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A solid sphere of uniform density and ra...

A solid sphere of uniform density and radius 4 units is located with its centre at the origin O of coordinates. Two sphere 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 if fig 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 points of circle `y^(2)+z^(2)q=36`

D

the gravitational potential is the same at all points on the circle `y^(2)+z^(2)=4`

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
The gravitational field intensity at point `O` is zero (as the cavities are symmetrical with respect to `O`).
Now the force acting on a test mass `m_(0)` placed at `O` is given by
`F=m_(0)E=m_(0)xx0=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 the axis. Since the spherical mass distribution behaves as if the whole mass is at its centre (for a point outside the sphere) and since all the points on the circle are equidistant from the centre of the sphere, the circle is a gravitational equipotential.
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Knowledge Check

  • Consider a solid sphere of density rho and radius 4R . Centre of the sphere is at origin. Two spherical cavities centred at (2R,0) and (-2R,0) are created in sphere. Radii of both cavities is R . In left cavity material of density 2rho is filled while second cavity is kept empty. What is gravitational field at origin.

    A
    `(4GrhopiR)/3`
    B
    `(3GrhopiR)/2`
    C
    `(GrhopiR)/3`
    D
    `(2GrhopiR)/3`

  • A solid sphere of uniform density and radius R applies a gravitational force of attraction equal to F_(1) on a particle placed at P, distance 2R from the centre O of the sphere. A spherical cavity of radius R//2 is now made in the sphere as shown in given figure. The sphere with cavity now applies a gravitational force F_(2) on same particle placed at P. The ratio F_(2)//F_(1) will be

    A
    `1//2`
    B
    `7//9`
    C
    3
    D
    7

  • A solid sphere of uniform density and radius R applies a gravitational force of attraction equal to F_(1) on a particle placed at P, distance 2 R from the centre O of the sphere. A spherical cavity of the radius R//2 is now made in the sphere as shown in the figure. The sphere with the cavity now applies a gravitational force F_(2) on the same particle placed at P. The ratio F_(2)//F_(1) will be

    A
    `(1)/(2)`
    B
    `(7)/(9)`
    C
    3
    D
    7