Four equal and parallel forces are acting on a rod (as shown in the figure) at distances of $20 \, cm, 40 \, cm, 60 \, cm$,and $80 \, cm$ respectively from one end of the rod. Under the influence of these forces,the rod -

$A$ couple produces

$A$ solid cone hangs from a frictionless pivot at the origin $O$,as shown. If $\hat{i}$,$\hat{j}$,and $\hat{k}$ are unit vectors,and $a, b$,and $c$ are positive constants,which of the following forces $\vec{F}$ applied to the rim of the cone at a point $P$ results in a torque $\vec{\tau}$ on the cone with a negative $z$-component $\tau_z$?

The torque of the force $\vec{F} = (2\hat{i} - 3\hat{j} + 4\hat{k}) \text{ N}$ acting at the point $\vec{r} = (3\hat{i} + 2\hat{j} + 3\hat{k}) \text{ m}$ about the origin is:

If $\vec{F} = (4\hat{i} - 10\hat{j})$ and $\vec{r} = (5\hat{i} - 3\hat{j})$,then calculate the torque $\vec{\tau} = \vec{r} \times \vec{F}$. (in $hat{k}$)

$A$ force $\vec{F} = (2\hat{i} - \hat{j} + 3\hat{k}) \text{ N}$ is acting at a point $(-1, 2, -3) \text{ m}$. Find its torque about the origin.