The density of Earth varies with the distance r from the center of Earth as ρ = C[1.22-(r/R)].?
where R is the radius of Earth and C is a constant.
A- Find C in terms of the total mass M and the radius R.
B- According to this model, what is the moment of inertia of Earth about an axis through its center? (Use the following as necessary: M and R.)
I need help, itried to solve but no sucess......
More Questions From This User See All
Answers & Comments
Verified answer
I think your formula is C[1.22 - (r/R)]. If this is not right, append additional info to correct me.
A. Use a spherical shell for the element of mass and integrate from 0 to R.
dM = (spherical shell volume)*density
spherical shell volume = 4*pi*r^2*dr
dM = (4*pi*r^2*dr)*C[1.22 - (r/R)]
Integrate: M = 4*pi*C[1.22*r^3/3 - r^4/(4*R)]
Evaluate: M = 4*pi*C*[R^3*1.88/12] = pi*C*[R^3*1.88/3]
Solve for C: C = 3*M/(1.88*pi*R^3)
B. You need to do an integration of r^2*dM over the volume of the sphere. dM is as above, spherical shells. so the integration will be from r = 0 to R.
d(Moment of Inertia) = r^2*(4*pi*r^2*dr)*C[1.22 - (r/R)]
Integrate to get: 4*pi*C*[1.22*r^5/5 - r^6/(6*R)]
Evaluate from 0 to R: 4*pi*C*R^5*2.32/30
Use C from above: 4*pi*[3*M/(1.88*pi*R^3)]*R^5*2.32/30
12*M*R^2*2.32/(1.88*30)
M*R^2*4.64/(1.88*5)
M*R^2*0.4936
So that is the moment of inertia if I did the math correctly. If I didn't then I hope I made the method clear enough.
A.where from comes this formula?
B.there is a formula only for homogeneous sphere to calculate this...