question_answer

A force of 10 N is needed to produce certain acceleration in a gold sphere of radius 10 cm. find the force required to produce same acceleration in sphere made of same material of radius 20 cm.

Answer:

Case-I	Case-II
\[{{F}_{1}}=10\]	\[{{F}_{2}}=?\]
\[{{a}_{1}}=a\,\,units\]	\[{{a}_{2}}=a\,\,units\]
\[{{r}_{1}}=10\,cm\]	\[{{r}_{2}}=20\,cm\]

\[F=ma\] .........(1) How to get mass (m) in terms of given data a, r and F? We know that, Mass\[{{v}_{1}}=v\] \[\Rightarrow m=V\times d=\left( \frac{4}{3}\pi {{r}^{3}}\times d \right)\] ??(2) (\[\because \]Volume of sphere\[=\frac{4}{3}\pi {{r}^{3}}\]) Substituting (2) in (1), we get, \[F=\frac{4}{3}\pi {{r}^{3}}d\times a\Rightarrow F\propto {{r}^{3}}\] (\[\because \]?d? and ?a? are same for both the spheres) \[\frac{{{F}_{2}}}{{{F}_{1}}}=\frac{r_{2}^{3}}{r_{1}^{3}}\Rightarrow \frac{{{F}_{2}}}{10}={{\left( \frac{20}{10} \right)}^{3}}\] \[\Rightarrow {{F}_{2}}=8\times 10=80\,\,N\]