How Do You Calculate the Moment of Inertia for a Cone?

[jforce93]

Homework Statement

Find the moment of inertia of a right circular cone of radius r and height h and mass m

Homework Equations

I = ∫r² dm
V = 1/3*π*r²*h

The Attempt at a Solution

Assume density is p

dm = p dv
divide both sides by dr
dm/dr = p dv/dr

dm/dr = p (d/dr * 1/3*π * r²*h)
so
dm/dr = p(2/3)*πrh
so:
dm = (2/3)pπrh dr

Sub that into the moment of inertia equation

∫(2/3)pπrh dr = I

I = (1/3) pπr²h
p = m/v
I = (1/3)(m/v)πr²h
I = v*(m/v)
I = m

What am I doing wrong?

 

[LawrenceC]

Hint: Set the problem up as a double integral problem. Lay the problem out with the cone's longitudinal axis being the x-axis. Use a ring for your dV.

If you need more hints, let me know.

 

[vela]

You forgot the factor of r² multiplying dm. What you found was \int dm, which unsurprisingly turns out to equal the mass of the cone.

About what axis are you supposed to be calculating the moment of inertia?