To solve this problem it is necessary to apply the concepts related to the Force from Hook's law as well as the definition of the period provided by the same definition.
We know that the Force can be defined as
[tex]F = xk \rightarrow mg = kx \Rightarrow k = \frac{mg}{x}[/tex]
Where
k = Spring constant
x = Displacement
g = Gravity
m = mass
At the same time the period of a spring mass system is defined as
[tex]T = 2\pi \sqrt{\frac{m}{k}}[/tex]
Where
m = Mass
k = Spring constant
Our values are given as,
m = 0.404kg
x = 0.666m
Replacing to find the value of the Spring constant we have that
[tex]k = \frac{mg}{x}[/tex]
[tex]k = \frac{(0.404)(9.8)}{0.666}[/tex]
[tex]k = 5.944N/m[/tex]
Now using the formula of the period we know that
[tex]T = 2\pi \sqrt{\frac{m}{k}}[/tex]
[tex]T = 2\pi \sqrt{\frac{0.404}{5.944}}[/tex]
[tex]T = 1.638s[/tex]
Finally, if the oscillation was 0.359m
The maximum height will be determined by the total length of that oscillation being equivalent to
[tex]h=2a[/tex]
[tex]h = 2*0.359[/tex]
[tex]h = 0.718m[/tex]
