$(1)$ You hang a mass m $=0\mathrm{.}22$ kg on a spring with spring constant $12$ N$/$m$.$ You pull the mass down a distance of $0\mathrm{.}04$ m from its equilibrium position and release it$.$ You measure the period to be

$0\mathrm{.}85$ s$.$

$1.$A If you instead pull the mass down $0\mathrm{.}12$ m from its equilibrium position and release it$,$ what would the new period be$?$ Justify your answer with your choice of mathematical expressions or prose $($or a combination of both$).(10$ pts$)$

$1.$B If you use a spring whose spring constant is one$-$fourth the original spring constant, how would the new period compare to the original period? Choose your answer from the options below and justify it with your choice of mathematical expressions or prose $($or a combination of both$).(10$ pts$)$

$$ The new period will be the same.

$$ The new period will be longer.

$$ The new period will be shorter.

$$ There$$s no way to tell!

$1.$C If you use the same spring as in part B but you use one$-$fourth the original mass instead, how would the new period compare to the original period? Choose your answer from the options below and justify it with your choice of mathematical expressions or prose $($or a combination of both$).(10$ pts$)$

$$ The new period will be the same.

$$ The new period will be longer.

$$ The new period will be shorter.

$$ There$$s no way to tell!