# 11. Physics | Gravitation | Gravitational Field Strength due to a Uniform Ring | Ashish Arora (GA)

let us discuss gravitational field strength

due to a uniform ring . again we will discuss it in 2 cases , if we talk about the case

1 , at a point . at centre of ring , if we talk about the gravitational field, at the

centre of ring , we can say we are given with a uniform ring . that means the total mass

of the ring is distributed uniformly at its sircumference which is of radius, say , r

, and if we talk about the centre of ring we can directly state that gravitational field

at the centre of ring is equal to zero . because for any elemental mass dm if we wish

to find the gravitational field at the centre it will be directed towards the dm , say it

is dg, and corresponding to this dm there always exist an element which is opposite

to this which will produce gravitational field , that is dg , in opposite direction and they

will cancel out , so at all points in a ring there exist , an opposite point diametrically

opposite point on the ring which will cancel the gravitational field, due to one point

. so we can say the net effect of all, the point

masses , at centre will be equal to zero so , the net gravitational field strength at

the centre of ring will be zero, similarly if we talk about, its second case . at an

axial point of ring .

and in this situation if we draw a 3 dimensional picture , say we are having a ring . which

is of mass m, and, the ring is placed along x and y axis , here y axis is vertical and

x axis is horizontally outward . and along z axis, there exist a point p .

located, at a distance z from the centre of the ring and we are required to find the gravitational

field strength due to the ring at point p, then in this situation , we can state that

we will consider a small element of length d l on the ring, due to which we find the

gravitational field at p , and due to this it will be directed towards this elemental

mass , say it is dg , then if we find the mass of this element dm , mass its elemental

mass dm can be written as m by 2 pie r into d l .

as the total mass of ring is distributed throughout its length that is its sircumference of length

2 pie r, so in this situation we can write , gravitational field strength . at p , due

to dm is , in this situation this dg is given as, g dm by , here we have to divide by the

square of this distance which is root of r square + z square , so it is r square + , z

square , this is the value of , gravitational field at p due to dm and if this angle is

alpha we can have 2 components of this gravitational field strength , one is d g coz alpha, and

other is in perpendicular direction to z axis which is d g , sine alpha .

so in this situation, if we just integrate , this elemental gravitational field for this

whole length, here we can state this d g sine alpha will cancel out each other because corresponding

to each element , there will be diametrically opposite element for which the normal components

will cancel each other , so here we can state net gravitational field strength .

at point p is , this is given as g at point p which is integration of dg coz alpha, if

we substitute the values this will be integration of g , dm we write as m by 2 pie r , multiplied

by d l, this is, r square + z square , multiplied by , coz alpha we can substitute as z by root

of r square, + , z square . and we will integrate this for the whole length

of the sircumference that is from zero to 2 pie r , integrating this the only variable

here is dl , so this dl will be integrated and this 2 pie r will cancel out and the result

we get will be , g m z by , r square , + z square to power, 3 by 2 , this is the result

we get , as the final result of the gravitational field strength at the axial point of a ring

located at a distance z from, centre of the ring .

## 4 Replies to “11. Physics | Gravitation | Gravitational Field Strength due to a Uniform Ring | Ashish Arora (GA)”

Thank you, you are the only person on youtube/online sources who give a through explanation

super

It's great

Good explanation👍