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?
- The magnitude of the z-component of $\vec{L}$ is $55\, ma ^{2} \omega$
- The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81\, ma ^{2} \omega$
- The center of mass the assembly rotates about the z-axis with an angular speed of $\omega / 5$
- The magnitude of angular momentum of the assembly about its center of mass is $17\, ma ^{2} \omega / 2$
The Correct Option is D
Solution and Explanation
$\omega_{1}=\frac{ a \omega}{\ell} \Rightarrow v _{ cm }=\omega_{1}\left(\ell+\frac{4 \ell}{5}\right)$
$v _{ cm }=\frac{ a \omega}{\ell}\left(\frac{9 \ell}{5}\right)$
$v _{ cm }=\frac{9 a \omega}{5}$
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
$=L_{cm} \cos \theta-L_{wrt\, cm} \sin \theta$
$=\frac{81}{5} \sqrt{24} m \omega a^{2} \cos \theta-\frac{17 ma ^{2}}{2} \omega \sin \theta$
$=\frac{81}{5} \sqrt{24}\left(\frac{\ell}{\sqrt{\ell^{2}+a^{2}}}\right) m \omega a^{2}-\frac{17 m a^{2}}{2} \omega\left[\frac{a}{\sqrt{\ell^{2}+a^{2}}}\right]$
$\Rightarrow \frac{81 \times 24}{25} m \omega a^{2}-\frac{17}{10} m a^{2} \omega$
$\Rightarrow\left(\frac{81 \times 24 \times 2-17 \times 5}{50}\right) m \omega a^{2}$
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Concepts Used:
Torque
Torque is a moment of force. Torque is measured as a force that causeque is also defined as the turning effect of force on the axis of rotation. Torque is chs an object to rotate about an axis and is responsible for the angular acceleration. Characterized with “T”.
How is Torque Calculated?
Torque is calculated as the magnitude of the torque vector T for a torque produced by a given force F
T = F. Sin (θ)
Where,
r - length of the moment arm,
θ - the angle between the force vector and the moment arm.
Read More: Torque
Types of Torque
Torque is of two types:
- Static torque
- Dynamic torque