## Efficiently calculating the magnitude of gravitational force

I am reading Kolenkow and Kleppner's Classical Mechanics and they have tried to calculate the gravitational force between a uniform thin spherical shell of mass $M$ and a particle of mass $m$ located at a distance $r$ from the center.

The shell has been divided into narrow rings.$R$ has been assumed to be the radius of the shell with thickness $t$ ($t<<R$). The ring at angle $\theta$ which subtends angle $d\theta$ has circumference $2\pi R\sin\theta$.The volume is $$dV=2\pi R^2t\sin \theta d\theta$$ and its mass is $$pdV=2\pi R^2t\rho\sin\theta d\theta$$

If $\alpha$ be the angle between the force vector and the line of centers, $dF=\frac{Gm\rho dV}{r'^2}\cos\alpha$ where $r'$ is the distance of each part of the ring from $m$.

Next, an integration has been carried out using $$\cos\alpha=\frac{r-R\cos\theta}{r'}$$ and $$r'=\sqrt{r'^2+R^2-2\pi R\cos\theta}$$

Question: I would like to avoid these calculations and I was wondering if there exists a better solution.

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 Mentor You can use Gauß' law and symmetry. But if you want to calculate it via an integral, I don't think there is an easier way.

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