A solid sphere of mass 'M' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance '3a' from the centre will be :

Gravitation PYQ — JEE Main 2019 · 9 Apr · Shift 1 · Q63 with Solution

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Correct answer: A

A solid sphere has mass $M$ and radius $a$ .
It is surrounded by a concentric spherical shell of thickness $2a$ and mass $2M$ .

So the shell extends from radius $a$ to radius $3a$ .

We need the gravitational field at distance $r = 3a$ from the centre.

For any spherically symmetric mass distribution, the gravitational field at radius $r$ depends only on the total mass enclosed within radius $r$ :

g(r)=\frac{G M_{\text{enclosed}}}{r^2}

directed toward the centre.

At $r=3a$ , the entire solid sphere and the entire shell are enclosed.

Thus,

M_{\text{enclosed}} = M + 2M = 3M

g = \frac{G(3M)}{(3a)^2}

g = \frac{3GM}{9a^2} = \frac{GM}{3a^2}

The value is

\boxed{\frac{GM}{3a^2}}

This corresponds to Option A.

Stored correct answer: A

Our derived answer: A

So they agree.