The moment of inertia of a semicircular ring | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Consider a small element of the semicircular ring of angular width $d	heta$ at an angle $	heta$ from the horizontal.The mass of this small eleme

Vedclass · Accepted Answer

$MR^2$

Vedclass · Answer

(A) Consider a small element of the semicircular ring of angular width $d	heta$ at an angle $	heta$ from the horizontal.The mass of this small element is $dm = \frac{M}{\pi R} \cdot R d	heta = \frac{M}{\pi} d	heta$.The moment of inertia of this small element about the centre is $dI = dm \cdot R^2$.To find the total moment of inertia $I$,we integrate $dI$ from $	heta = 0$ to $	heta = \pi$:$I = \int_{0}^{\pi} \frac{M}{\pi} d	heta \cdot R^2 = \frac{MR^2}{\pi} \int_{0}^{\pi} d	heta$.$I = \frac{MR^2}{\pi} [	heta]_{0}^{\pi} = \frac{MR^2}{\pi} (\pi - 0) = MR^2$.Thus,the moment of inertia is $MR^2$.

The moment of inertia of a semicircular ring of mass $M$ and radius $R$ about its centre is:

Similar Questions

The moment of inertia of a semicircular ring of mass $M$ and radius $R$ about an axis passing through its center and perpendicular to its plane is:

$A$ thin wire of length $l$ and mass $M$ is bent in the form of a semi-circle. What is its moment of inertia about an axis passing through the ends of the wire?

The ratio of radii of gyration of a circular ring and a circular disc of the same mass and radius,about an axis passing through their centres and perpendicular to their planes is

An iron circular disc $X$ has radius $R$ and thickness $t$. Another iron circular disc $Y$ has radius $4R$ and thickness $t/4$. The relationship between the moments of inertia $I_X$ and $I_Y$ of these two discs is .......

$A$ disc $X$ of radius $R$ and thickness $t$ is made of an iron plate. Another disc $Y$ of radius $4R$ and thickness $t/4$ is also made of the same iron plate. What is the relationship between their moments of inertia $I_x$ and $I_y$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module