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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
$ - mg\,{v_0}{t^2}\,\cos \,\theta\, \hat j$
$ mg\,{v_0}t\,\cos \,\theta\, \hat k$
$ - \frac{1}{2}\,mg\,{v_0}{t^2}\,\cos \,\theta \,\hat k$
$\frac{1}{2}\,mg\,{v_0}{t^2}\,\cos \,\theta \,\hat i$
Solution
$\overrightarrow L = m\left( {\overrightarrow r \times \overrightarrow v } \right)$
$\overrightarrow L = m\left[ {{v_0}\cos \theta t\hat i + ({v_0}\sin \theta t – \frac{1}{2}g{t^2})\hat j} \right]$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\, \times \left[ {{v_0}\cos \theta \hat i + \left( {{v_0}\sin \theta – gt} \right)\hat j} \right]$
$ = m{v_0}\cos \theta t\left[ { – \frac{1}{2}gt} \right]\hat k$
$ = – \frac{1}{2}mg{v_0}{t^2}\cos \theta \hat k$
