Consider a particle of mass $m$ having linear momentum $\vec p$ at position $\vec r$ relative to the origin $O$ . Let $\vec L$ be the angular momentum of the particle with respect the origin. Which of the following equations correctly relate $(s)\, \vec r,\,\vec p$ and $\vec L$ ?
Consider a particle of mass $m$ having linear momentum $\vec p$ at position $\vec r$ relative to the origin $O$ . Let $\vec L$ be the angular momentum of the particle with respect the origin. Which of the following equations correctly relate $(s)\, \vec r,\,\vec p$ and $\vec L$ ?
- A
$\frac{{d\vec L}}{{dt}} + \vec r\, \times \frac{{\overrightarrow {dp} }}{{dt}} = 0$
- B
$\frac{{d\vec L}}{{dt}}\, + \,\frac{{\overrightarrow {dr} }}{{dt}} \times \vec p = 0$
- C
$\frac{{d\vec L}}{{dt}}\, - \,\frac{{\overrightarrow {dr} }}{{dt}} \times \vec p = 0$
- D
$\frac{{d\vec L}}{{dt}}\, - \vec r \times \frac{{\overrightarrow {dp} }}{{dt}} = 0$
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- [IIT 2021]
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- [AIPMT 1991]
In the List-$I$ below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \neq \beta$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{ p }$ is the linear momentum, $\bar{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-$I$ with those quantities in List-$II$, which are conserved for that path.
List-$I$
List-$II$
$P$ $\dot{r}(t)=\alpha t \hat{t}+\beta t \hat{j}$
$1$ $\overrightarrow{ p }$
$Q$ $\dot{r}(t)=\alpha \cos \omega t \hat{i}+\beta \sin \omega t \hat{j}$
$2$ $\overrightarrow{ L }$
$R$ $\dot{r}(t)=\alpha(\cos \omega t \hat{i}+\sin \omega t \hat{j})$
$3$ $K$
$S$ $\dot{r}(t)=\alpha t \hat{i}+\frac{\beta}{2} t^2 \hat{j}$
$4$ $U$
$5$ $E$
In the List-$I$ below, four different paths of a particle are given as functions of time. In these functions, $\alpha$ and $\beta$ are positive constants of appropriate dimensions and $\alpha \neq \beta$. In each case, the force acting on the particle is either zero or conservative. In List-II, five physical quantities of the particle are mentioned: $\overrightarrow{ p }$ is the linear momentum, $\bar{L}$ is the angular momentum about the origin, $K$ is the kinetic energy, $U$ is the potential energy and $E$ is the total energy. Match each path in List-$I$ with those quantities in List-$II$, which are conserved for that path.
| List-$I$ | List-$II$ |
| $P$ $\dot{r}(t)=\alpha t \hat{t}+\beta t \hat{j}$ | $1$ $\overrightarrow{ p }$ |
| $Q$ $\dot{r}(t)=\alpha \cos \omega t \hat{i}+\beta \sin \omega t \hat{j}$ | $2$ $\overrightarrow{ L }$ |
| $R$ $\dot{r}(t)=\alpha(\cos \omega t \hat{i}+\sin \omega t \hat{j})$ | $3$ $K$ |
| $S$ $\dot{r}(t)=\alpha t \hat{i}+\frac{\beta}{2} t^2 \hat{j}$ | $4$ $U$ |
| $5$ $E$ |
- [IIT 2018]