7. (15 points) A uniformly charged sphere of radius $R$ is fixed on a horizontal surface at point $A$. The charge of the sphere is $Q$. At point $C$, which is located at a distance $L$ from the surface of the sphere, a charged sphere of radius $r$ and mass $m$ is floating. Its charge is $q$. It is known that $r<<R$. Determine the acceleration of the sphere immediately after a part of the material is removed at point $B$. It is known that $A B=S$. The removed material is a sphere of radius $r$. Points $A, B$, $C, O$ are located on the same vertical line. The acceleration due to gravity is $g$.

![](https://cdn.mathpix.com/cropped/2024_05_06_7a4bc87492c06fc34cd6g-05.jpg?height=394&width=329&top_left_y=2193&top_left_x=949)
Answer: $a=\frac{k q Q r^{3}}{m R^{3}(L+2 R-S)^{2}}$.

Solution. In the initial state for the ball: $F_{\text{el}}=m g$.

In the final state for the ball: $F_{\text{el}}-m g-F_{\text{rem}}=-m a$,

where $F_{\text{rem}}=k \frac{q q_{\text{rem}}}{(L+2 R-S)^{2}}$.

The removed charge: $q_{\text{rem}}=Q\left(\frac{r}{R}\right)^{3}$.

In the end, we get $a=\frac{k q Q r^{3}}{m R^{3}(L+2 R-S)^{2}}$.