A sphere of radius R has its centre at the origin. It has uniform mass density ρ0 except that there is a spherical hole of radius r=12R whose centre is at x=12R, as shown in figure. The gravitational field at x=2R is aa+20πGρ0R. Find a.

Let E be the gravitational field at x due to the complete sphere.If E1 be the field due to hole and E2 be the field due to the remaining portion, then we have          E=E1+E2⇒ E2=E−E1⇒ E2=GMx2−Gmx−R22    .........(1)where, M=43πR3ρ0 and m=43πR23ρ0Substituting the values in equation (1), we getE2=−πGρ0R361x−R22−8x2⇒E2=−πGρ0R3612R−R22−8(2R)2⇒E2=−πGρ0R3649R2−2R2⇒E2=−πGρ0R64−189⇒E2=1454πGρ0R⇒E2=727πGρ0R Since, E=aa+20πGρ0R⇒aa+20=727⇒a=7