Question

two similar solid spheres of radius R are placed in contact with each other the gravitational attraction between them is proportional to

Answer 1

if two similar solid sphere are in contact then distance between the centre of sphere equal sum of radius of sphere.

let two sphere which mass m and radius r in contact .
so,
distance between centre of sphere =2r

according to Newton's gravitational law ,
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F = Gm²/(2r)²
F =Gm²/4r²
but both sphere have similar so,
density equal
d =m/4/3πr³
m=4/3πr³.d
put this in above expression ,
F=G(4/3πr³d)²/4r²
={(4/9)Gπ²d²}.r⁴
here (4/9Gπ²d²) is constant let K
so,
F =K.r⁴
hence, force is directly proportional to radius⁴

Answer 2

To Find Gravitational Force ( /attraction; say F ) we have to find :
1) Mass of each sphere
2) Distance between their centers .
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# FINDING MASS :

Radius of each sphere = R
Volume of the sphere (V) = (4/3)πR³
Let the mass of each sphere is 'm' and density of material from which they are made be 'd' .
using m= dV ;
m = 4/3πR³d

# FINDING DISTANCE BETWEEN SPHERES :

As the spheres are kept in contact then the distance separating their centers will be = R+R = 2R
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Now applying Newton's law of Gravitation :
F = Gm²/(2R)²
Using the value of "m" in above equation we get ;
F = (4/9)π²Gd²R⁴
as π,G,d are constants .
=>
F is directly proportional to R⁴
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hope it helps !