Given,

$\Im_n\ =\ 0.21 seconds\ = 2π(\sqrt{(\frac {m}{k}}))$

$\sqrt m = \frac {0.21 \sqrt{k}}{2π}$

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(a) If the spring constant is increased by 50 percent,

$(\Im_n)_{new}\ =\ \frac {2π \sqrt{m}} {\sqrt{k_{new}}} = \frac {2π \sqrt{m}} {\sqrt{1.5k}} = 2π(\frac {0.21 \sqrt{k}} {2π}) \frac{1}{\sqrt {1.5k}} = 0.17146sec$

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(b) If the spring constant …