Moment of inertia of solid sphere

[quietrain]

Hi, i am trying to find the moment of inertia of a uniform density solid sphere about z-axis

I = integrate => x^2 dm

x = perpendicular distance from z-axis to anywhere in sphere
so by pythagorus theorem, r^2 - z^2 = x^2

since dm = p dV
and V = 4/3 (//pi)r^3
dV = 4(//pi)r^2 dr

so I = integrate=> r^2 - z^2 pdV

but the problem is z is a variable.

so how do i convert z?

assuming i put z = rcos θ,

then i will have a θ variable now.

i tried integrating θ from 0 to (pi) but the answer is wrong, its not 2/5mr^2

so what should i do?

thanks

 

[Vanadium 50]

Look at your volume element.

 

[DocZaius]

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

This is a great resource for figuring your way through the integration.

 

[quietrain]

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

This is a great resource for figuring your way through the integration.

i have seened that, but they considered slices of circular disk

and so they sum up the moment of inertia of each disk for the whole sphere

but i am sure the method that i am doing can work also.. just that i don't know how

 

[quietrain]

Look at your volume element.

whats wrong with the volume element?

 

[DocZaius]

If you are not summing slices, what are you summing in your integration? I suppose maybe you could take ever-widening, and ever-shortening cylinders centered about an axis. The first piece would be a line, and the last would be a circle. Is that what you are doing?

 

[quietrain]

i am trying to sum all the element volume

its like this thread https://www.physicsforums.com/showthread.php?t=159293

but now i understand i am missing out the rsin(theta) term and the r term for the integration