Question

A uniform solid sphere of mass $m$ and radius $r$ is surrounded symmetrically by a uniform thin spherical shell of radius $2r$ and mass $m$. Choose the correct statement

• A. The gravitational field at a distance of $1.5r$ from the centre is $\frac{2}{9}\frac{Gm}{r^2}$
• B. The gravitational field at a distance of $2.5r$ from the centre is $\frac{8}{25}\frac{Gm}{r^2}$
• C. The gravitational field at a distance of $1.5r$ from the centre is zero.
• D. The gravitational field between the sphere and spherical shell is uniform.

Answer 1

The correct option is B The gravitational field at a distance of $2.5r$ from the centre is $\frac{8}{25}\frac{Gm}{r^2}$

The net gravitational field at any point inside a spherical shell is zero

So, at a distance $1.5r$, field will only be due to solid sphere, which is given by,

$E=\frac{Gm}{(1.5r{)}^2}$

$\Rightarrow E=\frac{4}{9}\frac{Gm}{r^2}$

Hence option (a) and (c) are wrong.

Also, the gravitational field intensity depends on the distance from the center, so it would not be uniform between the spherical shell and sphere. Hence, option (d) is also wrong.

Gravitational Field intensity at a distance $2.5r$ will be due to both solid sphere and spherical shell, so

$E=\frac{Gm}{(2.5r{)}^2}+\frac{Gm}{(2.5r{)}^2}=\frac{8}{25}\frac{Gm}{r^2}$

Hence, the option (b) is the correct alternative.