49. Physics | Gravitation | Gravitational Self Energy of a Uniform Solid Sphere | by Ashish Arora

49. Physics | Gravitation | Gravitational Self Energy of a Uniform Solid Sphere | by Ashish Arora

now let’s calculate gravitational self energy
of a uniform solid sphere . which is of mass m, and radius r , we can say the self energy
of this solid will be the work done in bringing all the ,particles of this mass m from infinity
and assemble them in shape of a sphere of radius r .
so in this situation we can state if we calculate the density of this sphere it will be m by
4 by 3 pie r cube , and say gradually from infinity , elemental small masses d m , are
brought , and assembled at a point, gradually with a uniform density, and say at any instance
its radius is x . its mass is m, and on bringing a mass dm say its radius increases by dx.
so in this situation we can write this dm which is increasing its radius from x to x+
dx it will be rho into , 4 pie x square dx , will be the elemental mass dm , now, if
dm is brought from infinity to this point , we can state , work done, in bringing . d
m from , infinity to the . surface of , radius-x is , this work done dw can be written as dm
multiplied by , it is v-x minus, v-infinity , here v-x is the potential at the surface
of radius x and v-infinity is the potential at infinity which we can directly consider
as zero . so this dw can be written as dm multiplied
by potential at , the surface of radius x can be written as minus g-m by x , and here
if we substitute the values here this dw will be , it is minus g by x mass can be written
as density into 4 by 3 pie x cube , and dm we can write as density into 4 pie x square
dx, now for this situation the result will be , minus 16 pie square, by 3 , g rho square
, x to power4 dx . this will be dw , if we calculate , the total
work done in assembling the sphere , from, its radius x varies , from zero to r ,then
total work can be directly written as , total work in assembling, sphere. of radius r , this
can be given as w which is integration of dw , it will be , here in this situation it
is , negative of 16 pie square g , rho square by 3 , integration of x 4 dx , from, zero
to r . now here on integrating it , the integration
of x 4 will be x five by five , and we substitute the value of r. then the result will be, minus
16 by 15 . pie square g , rho square r to power 5 , now on substituting the value of
rho here , the result will be minus 16 by 15 .
pie square , g r to power 5 , and density we can write as , m by 4 by 3 pie r cube , and
its whole square , we use , and here we can see that pie cancelled out , r cube square
is, r to power 6, one r is left here , and this 4 square is 16 will also cancel out,
and. this 3 square is 9 , so final result here
is that , the work done will be , minus 3 by 5. g-m square by r , and this same can
be written as , self energy of a uniform solid sphere of mass m, and radius r .
and here we can say that when 3 by 5 g-m square by r , energy is supplied to a sphere its
total energy will become zero because, its existing self energy is given by this value,
and energy zero implies that all its particles , will dis assemble and reach to, infinity
, so here total self energy , means it is the amount of energy required to , dis assemble
or break , the sphere particles from, its initial shape to , infinity .

12 Replies to “49. Physics | Gravitation | Gravitational Self Energy of a Uniform Solid Sphere | by Ashish Arora”

1. Matt Walker says:

Nicely presented.ย  Thanks.

2. dhayyan sbdar says:

finished once

3. Harshvardhan Nigam says:

thank you very much sir

4. TRIXTORAX says:

sir i didnt understand the difference between self energy and potential energy.. please explain sir

5. Ritesh Uppal says:

In derivation of self energy for "hollow sphere" radius was taken constant and dm was not not written in any terms…like in case of solid sphere…dm is written in terms of x

6. Nandini Loomba says:

Sir I didn't understand one thing that why u have taken Vx = -Gm/x means hv u considered a sphere with radius X as solid or hollow ? As you have taken its mass as m so u might have considered it as solid because potential of solid sphere is – GM/X plz explain

7. Akash Singh says:

sir, what is the significance of self energy being negative ? means, what can we conclude from it

8. Needa khan says:

thanks a lot!!!!! ๐

Sir one doubt please…where can we consider infinity?…Is it there where grav. field strength is insignificant or what..

10. Sarthak Agrawal says:

Can similar thing be said that in case of electrostatics if in a uniformly charged solid sphere if we take out 3/5KQ^2/R OF ENERGY THEN WHOLE CONFIGURATION OF CHARGE WOULD BE DESTROYED

11. Khushi Gupta says:

Thank you sir

12. Rishikesh Bari says:

Chutiya saalla kitna English bol ra