A) \[\frac{7F}{9}\]
B) \[\frac{3F}{9}\]
C) \[\frac{2F}{9}\]
D) \[\frac{F}{9}\]
Correct Answer: A
Solution :
[a] \[F=\frac{GmM}{{{\left( 2R \right)}^{2}}}=\frac{GmM}{4{{R}^{2}}}\] ...(i) Mass of complete sphere is \[M=\frac{4\pi }{3}{{R}^{3}}\rho .\] Mass of the cut out portion is \[{{m}_{0}}=\frac{4\pi }{3}\left( \frac{R}{2} \right)\] Thus, \[{{m}_{0}}=\frac{M}{8}\] The distance between the centre of the cut out portion and mass \[m=2R-\frac{R}{2}=\frac{3R}{2}\]. Hence, the force of attraction between the cut out portion and mass m is \[f=\frac{G{{m}_{0}}m}{{{\left( 3R/2 \right)}^{2}}}=\frac{G\left( M/8 \right)m}{{{R}^{2}}/4}=\frac{GmM}{4{{R}^{2}}}\times \frac{2}{9}\] Using (i), we get \[f=\frac{2F}{9}\]. Therefore, the force of attraction between the remaining part of the sphere and mass \[m=F-f=F-\frac{2F}{9}=\frac{7F}{9}\]
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