Moment of Inertia of Hollow Sphere

  • #1

Homework Statement



Calculate the moment of inertia of a spherical shell (i.e. hollow sphere) of uniform surface density about an axis passing through its center.

Homework Equations





The Attempt at a Solution



Integral ( r^2 * dm)
Integral (r^2 * p * dV) ..... where p=density
p * Integral (r^2 * dV)

where V= (4/3)* pi * r^3
and dV= 4 * pi* r^2
therefore,

p * Integral (r^2 *4 * pi* r^2)
4*p*pi* Integral (r^4)
4*p*pi* ((r^5)/5) from 0 to r

p=density=m/V= m/(4/3 * pi * r^3)

4*m/(4/3 * pi * r^3)*pi* ((r^5)/5

(3*m*r^2)/5

but it should be (2*m*r^2)/3.
Please help!
 

Answers and Replies

  • #2
D H
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Integral ( r^2 * dm)
...
and dV= 4 * pi* r^2
The r in the first expression represents the distance from the axis of rotation to the point in question while the r in the second expression represents the distance from the center of the sphere to the point in question. Those are two different things.
 
  • #3
vela
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A hollow sphere is a two-dimensional surface, so you need to use [itex]dm = \sigma dA[/itex], where [itex]\sigma[/itex] is the surface density.
 

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