A small particle of mass $m$ is projected at an angle $\theta$ with the $\mathrm{x}$-axis with an initial…

A small particle of mass $m$ is projected at an angle $\theta$ with the $\mathrm{x}$-axis with an initial velocity $\mathrm{v}_0$ in the $\mathrm{x}-\mathrm{y}$ plane as shown in the figure. At a time $t < \frac{v_0 \sin \theta}{g}$, the angular momentum of the particle is where $\hat{\mathrm{i}}, \hat{\mathrm{j}}$ and $\hat{\mathrm{k}}$ are unit vectors along $\mathrm{x}, \mathrm{y}$ and $\mathrm{z}$-axis respectively.
  1. $-\operatorname{mgv}_0 \mathrm{t}^2 \cos \theta \hat{\mathrm{j}}$
  2. $\operatorname{mgv}_0 t \cos \theta \hat{\mathrm{k}}$
  3. $-\frac{1}{2} m g v_0 t^2 \cos \theta \hat{k}$
  4. $\frac{1}{2} m g v_0 t^2 \cos \theta \hat{i}$

Solution

$\begin{aligned} \overrightarrow{\mathrm{L}} & =m(\vec{r} \times \overrightarrow{\mathrm{v}}) \\ \overrightarrow{\mathrm{L}} & =m\left[\mathrm{v}_0 \cos \theta t \hat{\mathrm{i}}+\left(\mathrm{v}_0 \sin \theta \mathrm{t}-\frac{1}{2} g \mathrm{t}^2\right) \hat{\mathrm{j}}\right] \times\left[\mathrm{v}_0 \cos \theta \hat{\mathrm{i}}+\left(\mathrm{v}_0 \sin \theta-g \mathrm{~g}\right) \hat{\mathrm{j}}\right] \\ & =m \mathrm{v}_0 \cos \theta \mathrm{t}\left[-\frac{1}{2} g \mathrm{~g}\right] \hat{\mathrm{k}} \\ & =-\frac{1}{2} m g \mathrm{v}_0 \mathrm{t}^2 \cos \theta \hat{\mathrm{k}}\end{aligned}$

Asked in: JEE Main 2010

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