Question

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 1. g 2. 3. 4. 4g

Answer 1

Answer:

-4G

Step by step explanation:

The gravitational potential due to a mass 'M' at a distance 'd' from it is:

$V=\frac{-GM}{d}$.

Since there are infinite number of masses, therefore, the total gravitational potential is:

$-[\frac{GM}{1}+\frac{GM}{2}+\frac{GM}{4}+..... \infty]\\ \\=-GM[1+\frac{1}{2}+\frac{1}{4}=.....\infty]\\ \\=-2G[1+\frac{1}{2}+\frac{1}{2^2}+.... \infty] \\ \\=-2G[1-\frac{1}{2}]^{-1}\\ \\=\frac{-2G}{1-\frac{1}{2}}\\ \\ =-\frac{2G}{(\frac{1}{2})}\\ \\=-4G$

Answer 2

Given : infinite number of bodies, each of mass 2 kg are situated on x-axis at distance 1m, 2m, 4m, 8m, respectively from the origin.

To Find :  The resulting gravitational potential due to this system at the origin will be​

Solution

V = - Gm/d

m = 2 kg

d  = 1  , 2 ,  4 , 8  , ....

V = -(2G/1  +  2G/2   +  2G/4  +  2G/8    + .   ...    )

=> V   =- 2G(1 + 1/2  + 1/4  + 1/8  + ...  + ..  +   )

a = 1  r = 1/2  sum = 1/(1 -1/2)  = 2

=> V = -2G(2)

=> V =  -4G

resulting gravitational potential due to this system at the origin will be​ -4G

 

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