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 center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega / 5$
($B$) The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81 m a^2 \omega$
($C$) The magnitude of angular momentum of the assembly about its center of mass is $17 \mathrm{ma}^2 \mathrm{\omega} / 2$
($D$) The magnitude of the $z$-component of $\vec{L}$ is $55 \mathrm{ma}^2 \omega$
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 center of mass of the assembly rotates about the $z$-axis with an angular speed of $\omega / 5$
($B$) The magnitude of angular momentum of center of mass of the assembly about the point $O$ is $81 m a^2 \omega$
($C$) The magnitude of angular momentum of the assembly about its center of mass is $17 \mathrm{ma}^2 \mathrm{\omega} / 2$
($D$) The magnitude of the $z$-component of $\vec{L}$ is $55 \mathrm{ma}^2 \omega$
- [IIT 2016]
- A
$A,C$
- B
$A,B$
- C
$A,D$
- D
$D,C$
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The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $\mathrm{V}(\mathrm{r})=\mathrm{kr}^2 / 2$, where $\mathrm{k}$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $\mathrm{R}$ about the point $\mathrm{O}$. If $\mathrm{v}$ is the speed of the particle and $\mathrm{L}$ is the magnitude of its angular momentum about $\mathrm{O}$, which of the following statements is (are) true?
$(A)$ $v=\sqrt{\frac{k}{2 m}} R$
$(B)$ $v=\sqrt{\frac{k}{m}} R$
$(C)$ $\mathrm{L}=\sqrt{\mathrm{mk}} \mathrm{R}^2$
$(D)$ $\mathrm{L}=\sqrt{\frac{\mathrm{mk}}{2}} \mathrm{R}^2$
The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $\mathrm{V}(\mathrm{r})=\mathrm{kr}^2 / 2$, where $\mathrm{k}$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $\mathrm{R}$ about the point $\mathrm{O}$. If $\mathrm{v}$ is the speed of the particle and $\mathrm{L}$ is the magnitude of its angular momentum about $\mathrm{O}$, which of the following statements is (are) true?
$(A)$ $v=\sqrt{\frac{k}{2 m}} R$
$(B)$ $v=\sqrt{\frac{k}{m}} R$
$(C)$ $\mathrm{L}=\sqrt{\mathrm{mk}} \mathrm{R}^2$
$(D)$ $\mathrm{L}=\sqrt{\frac{\mathrm{mk}}{2}} \mathrm{R}^2$
- [IIT 2018]