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Consider a thin uniform spherical layer of mass \mathrm{M} and radius \mathrm{R}, if we consider a solid sphere of mass \mathrm{M} and radius \mathrm{R}, then the potential energy of gravitational interaction of matter forming this solid sphere is
 

Option: 1

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


Answers (1)

best_answer

\mathrm{ d U=-\frac{\mathrm{Gmdm}}{\mathrm{r}} }

\mathrm{ \Rightarrow \mathrm{dU}=-\frac{\mathrm{G}\left(\frac{4}{3} \pi r^3 \rho\right)\left(4 \pi r^2 \mathrm{dr} \rho\right)}{\mathrm{r}} }

\mathrm{ \Rightarrow \mathrm{dU}=-\frac{16 \pi^2 \mathrm{G} \rho^2}{3} \mathrm{r}^4 \mathrm{dr} }

\mathrm{ \Rightarrow \mathrm{U}=-\frac{16}{3} \pi^2 \mathrm{G}\left(\frac{\mathrm{M}}{\frac{4}{3} \pi \mathrm{R}^3}\right)^2 \int_0^{\mathrm{R}} \mathrm{r}^4 \mathrm{dr} }

\mathrm{ \Rightarrow \mathrm{U}=-\left(\frac{16}{3} \pi^2 \mathrm{G}\right)\left(\frac{\mathrm{M}^2}{\frac{16}{9} \pi^2 \mathrm{R}^6}\right)\left(\frac{\mathrm{R}^5}{5}\right) }

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

Hence option 3 is correct.





 

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