A force is acting at a point . The value of for which angular momentum about origin is conserved is [2015]

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Solution

Q.
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 [2015]

1 zero

2 1

3 - 1

4 2

Ans.
(3)

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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