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The Gravity Due to a Collection of Objects and Its Approximation

There is a beautiful mathematical theorem which says that for mass uniformly distributed on a sphere the center of gravitation is at the center of mass; i.e., the center of the sphere.

This page is to show how stringent the condition of uniform distribution is. It is not enough for the masses to be regularly distributed in discrete points. When a point is near a sphere of uniformly distributed matter the closest point has infinitesimal mass, whereas for the sphere of discrete mass points the nearest point may be of finite mass and for which the density is infinite.

To investigate this topic linear distributions of mass will be considered. This reduces the mathematical complexity to manageable proportions.

Consider a mass M distributed along a line of length L and its gravitational attraction for a unit mass located at a distance D from the right end of the line. In the following the gravitational constant G will be ignored. The analysis applies not just for gravitation but for any inverse distance squared law.

Suppose the mass is divided into n equal bodies that are evenly distributed along the line with a body at each end of the line. This means that there are (n-1) equal line segments of length δ=L/(n-1) separating them. The force on the unit mass located a distance D from the right end of the line is given by

F = Σi=0n (M/n)/(L+D-iδ)²

From the expression on the right a term M/(L+D)² can be factored to give

(F/M)(L+D)² = (1/n)Σi=0n1/(1-iε)²

where ε=δ/(L+D). Epsilon is not the crucial parameter. Epsilon can be expressed as

ε = (L/(n-1))/(L+D) = (1/(n-1))(1/(1+D/L)

The crucial parameter is thus (D/L). Let γ=1/(1+D/L). The force equation is then

(F/M)(L+D)² = (1/n)Σi=0n1/(1-iγ/(n-1)))²

For comparison consider the force the mass M would exert on the unit mass if it were all concentrated at the midpoint of the line. That force would be

F' = M/(L/2+D)²
or, equivalently
(F'/M)(L+D)² = [(L+D)/(L/2+D)]²
= [(1+D/L)/(½+D/L)]²

This relation can be expressed in terms of γ as

(F'/M)(L+D)² = (1/γ)/(1/γ - ½) = 1/(1 -½γ)

For a further comparison consider the force due to the mass uniformly distibuted along the line with a density σ=M/L. This force is given by

F" = ∫0L[σ/(L+D-x)²]dx
which can be expressed as
(F"/σ)(L+D)² = ∫0L[1/(1-x/(L+D))²]dx
or, equivalently
(F"/M)(L+D)² = (1/L)∫0L[1/(1-x/(L+D))²]dx

A change of the variable of integration from x to s=x/(L+D) results in

(F"/M)(L+D)² = ((L+D)/L)∫0L/(L+D)[1/(1-s)²]ds
or, equivalently
(F"/M)(L+D)² = ((1+D/L)∫01/(1+D/L)[1/(1-s)²]ds
and hence
(F"/M)(L+D)² = (1/γ)∫0γ[1/(1-s)²]ds

A further change in the variable of integration from s to z=1-s results in

(F"/M)(L+D)² = (1/γ)∫1-γ1[1/z²]dz
and hence
(F"/M)(L+D)² = (1/γ)[−1/z]1-γ1
which reduces to
(F"/M)(L+D)² = (1/γ)[1/(1-γ)-1] = 1/(1-γ)

Computational Results

The graph below shows the value of (1/n)Σi=0n1/(1-iγ/(n-1)))² for γ=0.9 and for ten different values of n.

The value is very slowly converging toward 1.0.

The functional dependence of (1/n)Σi=0n1/(1-iγ/(n-1)))² on γ is particularly of interest. The parameter γ is inversely related to the distance from the edge of the collection of bodies. For points close to the outside edge γ is close to unity. For points far away the value of γ decreases toward zero. The value of n for these computations is 20.

The final display shows a comparison of the true of the gravitational force with two approximations. One is for having the total mass of the collection concentrated at its center of mass. The second is when the mass is uniformly distributed over the length of the line segment. In the graph the true value is in red. The center of mass approximation is in green and the uniform distribution is in blue. Notably all three functions converge asymptotically to unity for values of γ going to zero; i.e., for the distance from the collection being large compared to the dimension or scale of the collection.

It is notable that the center of mass approximation is better than the uniformly distributed approximation. It is not a very good approximation for points near the edge of the collection. It is also notable that the center of mass approximation seems to converge to the true value and they jointly converge to unity.

(To be continued.)

(To be continued.)

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