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NEET PHYSICSMedium

Infinite number of bodies, each of mass 2 kg are situated on x-axis at distance 1m, 2m, 4m, 8m, respectively from the origin. The resulting gravitational potential due to this system at the origin will be:

A

-G

B

-8/3G

C

-4/3G

D

-4G

Step-by-Step Solution

The gravitational potential (VV) at a point due to a system of particles is the scalar sum of the potentials due to individual particles. V=GMiriV = -\sum \frac{GM_i}{r_i} Given that each mass M=2M = 2 kg and the distances are r1=1,r2=2,r3=4,r4=8,r_1 = 1, r_2 = 2, r_3 = 4, r_4 = 8, \dots The total potential at the origin is: V=G(21+22+24+28+)V = -G \left( \frac{2}{1} + \frac{2}{2} + \frac{2}{4} + \frac{2}{8} + \dots \right) V=2G(1+12+14+18+)V = -2G \left( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \right) The term in the bracket is an infinite geometric series with first term a=1a = 1 and common ratio r=1/2r = 1/2. The sum of an infinite GP is S=a1r=110.5=10.5=2S = \frac{a}{1-r} = \frac{1}{1 - 0.5} = \frac{1}{0.5} = 2. Substituting this back into the expression for VV: V=2G×2=4GV = -2G \times 2 = -4G.

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