Max Size of Object Formed by Aggregation Before Self-Gravity

[spinnaker]

Homework Statement

We know the terrestrial planets formed by aggregation of debris from the solar nebula. We want
to calculate the maximum size of object that can form by aggregation before self-gravity causes it to pull itself into a round shape.

Our analysis is assisted by considering Earth’s mountains. It is clear from Earth’s geology that of its many mountains, none are taller than 10 km high. This indicates that the strength of rock at the center of the base of a mountain cannot support a structure higher than 10 km. The pressure at the middle of the base of a mountain is equal to the weight of a column of rock (with unit cross sectional area) as high as the mountain. If a mountain were taller than 10 km, it would slump or begin to be overtaken by self-gravity. We want to use this fact to determine the maximum size of an object that can form without being overtaken by self-gravity.

Assume you have two cubes of rock (density of 3000 kg/m³) of equal mass (M) and size (length d) that are face to face. They are so large that the only force holding them together is pressure caused by gravitational attraction, but not so large that they flow into a spherical shape due to self-gravity.

Determine the maximum size of rectangular body that could be made from the two cubes.

Homework Equations

Force between cubes is F = GM²/d²

The Attempt at a Solution

V = d³
M = ρV = ρd³
F = GM²/d² = Gρ²d⁴

I think I have the formulas okay, but I don't know how to get a "maximum" size out of the equation.

 

[ZetaOfThree]

You can determine the maximum pressure that can be exerted by the rock from the information in the second paragraph. From there you can determine the maximum force, etc...

 

[spinnaker]

I'm not sure if I follow.

So a column of height h and radius r would have a volume of πr²h.

multiply that by the density of the rock (3000 kg/m3) and you get 3000πr²h, which is the mass of the column.

Assuming that's the mass, I don't see where the pressure comes in.

 

[ZetaOfThree]

The rock at the very bottom of the column has to support the weight of the column. The pressure is then the force that the rock exerts divided by the area over which it acts (the cross sectional area of the cylinder).

 

[paranormal]

Can someone help?

I would approach this problem by first identifying the key parameters and assumptions that need to be considered. In this case, we are looking at the maximum size of an object formed by aggregation before self-gravity takes over. This means we need to consider the strength of the material, the gravitational force between the objects, and the size of the objects.

Based on the given information, we can assume that the strength of the material is limited by the maximum height of Earth's mountains, which is 10 km. This means that the maximum force that the material can withstand is equal to the weight of a column of rock with a cross-sectional area equal to the base of the mountain.

Next, we can consider the gravitational force between the two cubes. This force is given by the formula F = GM^2/d^2, where G is the gravitational constant, M is the mass of the cubes, and d is the distance between their centers.

In order for the cubes to remain in a rectangular shape, the gravitational force between them must be equal to or less than the maximum force that the material can withstand. This means that we can set the two equations equal to each other and solve for d, the distance between the cubes:

Gρ^2d^4 = maximum force

d = (maximum force/Gρ^2)^1/4

This gives us the maximum distance between the cubes, which is equivalent to the maximum size of the rectangular body that can be formed from the two cubes. We can then use this distance to calculate the maximum size of the object, which is equal to twice the distance between the cubes (since the two cubes are face to face).

In summary, the maximum size of an object formed by aggregation before self-gravity takes over is determined by the strength of the material and the gravitational force between the objects. By setting these two forces equal to each other, we can solve for the maximum distance between the objects, which gives us the maximum size of the object.