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Question

The magnitude of gravitation field intensities at distance $r_{1}$ and $r_{2}$ from the centre of a uniform solid sphere of radius $R$ and mass $M$ and $I_{1}$ and $I_{2}$ respectively. Find the ratio of $I_{1} / I_{2}$ if (a) $r_{1} > R$ and $r_{2} > R$ and (b) $r_{1} < R$ and $r_{2} < R$ (c) $r_{1} > R$ and $r_{2} < R$ .

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Solution

Gravitational field intensity for a uniform spherical distribution of mass is given by:
$I = \frac{G M}{r^{2}}$ for $r >> R$ and $I = \frac{G M}{R^{3}} r$ for $r << R$ .
(a) $r_{1} > R$ and $r_{2} > R$
$\frac{I_{1}}{I_{2}} = \frac{(G M / r_{1}^{2})}{G M / r_{2}^{2})} = \frac{r_{2}^{2}}{r_{1}^{2}}$
(b) $r_{1} < R$ and $r_{2} < R$
$\frac{I_{1}}{I_{2}} = \frac{(G M / R^{3}) r_{1}}{(G M / R^{3}) r_{2}} = \frac{r_{1}}{r_{2}}$
(c) $r_{1} > R$ and $r_{2} < R$
$\frac{I_{1}}{I_{2}} = \frac{(G M / r_{1}^{2})}{(G M / R^{3}) r_{2}} = \frac{R^{3}}{r_{1}^{2} r_{2}}$ .

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The magnitude of gravitation field intensities at distance r1 and r2 from the centre of a uniform solid sphere of radius R and mass M and I1 and I2 respectively. Find the ratio of I1/I2 if (a) r1>R and r2>R and (b) r1<R and r2<R (c) r1>R and r2<R.

Gravitational field intensity for a uniform spherical distribution of mass is given by:I=GMr2 for r>>R and I=GMR3r for r<<R.(a) r1>R and r2>RI1I2=(GM/r21)GM/r22)=r22r21(b) r1<R and r2<RI1I2=(GM/R3)r1(GM/R3)r2=r1r2(c) r1>R and r2<RI1I2=(GM/r21)(GM/R3)r2=R3r21r2.