Question

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

Answer 1

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.