ANSWER
[tex]\begin{equation*} 0.249\text{ seconds} \end{equation*}[/tex]EXPLANATION
We want to find the period of the spring-mass system when the mass is changed.
The period of a mass-spring system is given by:
[tex]T=2\pi\sqrt{\frac{m}{k}}[/tex]where T = period
m = mass
k = spring constant
Substitute the given values for the 3.0 kg mass:
[tex]0.432=2\pi\sqrt{\frac{3.0}{k}}[/tex]Solve for the spring constant, k:
[tex]\begin{gathered} \frac{0.432}{2\pi}=\sqrt{\frac{3.0}{k}} \\ \\ 0.0688=\sqrt{\frac{3.0}{k}} \\ \\ 0.0688^2=\frac{3.0}{k} \\ \\ k=\frac{3.0}{0.0688^2} \\ \\ k=634.6\text{ kg/s}^2 \end{gathered}[/tex]Now, solve for the period, T, when m = 1.0:
[tex]\begin{gathered} T=2\pi\sqrt{\frac{1.0}{634.6}} \\ \\ T=2\pi *\sqrt{0.00158}=2\pi *0.0397 \\ \\ T=0.249\text{ seconds} \end{gathered}[/tex]That is its new period.
