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  • Consider a thin uniform spherical layer of mass and radius . The potential

Consider a thin uniform spherical layer of mass \mathrm{M} and radius \mathrm{R}. The potential energy of gravitational interaction of matter forming this shell is
 

Option: 1

-\frac{\mathrm{GM}^2}{\mathrm{R}}


 


Option: 2

-\frac{1}{2} \frac{\mathrm{GM}^2}{\mathrm{R}}


Option: 3

-\frac{3}{5} \frac{\mathrm{GM}^2}{\mathrm{R}}
 


Option: 4

-\frac{2}{3} \frac{\mathrm{GM}^2}{\mathrm{R}}


Answers (1)

best_answer

Let us consider the shell when a mass \mathrm{m} is already piled on it by the agency.

If \mathrm{V} is the potential on the shell, then

\mathrm{ \mathrm{V}=-\frac{\mathrm{Gm}}{\mathrm{R}} }

To add a mass \mathrm{dm} further we have

\mathrm{dW}=\mathrm{Vdm}

\mathrm{ \Rightarrow \mathrm{dW}=-\frac{\mathrm{Gm}}{\mathrm{R}} \mathrm{dm} }

\mathrm{ \Rightarrow \mathrm{W}=-\frac{\mathrm{G}}{\mathrm{R}} \int_0^{\mathrm{M}} \mathrm{mdm} }

\mathrm{ \Rightarrow \mathrm{W}=-\frac{1}{2} \frac{\mathrm{GM}^2}{\mathrm{R}} }=Potential Energy of Interaction.

Hence option 2 is correct.

Posted by

Pankaj Sanodiya

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