Question 268: Gravitation

Question

A uniform sphere has radius ' $R$ ' and mass ' $M$ '. The magnitude of gravitational field at distances ' $\mathrm{r}_1$ ' and ' $\mathrm{r}_2$ ' from the centre of the sphere are ' $E_1$ ' and ' $E_2$ ' respectively. The ratio $E_1: E_2$ is $\left(r_1>R\right.$ and $\left.r_2

Accepted Answer

Step 1: Gravitational field outside the sphere
For $ r_1 > R $, the gravitational field is as if all mass is concentrated at the center:
$ E_1 = \frac{GM}{r_1^2} $
Step 2: Gravitational field inside the sphere
For $ r_2 < R $, the gravitational field is due only to the mass inside radius $ r_2 $.
The enclosed mass is
$ M_{enclosed} = M \cdot \frac{r_2^3}{R^3} $
So the field is
$ E_2 = \frac{G M_{enclosed}}{r_2^2}       = \frac{G M}{R^3} \cdot r_2 $
Step 3: Ratio
$ \frac{E_1}{E_2}  = \frac{	frac{GM}{r_1^2}}{	frac{GM}{R^3} r_2} = \frac{GM}{r_1^2} \cdot \frac{R^3}{GM r_2} = \frac{R^3}{r_1^2 r_2} $
Final Answer:
$ E_1 : E_2 = \frac{R^3}{r_1^2 r_2} $
Ã¢ÂÂ Correct Option:
C

Answer

$\frac{R^2}{r_1^2 r_2}$

Answer

$\frac{R^3}{r_1 r_2}$

Answer

$\frac{R^3}{r_1^2 r_2}$

Answer

$\frac{R^3}{r_1 r_2^2}$

Gravitation

A uniform sphere has radius ' R ' and mass ' M '. The magnitude of gravitational field at distances ' \mathrm{r}_1 ' and ' \mathrm{r}_2 ' from the centre of the sphere are ' E_1 ' and ' E_2 ' respectively. The ratio E_1: E_2 is \left(r_1>R\right. and $\left.r_2

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