Moment of Inertia using density of Earth

[musiliu]

Homework Statement

The density of the Earth, at any distance r from its center, is approximately

p = [14.2 - 11.6 r/R] x 10^3 kg/m^3

where R is the radius of the Earth. Show that this density leads to a moment of inertia I = 0.330MR^2 about an axis through the center, where M is the mass of the Earth

Homework Equations

My teacher gave hints to the class and told us to:

--Find the mass M = integral(dm) where dm = pdV
--Break Earth into pieces with same r ---> spherical shells
so dI = (2/3)r^2 dm and I = (2/3) integral( (r^2) dm) where dm = pdV
-- dV = area x height = 4(pi r^2)dr

The Attempt at a Solution

calculated the integral for I and got
I = (2/3) [ (56800pi / 5)r^5 - (46400pi / 6R)r^6

I calculated the intral for M and got
M = (56800pi / 3)r^3 - (46400pi / 4R)r^4

now i am stuck and do not know how to show that I = 0.330MR^2
also i think my calculations are wrong..

please can someone help me figure out the problem..thanks

 

[musiliu]

ok, I don't need help on this problem anymore. A tutor helped me and i figured out that there are limits of integration from 0 to R..

thanks anyway.

 

[nlightner]

I would approach this problem by first checking the units of my calculations. The moment of inertia is typically measured in units of mass times distance squared (kgm^2), while the mass is measured in units of mass (kg). Therefore, the integral for I should have units of (kgm^2) rather than (m^5). Similarly, the integral for M should have units of mass (kg) rather than (m^3).

Next, I would double check my calculations to make sure they are correct. It may be helpful to write out the full integrals and double check my algebra.

Once I have verified my calculations, I would compare them to the given equation for the moment of inertia, I = 0.330MR^2. I would then manipulate my calculated integrals to see if I can simplify them to match the given equation. It may be helpful to use algebraic manipulation or substitution to get rid of any extra variables.

If I am still having trouble, I may consult with my peers or my teacher for additional guidance. As a scientist, it is important to double check calculations and ask for help when needed in order to ensure accuracy and understanding.