Infinite number of masses, each 1 kg , are placed along the x-axis at x=± text 1 text m,± 2m,± text 4 m,± text 8 text m,± text 16m.... The magnitude of the resultant gravitational potential in terms of gravitational constant G at the origin (x=0) is

Question

Infinite number of masses, each 1 kg , are placed along the x-axis at x=± text 1 text m,± 2m,± text 4 m,± text 8 text m,± text 16m.... The magnitude of the resultant gravitational potential in terms of gravitational constant G at the origin (x=0) is (A)  G/2  (B)  G  (C)  2G  (D)  4G

Tardigrade Admin · Accepted Answer

Gravitational potential V=GM( (1/r1)+(1/r2)+(1/r3)+...... ) =G× 1( (1/1)+(1/2)+(1/4)+(1/8)+(1/16)+.... ) =G( (1/1-1/2) ) ( ∵ sum of GP=(a/1-r) )

Q. Infinite number of masses, each $1 k g$ , are placed along the $x - a x i s$ at $x = \pm 1 m, \pm 2 m, \pm 4$ $m, \pm 8 m, \pm 16 m ....$ The magnitude of the resultant gravitational potential in terms of gravitational constant $G$ at the origin $(x = 0)$ is

$G /2$

$G$

$2 G$

$4 G$

$8 G$

Gravitational potential
$V = GM (\frac{1}{r _{1}} + \frac{1}{r _{2}} + \frac{1}{r _{3}} + ......)$
$= G \times 1 (\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ....)$
$= G (\frac{1}{1 - 1/2})$
$(∵ s u m o f GP = \frac{a}{1 - r})$

Q. Infinite number of masses, each 1kg , are placed along the x−axis at x=±1m,±2m,±4 m,±8m,±16m.... The magnitude of the resultant gravitational potential in terms of gravitational constant G at the origin (x=0) is

G/2

G

2G

4G

8G

Gravitational potential V=GM(r1​1​+r2​1​+r3​1​+......) =G×1(11​+21​+41​+81​+161​+....) =G(1−1/21​) (∵sumofGP=1−ra​)