Question #8a776

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Cosmic Defect
Nov 29, 2017

Definition: The radius-of-gyration is defined such that, for any geometry, the moment-of-inertia (#I#) is simply its mass (#M#) times the square of the radius-of-gyration (#K#).

#I \equiv M.K^2#

#K = \sqrt{I/M}#

[1] Circular Disc: For a circular disc of mass #M# and radius #R#,

[a] Its moment-of-inertia for rotation about an axis passing through its centre and perpendicular to its plane is -

#I_{z} = 1/2 MR^2 = M (R^2/2); \qquad K_z^2 = R^2/2 \rightarrow K_z = R/\sqrt{2}#

[b] Its moment-of-inertia about an axis passing through its centre and parallel to its plane is -
#I_{x} = 1/4 MR^2 = M (R^2/4); \qquad K_x^2 = R^2/4 \rightarrow K_x = R/2#

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