A) \[\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{8{{\left( 1-\frac{R}{2d} \right)}^{2}}} \right]\]
B) \[\frac{GMm}{{{d}^{2}}}\left[ 1+\frac{1}{8{{\left( 1-\frac{R}{2d} \right)}^{2}}} \right]\]
C) \[\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{4{{\left( 1+\frac{R}{2d} \right)}^{2}}} \right]\]
D) \[\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{8{{\left( 1+\frac{R}{2d} \right)}^{2}}} \right]\]
Correct Answer: A
Solution :
From the given figure For solid \[{{F}_{1}}=\frac{GMm}{{{d}^{2}}}\] For cavity \[{{F}_{2}}=\frac{\frac{Gm}{8}(m)}{{{\left( d-\frac{R}{2} \right)}^{2}}}\] \[=\frac{GMm}{8{{d}^{2}}{{\left( 1-\frac{R}{2d} \right)}^{2}}}\] So, resultant force \[F={{F}_{1}}-{{F}_{2}}\] \[=\frac{GMm}{{{d}^{2}}}-\frac{GMm}{8\,{{d}^{2}}{{\left( 1-\frac{R}{2d} \right)}^{2}}}\] \[=\frac{GMm}{{{d}^{2}}}\left[ 1-\frac{1}{8{{\left( 1-\frac{R}{2d} \right)}^{2}}} \right]\]You need to login to perform this action.
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