Question

The gravitational force between two objects is 49 N.How much must the distance between these objects be decreased so the the force between them becomes dounle.

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Answer 1

$\sf\;Here,F=49\;N\\\\\\ Let\;r=original\;distance\;b/w\;the\;objects\\\\\\ We know:-\\ \boxed{\sf\;F=\frac{Gm_1+m_2}{r^2}}---eq-1\\\\\\ Now\;F'=2F=2\times49=98\;N\\\\\\ Let\;r'=mew\;distance\;b/w\;the\;objects\\\\\\ Gravitational\;force\;b/w \;these\;is:-\\\\ F'=\frac{Gm_1m_2}{r^2}-----eq-2\\\\\\ DIVIDE\;1\;BY\;2\\ \frac{F}{F'}=\frac{r^2}{r}\\\\\\ \frac{r'}{r}=\sqrt{\frac{F}{F'} }\\\\\\ r'=\frac{r}{\sqrt{2}}$

Answer 2

Answer :-

The distance between the objects should be decreased by √2 times .

Explanation :-

It is given that the gravitational force between two objects is 49 N . Also, let the initial distance between them be 'r' . So according to Newton's Law of Gravitation, we get the equation as :-

⇒ F = GMm/r²

⇒ 49 = GMm/r² ---(1)

Now, let the new distance between the objects be 'R' . Force (F') becomes double i.e. 98 N . Required equation becomes :-

⇒ F' = GMm/R²

⇒ 98 = GMm/R² ---(2)

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On dividing eq.2 by eq.1, we get :-

⇒ 98/49 = GMm/R² ÷ GMm/r²

⇒ 2 = GMm/R² × r²/GMm

⇒ 2 = r²/R²

⇒ √2 = √(r²/R²)

⇒ √2 = r/R

⇒ R = r/√2