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(D) The formula for the time period of a spring-mass system is $T = 2 \pi \sqrt{\frac{m}{k}}$.For the initial mass $m$,the time period is $T = 3 	ext

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$\frac{9}{16} 	ext{ kg}$

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(D) The formula for the time period of a spring-mass system is $T = 2 \pi \sqrt{\frac{m}{k}}$.For the initial mass $m$,the time period is $T = 3 	ext{ s}$,so $3 = 2 \pi \sqrt{\frac{m}{k}}$.For the increased mass $m+1$,the time period is $T' = 5 	ext{ s}$,so $5 = 2 \pi \sqrt{\frac{m+1}{k}}$.Taking the ratio of the two equations:$\frac{T}{T'} = \frac{2 \pi \sqrt{\frac{m}{k}}}{2 \pi \sqrt{\frac{m+1}{k}}} = \sqrt{\frac{m}{m+1}}$.Substituting the given values:$\frac{3}{5} = \sqrt{\frac{m}{m+1}}$.Squaring both sides:$\frac{9}{25} = \frac{m}{m+1}$.Cross-multiplying gives:$9(m+1) = 25m \implies 9m + 9 = 25m$.$16m = 9 \implies m = \frac{9}{16} 	ext{ kg}$.

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