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]
- A
$A,C$
- B
$B,C$
- C
$A,D$
- D
$A,C,D$
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- [JEE MAIN 2024]