First slide

General Planar motion

Question

The angular momentum of the disc which spins with $\vec{ω} = \frac{3 v}{R} \hat{k}$ and its CM moves with a velocity $\vec{v} = v \hat{i}$ about O will be

Difficult

A

$\frac{3 m v R}{2} \hat{k}$
B

$\frac{m v R}{2} \hat{k}$
C

$- \frac{3 m v R}{2} \hat{k}$
D

$- \frac{m v R}{2} \hat{k}$

Solution

The angular momentum of the disc about O is
$\begin{array}{l} {\vec{L}}_{O} = m {\vec{r}}_{C} \times {\vec{v}}_{C} + I_{C} \vec{ω} \\ = m (x \hat{i} + R \hat{j}) \times v \hat{i} + \frac{1}{2} m R^{2} (\frac{3 v}{R} \hat{k}) \\ = m (x \hat{i} + R \hat{j}) \times v \hat{i} + \frac{1}{2} m R^{2} (\frac{3 v}{R} \hat{k}) = m v R (\hat{j} \times \hat{i}) + \frac{3}{2} m v R \hat{k} \\ = - m v R \hat{k} + \frac{3}{2} m v R \hat{k} = \frac{m v R}{2} \hat{k} \end{array}$

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