Moment of inertia of a solid sphere

[henry3369]

Homework Statement

Find the moment of inertia of a solid sphere of uniform mass density (like a billiard ball) about an axis through its center

Homework Equations

I = ∫rρdV

The Attempt at a Solution

I =ρ ∫r4πr²dr = ρ4π∫r⁴
Then I integrate this from 0 (the center) to R, so I = (ρ4π)*(R⁵/5)
And ρ = mv so ρ = M/(4/3)πR³ = 3M/4πR³. Put ρ into the equation for moment of inertia to get I = 3MR²/5.

My book tells me the answer is (2/5)MR^2. Where did I go wrong?

 

[PeroK]

Homework Statement

Find the moment of inertia of a solid sphere of uniform mass density (like a billiard ball) about an axis through its center

Homework Equations

I = ∫rρdV

The Attempt at a Solution

I =ρ ∫r4πr²dr = ρ4π∫r⁴
Then I integrate this from 0 (the center) to R, so I = (ρ4π)*(R⁵/5)
And ρ = mv so ρ = M/(4/3)πR³ = 3M/4πR³. Put ρ into the equation for moment of inertia to get I = 3MR²/5.

My book tells me the answer is (2/5)MR^2. Where did I go wrong?

Can you explain how you are setting up the integral? What shape are you integrating from 0 to R?

 

[henry3369]

Can you explain how you are setting up the integral? What shape are you integrating from 0 to R?

The surface area of a spherical shell from the center of the sphere to the outer shell of radius R.

 

[henry3369]

I figured it out. If I use spherical shells, the points from the shell aren't all the same distance form the axis of rotation. I was thinking about the distance from the center.

 

[PeroK]

I figured it out. If I use spherical shells, the points from the shell aren't all the same distance form the axis of rotation. I was thinking about the distance from the center.

Exactly!