Infinite number of bodies, each of mass 6 kg, are situated at distances, 1 m , 2 m , 4 m , 8 m , ldots respectively, on Y -axis, from the origin. The resultant gravitational field intensity at the origin is

Question

Infinite number of bodies, each of mass 6 kg, are situated at distances, 1 m , 2 m , 4 m , 8 m , ldots respectively, on Y -axis, from the origin. The resultant gravitational field intensity at the origin is (A) 4 G (B) 8 G (C) 9 G (D) 12 G

Tardigrade Admin · Accepted Answer

Fo=G(6 kg )[(1/12)+(1/22)+(1/42)+(1/82)+ ldots] =(6 G)[(1/1-((1)2)2)]=6 G × (4/3)=8 G

Q. Infinite number of bodies, each of mass $6 k g$ , are situated at distances, $1 m, 2 m, 4 m, 8 m, \dots$ respectively, on $Y$ -axis, from the origin. The resultant gravitational field intensity at the origin is

4 G

8 G

9 G

12 G

$F_{o} = G (6 k g) [\frac{1}{1 ^{2}} + \frac{1}{2 ^{2}} + \frac{1}{4 ^{2}} + \frac{1}{8 ^{2}} + \dots]$
$= (6 G) [\frac{1}{1 - ( \frac{1}{2} ) ^{2}}] = 6 G \times \frac{4}{3} = 8 G$

Q. Infinite number of bodies, each of mass 6kg, are situated at distances, 1m,2m,4m,8m,… respectively, on Y -axis, from the origin. The resultant gravitational field intensity at the origin is

4 G

8 G

9 G

12 G

Fo​=G(6kg)[121​+221​+421​+821​+…] =(6G)[1−(21​)21​]=6G×34​=8G

Q. Infinite number of bodies, each of mass $6 k g$ , are situated at distances, $1 m, 2 m, 4 m, 8 m, \dots$ respectively, on $Y$ -axis, from the origin. The resultant gravitational field intensity at the origin is

$F_{o} = G (6 k g) [\frac{1}{1 ^{2}} + \frac{1}{2 ^{2}} + \frac{1}{4 ^{2}} + \frac{1}{8 ^{2}} + \dots]$
$= (6 G) [\frac{1}{1 - ( \frac{1}{2} ) ^{2}}] = 6 G \times \frac{4}{3} = 8 G$