Question

Two thin circular discs of mass $m$ and $4\, m$, having radii of $a$ and $2 a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24}$ a through their centers. This assembly is laid on a firm and flat surface and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point 'O' is $\vec{ L }$ (see the figure). Which of the following statement(s) is(are) true?

• A. The magnitude of the z-component of $\vec{L}$ is $55\, ma ^{2} \omega$
• B. The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81\, ma ^{2} \omega$
• C. The center of mass the assembly rotates about the z-axis with an angular speed of $\omega / 5$
• D. The magnitude of angular momentum of the assembly about its center of mass is $17\, ma ^{2} \omega / 2$

Answer 1

The Correct Option is D

Top view

$\omega_{1}=\frac{ a \omega}{\ell} \Rightarrow v _{ cm }=\omega_{1}\left(\ell+\frac{4 \ell}{5}\right)$
$v _{ cm }=\frac{ a \omega}{\ell}\left(\frac{9 \ell}{5}\right)$
$v _{ cm }=\frac{9 a \omega}{5}$
angular momentum of COM about point of
$=\vec{r}_{c m} \times\left(m_{T} \vec{v}_{c m}\right)$
$= r _{ cm } m _{ T } v _{ cm}$
$=\frac{9 \ell}{5} \times(5 m )\left(\frac{9 a \omega}{5}\right)$
$=\frac{81}{5} a\ell m \omega$
$=\frac{81}{5} \times a \sqrt{24} a \times m \omega$
$=\frac{81}{5} \times \sqrt{24} a ^{2} m \omega$
Angular velocity of COM about $z$ axis
$\omega_{1}=\frac{ a \omega}{\ell}=\frac{ a \omega}{\sqrt{24} a }=\frac{\omega}{\sqrt{24}}$
$\omega_{z}=\omega_{1} \cos \theta$
$\omega_{z}=\frac{\omega}{\sqrt{24}} \times\left(\frac{\ell}{\sqrt{\ell^{2}+ a ^{2}}}\right)$
$=\frac{\omega \ell}{\sqrt{24} \times\left(\sqrt{25 a ^{2}}\right)}$
$\Rightarrow \frac{\omega \sqrt{24} a }{\sqrt{24} \cdot 5 a }=\frac{\omega}{5}$
Angular momentum about. $COM = I _{ cm } \omega$
$=\left(\frac{m a^{2}}{2}+\frac{4 m(2 a)^{2}}{2}\right) \omega$
$\Rightarrow\left(\frac{m a^{2}}{2}+8 m a^{2}\right) \omega$
$L _{ wrt\,cm }=\frac{17\, ma ^{2}}{2} \omega$
angular momentum about $O$ has component along $z$-axis
$=L_{cm} \cos \theta-L_{wrt\, cm} \sin \theta$
$=\frac{81}{5} \sqrt{24} m \omega a^{2} \cos \theta-\frac{17 ma ^{2}}{2} \omega \sin \theta$
$=\frac{81}{5} \sqrt{24}\left(\frac{\ell}{\sqrt{\ell^{2}+a^{2}}}\right) m \omega a^{2}-\frac{17 m a^{2}}{2} \omega\left[\frac{a}{\sqrt{\ell^{2}+a^{2}}}\right]$
$\Rightarrow \frac{81 \times 24}{25} m \omega a^{2}-\frac{17}{10} m a^{2} \omega$
$\Rightarrow\left(\frac{81 \times 24 \times 2-17 \times 5}{50}\right) m \omega a^{2}$