First slide

Moment of intertia

Question

A uniform rod of mass m and length l_o is rotating with a constant angular speed $ω$ about a vertical axis passing through its point of suspension. The moment of inertia of the rod about the axis of rotation if it makes an angle $θ$ to the vertical (axis of rotation) is

Moderate

A

$\frac{{ml}_{0}^{2} \sin^{2} θ}{12}$
B

$\frac{{ml}_{0}^{2} \sin^{2} θ}{6}$
C

$\frac{2 {ml}_{0}^{2} \sin^{2} θ}{3}$
D

$\frac{{ml}_{0}^{2} \sin^{2} θ}{3}$

Solution

We can observe each and every element of rod in rotating with different radius about the axis of rotation
Take an elementary mass dm of the rod.
$dm = \frac{m}{l_{0}} dl$
The moment of inertia of the elementary mass is given as $dl = (dm) r^{2}$
The moment of interita of the rod
= $I = \int dI \Rightarrow I = \int r^{2} dm$
Substituting $r = 1 \sin θ and dm = \frac{m}{l_{0}} . dl, we obtain$
$I = \int (l^{2} \sin^{2} θ) \frac{m}{l_{0}} dI$
= $\frac{{msin}^{2} θ}{l_{0}} \int_{0}^{l_{0}} l^{2} dl = \frac{m l_{0}^{2}}{3 l_{0}} \sin^{3} θ$
$\Rightarrow I = \frac{{ml}_{0}^{2} \sin^{2} θ}{3}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App