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Displacement of COM:Application

The magnitude...

Question

The magnitude of gravitational field at distances $r_{1}$ and $r_{2}$ from the centre of a uniform sphere of radius R and mass M is given by $F_{1}$ and $F_{2}$ respectively. Then which of the following is true?
( $I f r_{1} < R$ and $r_{2} < R$ )

\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}}

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\frac{F_{1}}{F_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}}

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Both (a) and (b)

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None of the above

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Solution

The correct option is A $\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}}$
Consider the following two cases
(i) When $r_{1} > R$ and $r_{2} > R$ i.e., the points are outside the sphere in this case, the gravitational field is given by $G M / r^{2}$ .
So, $F_{1} = \frac{G M}{r_{1}^{2}}$ and $F_{2} = \frac{G M}{r_{2}^{2}}$
$\frac{F_{1}}{F_{2}} = \frac{r_{2}^{2}}{r_{1}^{2}}$ i.e., option (b) is correct.
(ii) When $r_{1} < R$ and $r_{2} < R$ , i.e., the points are inside the sphere within the sphere of radius
Now, $M_{1} = \frac{4}{3} π r^{3} ρ$
$F_{1} = G . \frac{4}{3} π r_{1}^{3} ρ / r_{1}^{2}$
$F_{2} = G . \frac{4}{3} π r_{2}^{3} ρ / r_{2}^{2}$
$\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}}$
i.e., option (a) is also correct.

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The magnitude of gravitational field at distances $r_{1}$ and $r_{2}$ from the centre of a uniform sphere of radius R and mass M is given by $F_{1}$ and $F_{2}$ respectively. Then which of the following is true?
( $I f r_{1} < R$ and $r_{2} < R$ )