A) \[\frac{{{r}^{2}}g}{9\eta }(\rho -2\sigma )\]
B) \[\frac{{{r}^{2}}g}{9\eta }(2\rho -\sigma )\]
C) \[\frac{{{r}^{2}}g}{9\eta }(\rho -\sigma )\]
D) \[\frac{2{{r}^{2}}g}{9\eta }(\rho -\sigma )\]
E) \[\frac{{{r}^{2}}g}{18\eta }(\rho -2\sigma )\]
Correct Answer: A
Solution :
Net force on the ball = downward force - upward force\[=\frac{mg}{2}\] \[\frac{4}{3}\pi {{r}^{3}}(\rho -\sigma )-6\pi \eta rv=\frac{mg}{2}\] \[\frac{4}{3}\pi {{r}^{3}}(\rho -\sigma )g-6\pi \eta rv=\frac{1}{2}\left( \frac{4}{3}\pi {{r}^{3}}\rho \right)g\] \[{{r}^{2}}(\rho -\sigma )g-\frac{9}{2}\eta v=\frac{1}{2}{{r}^{2}}\rho g\] \[\frac{9}{2}v\eta ={{r}^{2}}(\rho -\sigma )g-\frac{1}{2}{{r}^{2}}\rho g\] \[=\frac{1}{2}{{r}^{2}}\rho g-{{r}^{2}}\sigma g\] \[\frac{9}{2}v\eta =\frac{1}{2}{{r}^{2}}g(\rho -2\sigma )\] \[v=\frac{{{r}^{2}}g}{9\eta }(\rho -2\sigma )\]You need to login to perform this action.
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