The Density of a Nucleus: A Question About Steps in Calculating Nuclear Density

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SUMMARY

The discussion centers on the calculation of nuclear density using the formula ρn = mn/Vn, where mn is approximately equal to ma. The participant identifies a potential error in the solution regarding the volume calculation, suggesting that the radius should be cubed, leading to the correct expression ρn = ρa/(10^-5)^3. This adjustment is crucial for accurate density calculations, as failing to cube the radius results in an incorrect density value of 108 kg/m³ instead of the expected result.

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Homework Statement
There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale—approximately 10^3 kg/m^3 . The nucleus of an atom has a radius about 10^-5 that of the atom and contains nearly all the mass of the entire atom. (a) What is the approximate density of a nucleus?
Relevant Equations
ρ = m/V
I have the solution to the problem but I think I found either a typo or steps that were not included. I think I have made sense of the problem but I would like to double check that I'm doing this right. Thanks.

ρn = mn/Vn

mn ≈ ma

Rn = 10-5Ra

This is were I am having issue. The solution says:

ρn = ma/10-5Va = ρa/10-5

ρn = 103 kg/m3/10-5

It says 10-5Va. Shouldn't it be (10-5)3Va because of the radius is a cubed value? Same for the ρn/10-5. Shouldn't it also be ρa/(10-5)3?

If those values are not cubed then I end up with

ρn = 103 kg/m3/10-5 = 108... which is not right.

But if I cube that value then I get the correct answer. I wrote up a similar problem using simpler numbers and my logic checked out. However, like I said, I just wanted to double check.
 
Last edited:
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You are correct. They should have cubed the 10-5.
 
TSny said:
You are correct. They should have cubed the 10-5.
awesome. thanks.
 

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