A) \[\frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{r}^{2}}\]
B) \[({{m}^{1}}+{{m}^{2}}){{r}^{2}}\]
C) \[\frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}-{{m}_{2}}}{{r}^{2}}\]
D) \[({{m}^{1}}-{{m}^{2}}){{r}^{2}}\]
Correct Answer: A
Solution :
[a] C is centre of mass of the dumb belt, \[{{r}_{1}}\]and \[{{r}_{2}}\]are distance of \[{{m}_{1}},{{m}_{2}}\]from C. Then M.O.I of dumb bell about the given axis \[I={{m}_{1}}r_{1}^{2}+{{m}_{2}}r_{2}^{2}\] \[r={{r}_{1}}+{{r}_{2,}}\] \[{{m}_{1}}{{r}_{1}}={{m}_{2}}{{r}_{2}}\] \[{{m}_{1}}{{r}_{1}}={{m}_{2}}r-{{m}_{2}}{{r}_{1}}\] \[{{r}_{1}}=\frac{{{m}_{2}}r}{{{m}_{1}}+{{m}_{2}}},{{r}_{2}}=\frac{{{m}_{1}}r}{{{m}_{1}}+{{m}_{2}}}\] \[I={{m}_{1}}\left[ \frac{{{m}_{2}}r}{{{m}_{1}}+{{m}_{2}}} \right]+{{m}_{2}}\left[ \frac{{{m}_{1}}r}{{{m}_{1}}+{{m}_{2}}} \right]\] \[I=\frac{{{m}_{2}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{r}^{2}}\]You need to login to perform this action.
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