48. Physics | Gravitation | Gravitational Self Energy of a Uniform Hollow Sphere | by Ashish Arora

48. Physics | Gravitation | Gravitational Self Energy of a Uniform Hollow Sphere | by Ashish Arora


let us calculate the gravitational self energy
of , a hollow sphere which is a uniform sphere of mass say m and radius r , and as we have
discussed, that gravitational self energy is the work done in assembling the whole body
, in bringing its constituent particles from infinity to a given shape and configuration
, say we are having a hollow sphere of mass m and radius r, and say it is made by, bringing
elemental masses d m, from infinity to surface of radius r , one by one, gradually say its
mass is increased from zero to small m and then it is increased up to , capital m,
and say if we wish to find the work done, in the whole process then it is taken as gravitational
self energy of the hollow sphere , so we can find it , say at any instant if its mass is
m and a d m element is brought from infinity, so we can say , when . mass of shell, is m
. and work done . in bringing . d m from, infinity to, r is , we know work done can
be written as d w is mass multiplied by the potential difference , so we can write it
as d m multiplied by potential on surface minus potential at infinity because this d
m is brought from infinity to this surface. as we have studied in previous section that
work done in displacing a mass in gravitational field is mass multiplied by the potential
difference , so potential at infinity we can consider as zero, we know that we take zero
reference at infinity , and the potential on this surface of sphere of mass m can be
written as , d m multiplied by, minus g m, by r , so in this situation the result will
be minus g-m , d m by, r . so we calculate this total work in bringing
this d m from infinity to the surface , here we can write the total work is integration
of dw which is equal to integration of minus. g m dm by r , and for mass it is integrated
from zero to , capital m , here work done will be , minus g by r is a constant, and
the integration of m is written as m square by 2, and the limits will be from zero to
m , so here work done will be minus g-m square by, 2 r, so this can be written as the gravitational
self energy of , a hollow uniform sphere of radius r and mass
m which is minus g-m square by, 2 r, this much amount of work is done in assembling,
mass m , in a spherical shell of radius r, which we treat as gravitational self energy
of this hollow sphere, here we can see minus g-m square by 2 r is
the amount of work done in bringing all the particles of the shell, from infinity to a
surface of radius r . now say for example we supply, g-m square by 2 r amount of energy
to this shell , its total energy will be zero and , we can state , energy is zero it implies
all its particles will dis assemble, and reach to infinity,
so in this situation we can directly state g-m square by 2 r is the energy to dis assemble
, a hollow spherical shell , from its surface of radius r, to infinity .

5 Replies to “48. Physics | Gravitation | Gravitational Self Energy of a Uniform Hollow Sphere | by Ashish Arora”

  1. Sir I did not get that how can we take Vs= -Gm/R it would only have been the case if a sphere of mass m is already formed and hence its potetIal at surface.."Which m is already there?" and are we finding self energy for complete sphere therefore mass from zero to M..no mass should be present initially..
    and if if u took an intermediate condition where of mass m it was already formed then the limits of dm should have been from zero to M-m because m was already present therefore adding only M-m to it for sphere of mass M…Please clearify my doubts..ll be highly thankful..

  2. Sir when to use this "self energy"…like in previous of your videos regarding the energy conservation…even spherical objects were taken (like a question where a tunnel is made in solid sphere…and to find the velocity to make it reach the centre)…but we didn't used "self energy" there!…I am very confused…bcoz after doing so many questions of energy conservation now this concept is confusing me😥..why we didn't use it there and when to use this concept?

  3. sir can we say Uself as a binding energy as on supplying this much energy the body particle will reach to infinity …???

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