Find the moment of inertia about a diameter of s spherical shell of uniform density, D, bounded by two concentric spheres of radii a and b, where a<b. Express your answer in terms of a, b and the mass of the spherical shell.

So far I have found my equation for row (p), and have manipulated the mass to equal the triple integral of the density D, and used that to form and equation for inertia. However I am unsure if I am going in the right direction. Thanks!

Sounds good so far. You can check your answer at

http://en.wikipedia.org/wiki/List_of_moments_of_inertia

More detailed drawings are found at

http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html

and the rationale of adjusting the moment after removing the inside sphere is discussed at

http://scienceworld.wolfram.com/physics/MomentofInertiaSphericalShell.html

To find the moment of inertia about a diameter of a spherical shell, you can use the formula:

I = (2/3) * m * R^2

where I is the moment of inertia, m is the mass of the shell, and R is the radius of the shell.

In this case, the spherical shell is bounded by two concentric spheres of radii a and b. The mass of the shell can be found by subtracting the mass of the inner sphere from the mass of the outer sphere. The mass of a spherical shell is given by:

m = D * V

where D is the density of the shell and V is the volume.

The volume of the shell can be found by subtracting the volume of the inner sphere from the volume of the outer sphere. The volume of a sphere is given by:

V = (4/3) * π * r^3

where r is the radius of the sphere.

First, let's find the mass of the outer shell:

m_outer = D * V_outer
= D * ((4/3) * π * b^3)

Next, let's find the mass of the inner shell:

m_inner = D * V_inner
= D * ((4/3) * π * a^3)

Now, we can find the mass of the shell:

m = m_outer - m_inner
= D * ((4/3) * π * b^3) - D * ((4/3) * π * a^3)
= D * (4/3) * π * (b^3 - a^3)

Finally, substitute the mass of the shell in the formula for moment of inertia:

I = (2/3) * m * R^2
= (2/3) * D * (4/3) * π * (b^3 - a^3) * R^2

Therefore, the moment of inertia about a diameter of a spherical shell of uniform density, bounded by two concentric spheres of radii a and b, is given by:

I = (8/9) * D * π * (b^3 - a^3) * R^2

where R is the radius of the shell.