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-
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.
$$ - mg\,{v_0}{t^2}\cos \theta \hat j$$
B.
$$mg\,{v_0}{t}\cos \theta \,\hat k$$
C.
$$ - \frac{1}{2}mg\,{v_0}{t^2}\cos \theta \,\hat k$$
D.
$$\frac{1}{2}mg\,{v_0}{t^2}\cos \theta \,\hat i$$
Answer :
$$ - \frac{1}{2}mg\,{v_0}{t^2}\cos \theta \,\hat k$$
Solution :
$$\eqalign{ & \vec L = m\left( {\vec r \times \vec v} \right) \cr & \vec L = m\left[ {{v_0}\,\cos \theta \,t\,\hat i + \left( {{v_0}\sin \theta \,t - \frac{1}{2}g{t^2}\,} \right)\hat j} \right] \times \left[ {{v_0}\,\cos \theta \,\hat i + \left( {{v_0}\sin \theta - gt\,} \right)\hat j} \right] \cr & \vec L = m{v_0}\,\cos \theta \,t\left[ { - \frac{1}{2}gt} \right]\hat k \cr & \vec L = - \frac{1}{2}\,mg\,{v_0}\,{t^2}\cos \theta \,\hat k \cr} $$
$$\eqalign{ & \vec L = m\left( {\vec r \times \vec v} \right) \cr & \vec L = m\left[ {{v_0}\,\cos \theta \,t\,\hat i + \left( {{v_0}\sin \theta \,t - \frac{1}{2}g{t^2}\,} \right)\hat j} \right] \times \left[ {{v_0}\,\cos \theta \,\hat i + \left( {{v_0}\sin \theta - gt\,} \right)\hat j} \right] \cr & \vec L = m{v_0}\,\cos \theta \,t\left[ { - \frac{1}{2}gt} \right]\hat k \cr & \vec L = - \frac{1}{2}\,mg\,{v_0}\,{t^2}\cos \theta \,\hat k \cr} $$
