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- Homework Statement
- Find the gravitational potential energy of a sphere of radius R and uniform density ##\rho##, due to the gravitational field of itself.

- Relevant Equations
- shell theorem, ##U=-G\frac{Mm}{r}##

My attempt was to consider spherical shells of radius ##r## (##r\leq R##))and thickness ##dr## and then the potential energy of this shell would be in the field only of the "residual" sphere of radius ##r## (a result also known as shell theorem) $$U_{dr}=G\frac{\rho\frac{4}{3}\pi r^3 \rho 4\pi r^2}{r}dr$$ and the whole gravitational potential energy of the sphere within the field of itself would be $$U=\int_0^R U_{dr}$$.

What do you think?

What do you think?