Answer:
\[\Rightarrow \]\[W=mg\]\[\Rightarrow \]\[W\propto g\] (\[\because \]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. Case-I Case-II \[{{R}_{1}}=R\] \[{{R}_{2}}=R/2\] \[{{M}_{1}}=M\] \[{{M}_{2}}=M\] \[{{W}_{2}}{{W}_{1}}=?\]
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