$\backslash$#$2$ A mass connected to a spring that is undergoing horizontal simple harmonic motion has the position$-$versus$-$time graph shown.

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x$(\backslash$mathrm$\{$~cm$\})\_\{\backslash$uparrow$\}$

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$($a$.)$ What is the period of the motion? Hint: There's a more precise way than just estimating where the cycle ends. The period is not $2\mathrm{.}5$ s $.$

$($b$.)$ What is the amplitude of the motion?

Here is the original graph again:

$($c$.)$ If a mass which is nine times larger than the original is attached to the same spring but no other changes are made to the system, plot the position.

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x$(\backslash$mathrm$\{$~cm$\})$

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$($d$.)$ If the original mass is attached to a spring whose constant is four times larger than the first but no other changes are made to the system, plot the position.

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x$(\backslash$mathrm$\{$~cm$\})\_\{\backslash$uparrow$\}$

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$($e$.)$ If the original mass is attached to the original spring but the amplitude is cut in half with no other changes are made to the system, plot the position.

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x$(\backslash$mathrm$\{$~cm$\})\_\{4\}$

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