The percentage uncertainty in the mass of the sphere is 0.000625%
The given parameters;
Density is given as the ratio of mass to volume;
[tex]\rho = \frac{mass}{volume} \\\\\% \ uncertainty \ in \ density = \frac{uncertainty \ in \ mass}{uncertainty \ in \ volume} \times 100\% \\\\uncertainty \ in \ mass = uncertainty \ in \ density \times uncertainty \ in \ volume[/tex]
The percentage uncertainty in the volume of the sphere is calculated as;
[tex]V_1 = \frac{4}{3} \pi r^3\\\\V_1 = \frac{4}{3} \pi(4)^3\\\\\V_1 = 268.12 \ cm^3[/tex]
[tex]V_2 = \frac{4}{3} \pi (0.2)^3\\\\V_2 = 0.0335 \ cm^3[/tex]
[tex]\% \ uncertainty \ in \ volume = \frac{absolute \ uncertainty }{Actual \ measurement} \times 100\%\\\\\% \ uncertainty \ in \ volume = \frac{V_2}{V_1} \times 100\%\\\\\% \ uncertainty \ in \ volume = \frac{0.0335}{268.12} \times 100\% = 0.0125 \ \%[/tex]
The percentage uncertainty in the mass of the sphere is calculated as;
[tex]5\% = \frac{\% \ uncertainty \ in \ mass}{\% \ uncertainty \ in \ volume } \times 100\%\\\\5 = \frac{\% \ mass}{0.0125} \times 100\\\\100(\% mass ) = 5(0.0125)\\\\\% \ mass = \frac{0.0625}{100} \\\\\% \ mass = 0.000625 \%[/tex]
Thus, the percentage uncertainty in the mass of the sphere is 0.000625%
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