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} \quad\left(\because\frac{d\vec{r}}{dt} \times\vec{p} = 0\right) $
$\frac{d\vec{L}}{dt}-\vec{r} \times\frac{d\vec{p}}{dt} = 0$