Question

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 to the origin. Which of the following equations correctly relate(s) $\vec{r},\vec{P}$ and $\vec{L}$ ?

• A. $\frac{d\vec{L}}{dt}+\vec{r}\times \frac{d\vec{p}}{dt}=0$
• B. $\frac{d\vec{L}}{dt}+\frac{d\vec{r}}{dt}\times \vec{p}=0$
• C. $\frac{d\vec{L}}{dt}-\frac{d\vec{r}}{dt}\times \vec{p}=0$
• D. $\frac{d\vec{L}}{dt}-\vec{r}\times \frac{d\vec{p}}{dt}=0$

Answer 1

Answer: (D)

As $\vec{L}=\vec{r}\times \vec{p}$
Differentiate both sides with respect to time, we get
$\frac{d\vec{L}}{dt}=\frac{d}{dt}\left(\vec{r}\times \vec{p}\right)$
$=\frac{d\vec{r}}{dt}\times \vec{p}+\vec{r}\times \frac{d\vec{p}}{dt}$
$=\vec{r}\times \frac{d\vec{p}}{dt}\left(\because \frac{d\vec{r}}{dt}\times \vec{p}=0\right)$
$\frac{d\vec{L}}{dt}-\vec{r}\times \frac{d\vec{p}}{dt}=0$