Simple Density Law: Prove Central Density 4x Mean

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Homework Help Overview

The problem involves demonstrating that a spherical body with a specific density function has a central density that is four times greater than its mean density. The density function is defined in terms of the radius of the sphere.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants express uncertainty about how to begin the problem and seek guidance on approaching it. Questions arise regarding the relationship between the density function and mass computation.

Discussion Status

Some participants have offered hints about calculating mass using integrals and the relationship between density and volume. Others are exploring the implications of the density function and considering how to apply mathematical principles to the problem.

Contextual Notes

There is a lack of clarity regarding the necessary equations and methods for integrating the density function to find mass. Participants are encouraged to refer to their notes or textbooks for further information.

seto6
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Homework Statement


Show that a spherical body with density
p (r) =Pc(1−x)
where x = r/R and R its radius, has a central density pc that is 4 times larger than the mean density (< p > =mass/volume).


Homework Equations



not sure

The Attempt at a Solution



im not sure how to start this problem... can someone give me a hit on doing this question. please!
 
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help?please
 
can some1 tell me how to approch this problem ...
 
If you know the density of the object at each point, do you know how to compute its mass?
 
nop.
 
seto6 said:
nop.

Not knowing any relation between the density function and the mass might be the root of your problem. Could you try and look something up?
 
sure
 
i know that p=m/v and v=(4/3) pi R^2

umm could you post some links b/c I am googleing it not much tho
 
It might be better to check your notes or the textbook. In general you find the mass by doing the triple integral of density*dV. I'll give you a hint that if the density is spherically symmetric then the mass is the integral from 0 to R of density*4*pi*r^2*dr. Does that sound familiar? If you can show that form follows from the density*dV formula by using spherical coordinates, I'll give you extra virtual points.
 
  • #10
let me give it ago and get back
 
  • #11
OMG! thank you soo much... i figured it out! and showed it too..RESPECT TO YOU :smile:
 
Last edited:

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