question_answer

If the mass of moon is \[\frac{M}{81}\], where \[M\] is the mass of earth, find the distance of the point from the moon, where gravitational field due to earth and moon cancel each other. Given that distance between earth and moon is\[60R\], where \[R\] is the radius of earth.

A) \[6R\]                                   

B) \[8R\]

C)  \[2R\]                                  

D)  \[4R\]

Correct Answer: A

Solution :

Key Idea: Where gravitational field due to earth and moon cancel each other, there the gravitational force is equal. From Newton's law of gravitation the force of attraction between any two material particles is given by where \[{{m}_{1}},\,\,{{m}_{2}}\] are masses and \[r\] is the distance between the two. Since gravitational fields cancel each other the force of attraction is same and opposite. \[i.e.,\]                 \[{{F}_{1}}={{F}_{2}}\]                 \[\frac{G\left( \frac{M}{81} \right)m}{{{x}^{2}}}=\frac{GM\times m}{{{(60\,\,R-x)}^{2}}}\] \[\Rightarrow \]                       \[\frac{1}{81{{x}^{2}}}=\frac{1}{{{(60\,\,R-x)}^{2}}}\] Taking square root of the above expression, we have                 \[\frac{1}{9x}=\frac{1}{60R-x}\] \[\Rightarrow \]               \[9x=60R-x\] \[\Rightarrow \]                 \[x=6R\] Hence, distance of that point from moon is\[6R.\]