Answer
(a) $T' = 2.83~s$
(b) $T' = 1~s$
(c) $T' = 2~s$
Work Step by Step
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.
