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The moment of inertia about the y-axis of the thin semicircular region of constant density is given below.
[tex]\rm I_y = \dfrac{1}{8} \times \pi r^4[/tex]
What is rotational inertia?
Any item that can be turned has rotational inertia as a quality. It's a scalar value that indicates how complex it is to adjust an object's rotational velocity around a certain axis.
Then the moment of inertia about the y-axis of the thin semicircular region of constant density will be
[tex]\rm I_x = \int y^2 dA\\\\I_y = \int x^2 dA[/tex]
x = r cos θ
y = r sin θ
dA = r dr dθ
Then the moment of inertia about the x-axis will be
[tex]\rm I_x = \int _0^r \int _0^{\pi} (r\sin \theta )^2 \ r \ dr \ d\theta\\\\\rm I_x = \int _0^r \int _0^{\pi} r^3 \sin ^2\theta \ dr \ d\theta[/tex]
On integration, we have
[tex]\rm I_x = \dfrac{1}{8} \times \pi r^4[/tex]
Then the moment of inertia about the y-axis will be
[tex]\rm I_y = \int _0^r \int _0^{\pi} (r\cos\theta )^2 \ r \ dr \ d\theta\\\\\rm I_y = \int _0^r \int _0^{\pi} r^3 \cos ^2\theta \ dr \ d\theta[/tex]
On integration, we have
[tex]\rm I_y = \dfrac{1}{8} \times \pi r^4[/tex]
Then the moment of inertia about O will be
[tex]\rm I_o = I_x + I_y\\\\I_o = \dfrac{1}{8} \times \pi r^4 + \dfrac{1}{8} \times \pi r^4\\\\I_o = \dfrac{1}{4} \times \pi r^4[/tex]
More about the rotational inertia link is given below.
https://brainly.com/question/22513079
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