Question

A small particle of mass $m$ is projected at an angle $\theta$ with the $x$-axis with an initial velocity $v_0$ in the $x-y$ plane as shown in the figure. At a time $t < \frac{v_{0} \,sin\,\theta}{g}, $ the angular momentum of the particle is
where $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along $x, y$ and $z-axis$ respectively.

• A. $-mgv_{0} t^{2} \,cos\,\theta\,\hat{j}$
• B. $-mgv_{0} t \,cos\,\theta\,\hat{k}$
• C. $-\frac{1}{2}mgv_{0} t^{2} \,cos\,\theta\,\hat{k}$
• D. $\frac{1}{2}mgv_{0} t^{2} \,cos\,\theta\,\hat{i}$

Answer 1

Answer: (C)

$\vec{L} = m\left(\vec{r}\times\vec{v}\right)$
$\vec{L} = m\left[v_{0}\,cos\,\theta t\,\hat{i} +\left(v_{0}\,sin\,\theta t-\frac{1}{2}gt^{2}\right)\hat{j}\right]\times\left[v_{0}\,cos\,\theta \hat{i}+\left(v_{0}\,sin\,\theta-gt\right)\hat{j}\right]$
$= mv_{0}\,cos\,\theta t\left[-\frac{1}{2}gt\right]\hat{k}$
$-\frac{1}{2}mgv_{0} t^{2} \,cos\,\theta\,\hat{k}$