GATE-2018 (24) :Wyllie – Time average equation

CSIR -NET 2012

105. Three masses M,2M and 3M are distributed inside three spheres A,B and C of radii R,2R and 3R respectively. The total normal gravitational fluxes g_A, g_B and g_C over A, B and C respectively are connected by the relation

A) g_A= g_B =g_C                                    B) 3g_A= 2g_B= g_C

C)6g_A=3g_B=2g_C                                D) 18g_A=9g_B=g_C

Solution:

Gravitational fluxes :

g_A= GM_A/ r_A²--------------------(1)

g_B= GM_B/ r_B² ---------------------(2)

g_C= GM_C/ r_C²---------------------(3)

(GM_A/ r_A²) = (GM_B/ r_B²)= (GM_C/ r_C²) ------------------(4)

Given data

M_A=M ,

M_B=2M

M_C= 3M

r_A= R

r_B= 2R

r_C=3R

Substitute above values in eq(4)

==>(GM/ R²) = (G2M/ 4R²)= (G3M/ 9R²)

==> (GM/ R²) = (GM/ 2R²)= (GM/ 3R²)

==> g_A =( g_B /2) = (g_C /3)

==> 6g_A = 3g_B = 2g_C