Include a free body

diagram with positive axis labeled. What is the general solution of this equation?

b$)$ Write down the equation describing the skater's position as a function of time $($assume

$t=0$ at $0\mathrm{.}22m$ and motion is in the negative $x-$direction$).$ Your solution should have

numeric values for the variables.

c$)$ Graphically sketch $($at least one full cycle$)$ your answer in b$)$ above. Be sure to include

numbers on each axis.

d$)$ How much time elapses for the skater to travel the distance from the starting position,

$x=0\mathrm{.}22m,$ directly to equilibrium $(x=0)?$

e$)$ What is the skater's speed at a time of $1\mathrm{.}2s?$