$A$ couple produces

For a particle moving in a circular path under the influence of a central force,the angular momentum remains constant because .......

In rotational motion,which quantity is analogous to force in linear motion?

Find the torque of a force $\vec F = -3\hat i + \hat j + 5\hat k$ acting at the point $\vec r = 7\hat i + 3\hat j + \hat k$ with respect to the origin.

$A$ force $\overrightarrow{F} = (\hat{i} + 2\hat{j} + 3\hat{k}) \text{ N}$ acts at a point $\vec{r}_1 = (4\hat{i} + 3\hat{j} - \hat{k}) \text{ m}$. The magnitude of torque about the point $\vec{r}_2 = (\hat{i} + 2\hat{j} + \hat{k}) \text{ m}$ is $\sqrt{x} \text{ N-m}$. The value of $x$ is $........$

$A$ force of $-P \hat{k}$ acts on the origin of the coordinate system. The torque about the point $(2, -3)$ is $P(a \hat{i} + b \hat{j})$. If the ratio $\frac{a}{b} = \frac{x}{2}$,find the value of $x$.