Expected Moment (Part 2)

Classical Mechanics Level 4

A uniform circular disk of mass M M and radius R R is rotated around a random perpendicular axis passing through its body (normal to the surface). If the expected moment of inertia is α M R 2 \alpha M R^2 , what is the value of α \alpha ?

Note: The probability distribution is uniform as a function of area.


The answer is 1.000.

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Steven Chase by Steven Chase , 37, USA

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2 solutions

Mark Hennings
Nov 11, 2017

Relevant wiki: Moment of Inertia

The moment of inertia about an axis normal to the disk and a distance r r from the centre of the disk is, by the Parallel Axes Theorem , equal to 1 2 M R 2 + M r 2 = 1 2 M ( R 2 + 2 r 2 ) \tfrac12MR^2 + Mr^2 \; = \; \tfrac12M(R^2 + 2r^2) and so the expected moment of inertia is 1 π R 2 ∫ 0 R 1 2 M ( R 2 + 2 r 2 ) 2 π r d r = M R − 2 [ 1 2 R 2 r 2 + 1 2 r 4 ] 0 R = M R 2 \frac{1}{\pi R^2} \int_0^R \tfrac12M(R^2 + 2r^2) 2\pi r\, dr \; = \; MR^{-2}\Big[\tfrac12R^2r^2 + \tfrac12r^4\Big]_0^R \; = \; MR^2 making the answer 1 \boxed{1} .

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Nj Star
Feb 13, 2020

1 Helpful 1 Interesting 1 Brilliant 0 Confused

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