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

(a) $T' = 2.83~s$ (b) $T' = 1.41~s$ (c) $T' = 2.00~s$ (d) $T' = 1.41~s$
We can write an expression for the period $T$ of a mass oscillating on a spring. $T = 2\pi~\sqrt{\frac{m}{k}} = 2.00~s$ (a) We can find the period $T'$ when the mass is doubled. $T' = 2\pi~\sqrt{\frac{2m}{k}}$ $T' = \sqrt{2}\times 2\pi~\sqrt{\frac{m}{k}}$ $T' = \sqrt{2}\times T$ $T' = (\sqrt{2})(2.00~s)$ $T' = 2.83~s$ (b) We can find the period $T'$ when the mass is halved. $T' = 2\pi~\sqrt{\frac{m/2}{k}}$ $T' = \sqrt{\frac{1}{2}}\times 2\pi~\sqrt{\frac{m}{k}}$ $T' = \sqrt{\frac{1}{2}}\times T$ $T' = (\sqrt{\frac{1}{2}})(2.00~s)$ $T' = 1.41~s$ (c) Since the period does not depend on the oscillation amplitude, the period of $T = 2.00~s$ remains the same. (d) We can find the period $T'$ when the spring constant is doubled. $T' = 2\pi~\sqrt{\frac{m}{2k}}$ $T' = \sqrt{\frac{1}{2}}\times 2\pi~\sqrt{\frac{m}{k}}$ $T' = \sqrt{\frac{1}{2}}\times T$ $T' = (\sqrt{\frac{1}{2}})(2.00~s)$ $T' = 1.41~s$