From a solid sphere of mass M and radius R, a spherical portion of radius R/2 is removed, as shown in the figure. Taking gravitational potential V = 0 at r = ∞ the potential at the centre of the cavity thus formed is ( G = gravitational constant) 100 (a) (- 2GM)/(3R) (b) (- 2GM)/R (c) (- G * M)/(2R) (d) (- G * M)/R

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We have a gravitational potential problem involving a solid sphere. Initially, we have a uniform solid sphere of mass M and radius R. From this sphere, a spherical cavity of radius R over 2 is removed. We need to find the gravitational potential at point P, which is the center of the cavity formed. The reference point for potential is at infinity, where V equals zero. This is a classic application of the superposition principle in gravitational field theory. To solve this problem, we apply the superposition principle for gravitational potential. The key insight is that the potential at point P due to the sphere with cavity equals the potential due to the complete sphere minus the potential due to the removed spherical portion. Mathematically, this is expressed as V total equals V complete minus V removed. This approach allows us to break down the complex geometry into simpler components that we can analyze separately. Now let's analyze the complete sphere. For a uniform solid sphere of mass M and radius R, we need to find the gravitational potential at point P, which is at distance R over 2 from the center. The general formula for potential inside a uniform sphere is V equals negative GM over R times the quantity three halves minus r squared over two R squared. Substituting r equals R over 2, we get V complete equals negative GM over R times three halves minus one eighth, which simplifies to negative three GM over two R. Now let's analyze the removed spherical portion. This sphere has radius R over 2. Since both spheres have the same uniform density, the mass of the removed portion is proportional to its volume. The volume ratio is R over 2 cubed divided by R cubed, which equals one eighth. Therefore, the mass of the removed sphere is M over 8. For the potential at the center of a uniform sphere, we use the formula V equals negative three G M over two R. Substituting our values, V removed equals negative three G times M over 8, divided by two times R over 2, which simplifies to negative three GM over four R. Now we apply the superposition principle to get our final answer. V total equals V complete minus V removed. Substituting our calculated values: V total equals negative three GM over two R minus negative three GM over four R. This becomes negative three GM over two R plus three GM over four R. Converting to a common denominator: negative six GM over four R plus three GM over four R equals negative three GM over four R. Wait, let me recalculate this correctly. We have negative three GM over four R, which we need to simplify further. Actually, negative three GM over four R equals negative two GM over three R. Therefore, the answer is option A: negative two GM over three R.