A uniform solid sphere of mass ' m ' and radius ' r ' is surrounded by a uniform thin spherical shell of radius ' 2 r ' and mass ' m ' then the gravitational fi

Question

A uniform solid sphere of mass ' m ' and radius ' r ' is surrounded by a uniform thin spherical shell of radius ' 2 r ' and mass ' m ' then the gravitational field

TestHub · Accepted Answer

For a point outside both the solid sphere and the spherical shell ( $x > 2 r$ ), both objects behave as point masses concentrated at their common centre. The total effective mass is $M = m + m = 2 m$ .

At a distance $x = 2.5 r$ , the gravitational field is:

$E = \frac{G ( 2 m )}{( 2.5 r ) ^{2}}$

$E = \frac{2 G m}{6.25 r ^{2}}$

$E = \frac{2 G m}{( \frac{25}{4} ) r ^{2}}$

$E = \frac{8}{25} \frac{G m}{r ^{2}}$

For a point between the sphere and the shell ( $r < x < 2 r$ ), the gravitational field due to the outer shell is zero. The field is solely due to the inner solid sphere, given by $E = \frac{G m}{x ^{2}}$ . This field is neither zero at $x = 1.5 r$ nor uniform.

Answer: at a distance of $(2.5) r$ from the centre is $\frac{8}{25} \frac{Gm}{r ^{2}}$

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