Question

Two thin circular discs of mass $m$ and $4m$, having radii of $a$ and $2a$, 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 $55m{a}^2\omega$
• B. The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81m{a}^2\omega$
• 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 $17m{a}^2\omega /2$

Answer 1

Answer: (C), (D)

Top view

${\omega }_1=\frac{a\omega }{\ell }\Rightarrow v_{cm}={\omega }_1\left(\ell +\frac{4\ell }{5}\right)$
$v_{cm}=\frac{a\omega }{\ell }\left(\frac{9\ell }{5}\right)$
$v_{cm}=\frac{9a\omega }{5}$
angular momentum of COM about point of
$={\vec{r}}_{cm}\times \left(m_T{\vec{v}}_{cm}\right)$
$=r_{cm}{m}_T{v}_{cm}$
$=\frac{9\ell }{5}\times (5m)\left(\frac{9a\omega }{5}\right)$
$=\frac{81}{5}a\ell m\omega$
$=\frac{81}{5}\times a\sqrt{24}a\times m\omega$
$=\frac{81}{5}\times \sqrt{24}{a}^{2}m\omega$
Angular velocity of COM about $z$ axis
${\omega }_1=\frac{a\omega }{\ell }=\frac{a\omega }{\sqrt{24}a}=\frac{\omega }{\sqrt{24}}$
${\omega }_z={\omega }_1\cos \theta$
${\omega }_z=\frac{\omega }{\sqrt{24}}\times \left(\frac{\ell }{\sqrt{{\ell }^2+a^2}}\right)$
$=\frac{\omega \ell }{\sqrt{24}\times \left(\sqrt{25a^2}\right)}$
$\Rightarrow \frac{\omega \sqrt{24}a}{\sqrt{24}\cdot 5a}=\frac{\omega }{5}$
Angular momentum about. $COM=I_{cm}\omega$
$=\left(\frac{m{a}^2}{2}+\frac{4m(2a{)}^2}{2}\right)\omega$
$\Rightarrow \left(\frac{m{a}^2}{2}+8m{a}^2\right)\omega$
$L_{wrtcm}=\frac{17m{a}^2}{2}\omega$
angular momentum about $O$ has component along $z$-axis
$=L_{cm}\cos \theta -L_{wrtcm}\sin \theta$
$=\frac{81}{5}\sqrt{24}m\omega a^2\cos \theta -\frac{17m{a}^2}{2}\omega \sin \theta$
$=\frac{81}{5}\sqrt{24}\left(\frac{\ell }{\sqrt{{\ell }^2+a^2}}\right)m\omega a^2-\frac{17m{a}^2}{2}\omega \left[\frac{a}{\sqrt{{\ell }^2+a^2}}\right]$
$\Rightarrow \frac{81\times 24}{25}m\omega a^2-\frac{17}{10}m{a}^2\omega$
$\Rightarrow \left(\frac{81\times 24\times 2-17\times 5}{50}\right)m\omega a^2$