Question

consider a ring of mass m and radius r. maximum gravitational intensity on the axis on the ring has value?​

Answer 1

Maximum gravitational intensity on the axis on the ring has value,

E = 2Gm / [R².(3)³/²]

Formula for gravitational field intensity on its axis is

E = Gmx / [(x² +R²)³/²]    --------- Eq.1

For the maximum value of E, dE / dx = 0

dE       Gm[(x² +R²)³/² − 3x/2[(x² +R²)¹/²] × 2x

-----  =   ----------------------------------------------------  = 0

dx                              (x² +R²)³

Thus, x = R/√2

Putting value of x in Eq.1

E = GmR/√2

    ---------------

   [(R/√2)² + R²]³/²

E = 2Gm / [R².(3)³/²]

Answer 2

Explanation:

first consider a mass dm

then find the components of intensity at a distance x from centre , on the axis

the y component will be cancelled

integrate the x component to have value of gravitational intensity

for the maximum value , use the concept of maxima and minima

put dI / dX= 0

differentiate R.H.S and put it equal to 0

find X and put its value to have answer