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. 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

  2. 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

  3. 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

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