Question

A solid sphere of uniform density and radius $4$ units is located with its centre at the origin O of coordinates. Two spheres of equal radii $1$ unit, with their centres at A$(-2,0,0)$ and B$(2,0,0)$, respectively, are taken out of the solid leaving behind spherical cavities as shown in the figure. Then.

• A. The gravitational force due to this object at the origin is zero
• B. The gravitational force at point $B(2,0,0)$ is zero
• C. The gravitational potential is the same at all the points of circle $y^2+z^2=36$
• D. The gravitational potential is the same at all points on the circle $y^2+z^2=4$

Answer 1

Answer: (A), (C), (D)

The correct options are
A The gravitational force due to this object at the origin is zero
C The gravitational potential is the same at all the points of circle $y^2+z^2=36$
D The gravitational potential is the same at all points on the circle $y^2+z^2=4$

Let ${\overrightarrow{F}}_A=$ Gravitational force due to sphere $A$

${\overrightarrow{F}}_B=$ Gravitational force due to sphere $B$

${\overrightarrow{F}}_R=$ Gravitational force due to remaining portion after cavities are mode.

Then from super position principle we can see that ${\overrightarrow{F}}_A+{\overrightarrow{F}}_B+{\overrightarrow{F}}_R=0$ as the force due to center sphere is zero at center. Now since ${\overrightarrow{F}}_A+{\overrightarrow{F}}_B=0$ due to symmetry.

Hence ${\overrightarrow{F}}_R=0$ so option $\left(A\right)$ correct.

Now at $B$

Field due to entire sphere is given by

$\overrightarrow{F}=\frac{GM}{R^3}r=\frac{GM}{64}2=\frac{GM}{32}$

where as ${\overrightarrow{F}}_A=\frac{GM}{4^2}=\frac{GM}{16}=\frac{GM}{16\times 64}=\frac{GM}{1024}$

where $M$ is mass of sphere $A=\frac{M}{64}$ and ${\overrightarrow{F}}_B=0$

For super position principle we have

${\overrightarrow{F}}_A+{\overrightarrow{F}}_B+{\overrightarrow{F}}_R=\overrightarrow{F}\Rightarrow {\overrightarrow{F}}_R=\overrightarrow{F}-{\overrightarrow{F}}_A=\frac{GM}{32}\hat{i}$

$\frac{GM}{1024}\hat{i}=\frac{31GM}{1024}\hat{i}\ne 0$ Hence option (B) not correct regarding potential at point on $y^2+z^2=36$ we can see that radius of circle is $6$ units now ever all points on it are symmetry located from remaining sphere. Potential same at every point on circle. Hence potential must be same at every point on circle same logic holds for $y^2+z^2=4$ so option (C) or (D) correct.

So, (A) (C) (D) correct.