First slide

Law of conservation of angular momentum

Question

A force $\vec{F} = α \hat{i} + 3 \hat{j} + 6 \hat{k}$ is acting at a point $\vec{r} = 2 \hat{i} - 6 \hat{j} - 12 \hat{k}$ .
The value of $α$ for which angular momentum about origin is conserved is

Moderate

A

zero $(c) : F o r t h e c o n s e r v a t i o n o f a n g u l a r m o m e n t u m a b o u t o r i g i n, t h e t o r q u e $ \ v e c {\ t a u} $ a c t i n g o n t h e p a r t i c l e w i l l b e z e r o .$
B

1
C

-1
D

2

Solution

For the conservation of angular momentum about origin, the torque $\vec{τ}$ acting on the particle will be zero.
By definition $\vec{τ} = \vec{r} \times \vec{F}$
$Here, \vec{r} = 2 \hat{i} - 6 \hat{j} - 12 \hat{k} and \vec{F} = α \hat{i} + 3 \hat{j} + 6 \hat{k}$
$∴ \vec{τ} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & - 6 & - 12 \\ α & 3 & 6 \end{matrix}|$
$= \hat{i} (- 36 + 36) - \hat{j} (12 + 12 α) + \hat{k} (6 + 6 α)$
$= - \hat{j} (12 + 12 α) + \hat{k} (6 + 6 α)$
$But \vec{τ} = 0$
$∴ 12 + 12 α = 0 or α = - 1$
$and 6 + 6 α = 0 or α = - 1$

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