Gravitational Self energy of a Uniform hollow and solid sphere – Definition and Derivation .

Self energy and Gravitational self energy

It is the energy possessed by a body because of the interaction forces within the body . This can be defined as the work done by external agent in assembling all the particles of a body in a definite shape and size or it is the work done in creating a body . we are able to say that , if the forces within the body are engaging , work tired collection is by the body itself and self energy of the body will be negative and if the forces within the body are repulsive then some external work are tired collection the body and its self energy are positive .

In case of Gravitational self energy we consider Gravitational force , gravitational self energy of an object is concerned obviously it is always negative because gravitational force is attractive forces .

Gravitational self energy of a uniform hollow sphere

Fig. 26.1

Let’s Calculate the gravitational self energy of a hollow sphere of mass M and radius R , as shown in figure 26.1 .
To assemble this we assume that we bring several mass elements dm step by step from infinity to a spherical surface of radius R with centre O . Every dm is uniformly on this surface .

Now we consider an intermediate situation when of surface becomes m and a further dm is brought from infinity to the surface thus we can write the work done in bringing this elemental mass dm from infinity to a distance R from O as

dW= Gmdm/R ….( i )

Here we can assume in a hollow sphere mass m is behaving as it is concentrated at O for outer points .

Now total work done in increasing the mass from 0 to M can be obtained by integrating the expression in equation ( i ) within proper limits as

W = dw = ∫Gmdm/R
( limit of dm from 0 to M )

W = GM²/2R … ( ii )

Here expression in equation ( ii ) is the work done in assembling the hollow sphere in space . As this work is done by the gravitational attractive forces of the body , this work is done by itself in assembling thus gravitational self energy of a hollow sphere of mass M and radius R is given as

Uself = – GM²/2R

Gravitational self energy of a uniform solid sphere

Let calculate self energy of a solid sphere of mass M and radius R . For it we’ll create (assemble) a solid sphere in space . For this we bring several mass elements dm from infinity and start assembling at a point O in such way that the size of the assembled mass increases gradually layer by layer .

Now consider an intermediate situation when the radius of assembled matter increased to ‘x’ and the mass becomes ‘m’ if a further mass dm is brought from infinity to its surface which increases the radius of sphere by dx as shown in figure 26.2

Fig. 26.2

then work done in this process is

dW = Gmdm …. ( i )

If ρ be the density of sphere
[ ρ = M/(4πR³/3) ]

m = ρ×(4πx³/3)

dm = ρ×(4πx²dx)

and
Now from equation- ( i )

dW= G[ρ×(4πx³/3)][ρ×(4πx²dx)]/x

dW =16π²Gρ²x⁴dx/3

Now we can find the total work done in assembling this sphere to a radius R by integrating the above expression within proper limits as

W= ∫ dW = ∫16π²Gρ²x⁴dx/3
( limit of x from 0 to R )

W = 16π²Gρ²R⁵/15

As we know the density of sphere is given as

ρ = M/(4πR³/3)

we have ,

W = 16π²G²[M/(4πR³/3)]²R⁵/15

W = 3GM²/5R …( ii )

Thus we get the above work ( In equation (ii) ) is done by gravitational forces in assembling a solid sphere of mass M and radius R . Thus the self energy of the solid sphere of mass M and radiur R is given as

Uself = – 3GM²/5R

Note – The magnitude of self energy is the amount of energy required to split a solid sphere into its constituent particles and separating them to infinity .

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