## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

(a) $T' = 2.83~s$ (b) $T' = 1~s$ (c) $T' = 2~s$
We can write an expression for the period $T$ of a mass oscillating on a spring. $T = 2\pi~\sqrt{\frac{m}{k}} = 2~s$ (a) We can find the period $T'$ when the mass is doubled. $T' = 2\pi~\sqrt{\frac{2m}{k}}$ $T' = \sqrt{2}\times 2\pi~\sqrt{\frac{m}{k}}$ $T' = \sqrt{2}\times T$ $T' = (\sqrt{2})(2~s)$ $T' = 2.83~s$ (b) We can find the period $T'$ when the value of the spring constant is quadrupled. $T' = 2\pi~\sqrt{\frac{m}{4k}}$ $T' = \frac{1}{2}\times 2\pi~\sqrt{\frac{m}{k}}$ $T' = \frac{1}{2}\times T$ $T' = (\frac{1}{2})(2~s)$ $T' = 1~s$ (c) Since the period does not depend on the oscillation amplitude, the period of $T = 2~s$ remains the same.