find gravitational field inside a ring of radius r at adistance x from the centre in the plane of ring

dEG IS GRAVITATIONAL FIELD ASSUME A SMALL ELEMENT OF MASS ''dM'' ON THE RING GRAVITATIONAL FIELD DUE TO THE OBJECT ON THE SMALL ELEMENT IS dEG BREAK THEM INTO COMPONENTS dEGCOS@ , AND dEG SIN@

SINCE THERE IS SYMMETRY OF THE HORIZONTAL COMPONENT (dEGSIN@) ON BOTH THE SIDES OF THE OBJECT SO HORIZONTAL COMPONENT WILL GET CANCELLED SO YOU ANSWER WILL BE dEGCOS@

dEG=G(dM)/(DISTANCE)²   NOW THE DISTANCE IS (R²+X²)^1/2 LET IT BE D NOW dEGCOS@= G(dM)COS@/D²

INTEGRATE THIS FUNCTION YOU GET EG=GMCOS@/D² SINCE INTEGRATION IS SUMMATION THEREFORE INTEGRATION OF dM WILL BE THE TOTAL MASS M

FROM THE DIAGRAM USING PHYTHOGORUS THEORM  COS@=X/D

SUBSTITUTING YOU GET EG=GMX/D³

OR EG = GMX/(X²+R^{2)3/2 PLZZZZZ approve if it helped you}