# Gravity outside a cylinder.

## Main Question or Discussion Point

With reference to this web page http://principles.ou.edu/earth_figure_gravity/index.html [Broken]

I calculate the force of gravity tangential to the long axis of a cylinder to be proportional to 4GM/R where M is the mass per unit length of the cylinder. Does that seem correct?

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pam
F is perpendicular to the long axis of a cylinder.

ZapperZ
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With reference to this web page http://principles.ou.edu/earth_figure_gravity/index.html [Broken]

I calculate the force of gravity tangential to the long axis of a cylinder to be proportional to 4GM/R where M is the mass per unit length of the cylinder. Does that seem correct?
No it isn't.

Did you use the Gauss's Law equivalent to calculate g?

Zz.

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No it isn't.

Did you use the Gauss's Law equivalent to calculate g?

Zz.
I was trying to use the method demonstrated on that web page using an an imaginary Gaussian surface and the integral form of Poisson's equation. M is the mass contained within the imaginary volume of the Gaussian surface. For a cylinder I assumed the mass is proportional to $$2 \pi R^2 p L$$ where L, R are length and radius of the cylinder and p is the density per unit volume of the cylinder. The integral of the Gaussian volume is $$\pi r L$$.

Using the formula given,

$$g(\pi r L) = 4\pi GM$$

$$g = \frac{4GM}{r L} = \frac{4G (2 \pi R^2 p L)}{ r L} = \frac{8 \pi G R^2 p}{r}$$

Where R and p are constants.

I am not sure if that is correct. I was just trying to interpret the method shown. I would like to know what the correct solution is.

What I am trying to establish is if the force of gravity perpendicular to the long axis of a cylinder is proportional to 1/r rather than the usual 1/(r^2) when considering a spherical massive body.

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This paper seems to agree that the force of gravity perpendicular to a cylinder is proportional to 1/R.

http://www.pgccphy.net/103/gravity.pdf

Interestingly the force equation is independent of the length of the cylinder and so applies equally to the force of gravity perpendicular to the axis of symmetry of a thin disk. According to this, an object leaving a solar system that had a lot of mass at the perimeter of the system (in the form of a disk of meteorites for example) would experience greater gravitational acceleration than GM/(R^2) where R is the total mass of the solar system.

For a disk of uniform density the force of gravity on a test particle at a radius that is less than the radius of the disk appears to be independent of the distance from the centre of the disk. For a galaxy that happened to have an ideal disk shape and an even distribution of mass would have much greater orbital velocities consistent with F= GMm. For a more realistic galaxy with less than a perfect disk shape and greater density towards the centre the orbital velocities would be consistent with GMm/R > F > GMm/(R^2).

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rcgldr
Homework Helper
For an infinitely long cylinder, gravity diminishes relative to 1/r. For a infinitely large plane, gravity remains constant regardless of position. A long cylinder or a large plane closely approximate this if distances from the cylinder or plane are relatively small compared to the size of the cylinder or plane.

As an extension to the question in the OP what is the gravitational potential for a hypothetical body that has a gravitational field that attenuates relative to 1/r ?

I read somewhere that there may be difficulties with using the gaussian method to determine the force of gravity. Is it possible to calculate the gravity/distance relationship for a flat slab by another method such as integrating the total forces of infinitesimal "point" masses that make up up the slab?

I believe Newton used a similar method to calculate the gravitational forces of spherical shell when he concluded that the gravity of a large sphere is equivalent to that of a point particle of the same mass.