Gravitation 1 Question 4
7. The magnitudes of the gravitational field at distance $r _1$ and $r _2$ from the centre of a uniform sphere of radius $R$ and mass $M$ are $F _1$ and $F _2$, respectively. Then
$(1994,2 M)$
(a) $\frac{F _1}{F _2}=\frac{r _1}{r _2}$ if $r _1<R$ and $r _2<R$
(b) $\frac{F _1}{F _2}=\frac{r _2^{2}}{r _1^{2}}$ if $r _1>R$ and $r _2>R$
(c) $\frac{F _1}{F _2}=\frac{r _1^{3}}{r _2^{3}}$ if $r _1<R$ and $r _2<R$
(d) $\frac{F _1}{F _2}=\frac{r _1^{2}}{r _2^{2}}$ if $r _1<R$ and $r _2<R$
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Answer:
Correct Answer: 7. (c)
Solution:
- For $r \leq R, F=\frac{G M}{R^{3}} \cdot r$
or $F \propto r$
$$ \frac{F _1}{F _2}=\frac{r _1}{r _2} \quad \text { for } \quad r _1<R $$
and $\quad r _2<R$
and for $r \geq R, F=\frac{G M}{r^{2}}$
or $\quad F \propto \frac{1}{r^{2}}$
i.e. $\quad \frac{F _1}{F _2}=\frac{r _2^{2}}{r _1^{2}}$ for $r _1>R$
and $\quad r _2>R$
