Question

a. State and prove perpendicular axis theorem.b. If a thin circular ring and thin flat circular disk of same mass have same moment of inertia about their respective diameters as axes, then find the ratio of the radii.

Answer 1

(a) The theorem of perpendicular axes states that the moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.
A physical body with center O and a point mass m, in the x–y plane at (x, y) is shown in figure.

Moment of inertia about x-axis, Ix = mx²
Moment of inertia about y-axis, Iy = my²
Moment of inertia about z-axis, Iz = mz²
Ix + Iy = mx² + my²
⇒ Ix + Iy = m(x² + y²)
We can write the following equation as,
⇒ Ix + Iy = m{$\sqrt{(x^2+y^2)}$}²
⇒ Ix + Iy= Iz
Hence, the theorem is proved.


(b) moment of inertia of ring about an axis passing through diameter of it is given by,
I = mr²/2

moment of inertia of circular disk about an axis passing through diameter of it is given by, I = mR²/4.

A/c to question, mr²/2 = mR²/4

or, r² = R²/2

Taking square root both sides,

r = R/√2

so, ratio of radii of ring to disk = 1 : √2

Answer 2

Answer:here you go

Explanation:

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