An uniform sphere of mass and radius exerts a force of on a small mass placed at a distance of from the centre of the sphere. A spherical portion of diameter is cut from the sphere as shown in the fig. The force of attraction between the remaining part of the disc and the mass is:
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Official Solution
Correct Option: (3)
The gravitational force between two masses and is given by Newton's law of gravitation:
where:
- is the gravitational constant,
- is the mass of the sphere,
- is the mass placed at distance from the center. In this case, the mass is at a distance of from the center of the sphere, and the force exerted on is . When a portion of the sphere with a diameter is cut out, we must determine the new force exerted by the remaining part of the sphere on the mass . The gravitational force exerted by the remaining portion of the sphere is proportional to the amount of mass left in the sphere. The mass of the cut portion is proportional to its volume, which is a spherical section with a radius . The remaining mass is proportional to the volume of the remaining sphere, which is roughly of the original sphere's mass. Thus, the force of attraction between the remaining portion of the sphere and the mass is times the original force, i.e., the new force is:
Thus, the correct answer is .
