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Alexus [3.1K]
4 years ago
8

A lamina with constant density rho(x, y) = rho occupies the given region. Find the moments of inertia Ix and Iy and the radii of

gyration x double bar and y double bar. The part of the disk x2 + y2 ≤ a2 in the first quadrant
Mathematics
1 answer:
jenyasd209 [6]4 years ago
3 0

Answer:

Ix = Iy = \frac{ρπR^{4} }{16}

Radius of gyration x = y =  \frac{R}{4}

Step-by-step explanation:

Given: A lamina with constant density ρ(x, y) = ρ occupies the given region x2 + y2 ≤ a2 in the first quadrant.

Mass of disk = ρπR2

Moment of inertia about its perpendicular axis is \frac{MR^{2} }{2}. Moment of inertia of quarter disk about its perpendicular is \frac{MR^{2} }{8}.

Now using perpendicular axis theorem, Ix = Iy = \frac{MR^{2} }{16} = \frac{ρπR^{4} }{16}.

For Radius of gyration K, equate MK2 = MR2/16, K= R/4.

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