question_answer

If the radius of the Earth was half of the present value, mass remaining same, the weight of the object changes ___ times.

Answer:

               

Case-I	Case-II
\[{{R}_{1}}=R\]	\[{{R}_{2}}=R/2\]
\[{{M}_{1}}=M\]	\[{{M}_{2}}=M\]
\[{{W}_{2}}{{W}_{1}}=?\]

\[\Rightarrow \]\[W=mg\]\[\Rightarrow \]\[W\propto g\] (\[\because \]m is constant) \[\Rightarrow \] \[\frac{{{W}_{2}}}{{{W}_{1}}}=\frac{{{g}_{2}}}{{{g}_{1}}}\] How to get \[\frac{{{g}_{2}}}{{{g}_{1}}}?\] \[g=\frac{GM}{{{R}^{2}}}\] \[\Rightarrow \] \[g\propto \frac{1}{{{R}^{2}}}\](GM are constant) \[\Rightarrow \] \[\frac{{{g}_{2}}}{{{g}_{1}}}=\frac{R_{1}^{2}}{R_{2}^{2}}=\frac{{{(R)}^{2}}}{{{(R/2)}^{2}}}=4\] \[\therefore \] \[\frac{{{W}_{2}}}{{{W}_{1}}}=4\] \[\Rightarrow \] \[{{W}_{2}}=4{{W}_{1}}\] \[\therefore \]Change in weight\[={{W}_{2}}{{W}_{1}}\] \[=4{{W}_{1}}{{W}_{1}}=3{{W}_{1}}\] \[\therefore \]The weight of the object changes 3 times.