I have a problem (in two parts) where I believe I am tackling the solution with the correct methodology but somewhere along the line I must be going wrong because cannot match my answers to the ones in the book, in fact I am way out (by a factor of 10 to the power 11 or 12 for each).

The problem reads as follows -

__Problem__The moment of inertia of a sphere with uniform density about an axis through it's centre is ..

**2/5 M R^2) = 0.400 M R^2.**

Satellite observations show the Earth's moment of inertia is ..

__0.3380 M R^2)__The geophysical data suggests that the earth consists of 5 main regions ..

__1. Inner Core (R=0 to R=1220km) of average density 12,900 kg/m^3__

2. Outer Core (R=1220 to R=3480km) average density 10,900 kg/m^3

3. Lower Mantle (R=3480 to R=5700km) average density 4,900 kg/m^3

4. Upper Mantle (R=5700 to R=6350km) average density 3,600 kg/m^3

5. Outer Crust (R=6350 to R=6370km) average density 2,400 kg/m^32. Outer Core (R=1220 to R=3480km) average density 10,900 kg/m^3

3. Lower Mantle (R=3480 to R=5700km) average density 4,900 kg/m^3

4. Upper Mantle (R=5700 to R=6350km) average density 3,600 kg/m^3

5. Outer Crust (R=6350 to R=6370km) average density 2,400 kg/m^3

(a)

**Show that the moment of inertia about a diameter of a uniform spherical**

shell of inner radius = R(1) and outer radius = R(2), and density 'Rho' is ..

shell of inner radius = R(1) and outer radius = R(2), and density 'Rho' is ..

__I = (8/15 Pi) x Rho x ((R(2)^5) - (R(1)^5))__**[Hint - form the shell by superposition of a sphere of density 'Rho' and**

smaller sphere of density '-Rho' ]

smaller sphere of density '-Rho' ]

**(b) Check the given data by using them to calculate the mass of the earth.**

**(c) Use the given data to calculate the earths moment of inertia in terms of**

'M R^2'.

'M R^2'.

I was able to do part (a) without much problem, but it is parts (b) and (c)

I am unable to solve to match the answers given in the book.

For part (b) I used the fact that

**Mass = density x volume**and volume of

a sphere is ..

**4/3 Pi R^3.**

Now since we are using spherical shells one within the other then to calculate

the volumes we multiply

**(4/3) x Pi x ((R(2)^3 - R(1)^3)**for each of the 5 'shells'

giving ..

__M = (4/3) Pi x Rho x((R(2)^3 - R(1)^3) for each 'shell'.__I then summed each incremental mass to get a total mass !

However my arithmetic gave individual masses of the order a number between

1 and 10 x 10^15 and when summed gave a larger number of the same order

**(number x 10^15)**.. the book gives the total mass as

**5.97 x 10^24**!!

As you can see I am way out .. in the order of 10^9.

Similarly for part (c) .. M R^2 when taking the official figures for the earth is

__M= 5.97 x 10^24 and R=6.38 x 10^6 .. so M R^2 = 4.07 x 10^37kgm^2__I used the expression ...

**for each 'shell' and then summed them. he correct answer should show.. (**

__I = (8/15) x Pi x Rho ((R(2)^5) - (R(1)^5))__**my answer/(4.07 x 10^37)) = 0.400 approximately**, but my answer is of the order of ..

**number (between 1 and 9.99) x 10^23**.. which again is way way out.

Can anyone help show me where I am going wrong .. is it my method or is it my arithmetic ??

Regards,

Jackthehat.

shell