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 fig.Then

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the gravitational force due to this object at the origin is zero

the gravitational force at the point B(2,0,0) is zero

the gravitational potential is the same at all points of circle $y^{2}$ + $z^{2}$ =36

the gravitational potential is the same at all points on the circle $y^{2}$ + $z^{2}$ =4

answer is A, C, D.

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Detailed Solution

The gravitational field (E) intensity at the point O. i.e., centre of a solid sphere is zero. Force acting on a test mass $m_{o}$ placed at O.

$F = m_{o} E = m_{o} \times 0 = 0$

Now, $y^{2} + z^{2} = 36$ represents the equation of a circle with centre (0,0,0) and radius 6 units the plane of the circle is perpendicular to x-axis.

The gravitational field due to masses at A and B at ‘O’ is equal and opposite. As the plane of these circles is Y-Z $⊥$ to X-axis so potential at any point on these two circles will be equal due to mass M and masses M and m.

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