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A particle of mass m is projected from origin O with speed u at an angle $θ$ with positive x-axis. Positive y-axis is in vertically upward direction. Find the angular momentum of particle at any time t about O before the particle strikes the ground again.

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(3) $- \frac{1}{2} m (ucosθ) g t^{2} \hat{k}$

(1) $- \frac{5}{2} m (usinθ) g t^{2} \hat{k}$

(2) $\frac{1}{2} m (usinθ) g t^{2} \hat{k}$

(4) $\frac{3}{2} m (ucosθ) g t^{2} \hat{k}$

answer is C.

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Detailed Solution

$L = m (r \times v)$

Here, $\vec{r} = x \hat{i} + y \hat{j} = (ucosθ) r \hat{i} + (utsinθ - \frac{1}{2} {gt}^{2}) \hat{j}$

and $\vec{v} = v_{x} \hat{i} + v_{y} \hat{j} = (ucosθ) \hat{i} + (usinθ - gt) \hat{j}$

$L = m (\vec{r} \times \vec{v})$

$= - \frac{1}{2} m (ucosθ) g t^{2} \hat{k}$

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