Find the moment of inertia of the system of p | vedclass

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

Question

(B) The moment of inertia $I$ of a system of discrete particles about an axis is given by the sum of the products of the mass of each particle and the

Vedclass · Accepted Answer

$\frac{m{l^2}}{6}n(n + 1)(2n + 1)$

Vedclass · Answer

(B) The moment of inertia $I$ of a system of discrete particles about an axis is given by the sum of the products of the mass of each particle and the square of its perpendicular distance from the axis: $I = \sum m_i r_i^2$.Here,we have $n$ particles each of mass $m$ located at distances $\ell, 2\ell, 3\ell, \dots, n\ell$ from the axis $OA$.Therefore,the total moment of inertia is:$I = m(\ell)^2 + m(2\ell)^2 + m(3\ell)^2 + \dots + m(n\ell)^2$$I = m\ell^2 (1^2 + 2^2 + 3^2 + \dots + n^2)$Using the standard formula for the sum of squares of the first $n$ natural numbers,$\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$,we get:$I = m\ell^2 \left[ \frac{n(n+1)(2n+1)}{6} \right]$$I = \frac{m\ell^2}{6} n(n+1)(2n+1)$

Find the moment of inertia of the system of particles about the axis $OA$ passing through point $O$.

Similar Questions

Two spheres of equal masses,one of which is a thin spherical shell and the other a solid sphere,have the same moment of inertia about their respective diameters. The ratio of their radii is

Three point masses,each of mass $m$,are placed at the corners of an equilateral triangle of side $a$. The moment of inertia of this system about an axis passing along one side of the triangle is:

If the length of a thin uniform rod is $L$ and the radius of gyration of the rod about an axis perpendicular to its length and passing through one end is $K$, then $K: L=$

The figure shows an isosceles triangular plate of mass $M$ and base $L$. The angle at the apex is $90^o$. The apex lies at the origin and the base is parallel to the $X$-axis. The moment of inertia of the plate about the $X$-axis is

$A$ solid sphere and a solid cylinder have the same mass and same radius. The ratio of the moment of inertia of the solid sphere about its diameter and the moment of inertia of the solid cylinder about its axis is

Vedclass Test Series

Exam Paper Generator

Online Exam Module