View Single Post
Richard Nash
Richard Nash is offline
Dec9-12, 10:31 PM
P: 1
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 [itex]M[/itex] and a particle of mass [itex]m[/itex] located at a distance [itex]r[/itex] from the center.

The shell has been divided into narrow rings.[itex]R[/itex] has been assumed to be the radius of the shell with thickness [itex]t[/itex] ([itex]t<<R[/itex]). The ring at angle [itex]\theta[/itex] which subtends angle [itex]d\theta[/itex] has circumference [itex]2\pi R\sin\theta[/itex].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 [itex]\alpha[/itex] be the angle between the force vector and the line of centers, [itex]dF=\frac{Gm\rho dV}{r'^2}\cos\alpha [/itex] where [itex]r'[/itex] is the distance of each part of the ring from [itex]m[/itex].

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.
Phys.Org News Partner Physics news on
Physicists consider implications of recent revelations about the universe's first light
Vacuum ultraviolet lamp of the future created in Japan
Grasp of SQUIDs dynamics facilitates eavesdropping