$A$ body of mass $10 \, kg$ is hanging at rest with the help of a rope wrapped around a cylinder of diameter $0.2 \, m$. The torque applied about the horizontal axis of the cylinder is ....... $N-m$.

When a force of $6.0 \, N$ is exerted at $30^{\circ}$ to a wrench at a distance of $8 \, cm$ from the nut,it is just able to loosen the nut. What force $F$ would be sufficient to loosen it,if it acts perpendicularly to the wrench at $16 \, cm$ from the nut (in $, N$)?

Find the torque about the origin when a force of $3 \hat{j} \text{ N}$ acts on a particle whose position vector is $2 \hat{k} \text{ m}$.

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 -

$A$ triangular plate is shown in the figure. $A$ force $\overrightarrow{F} = 4 \hat{i} - 3 \hat{j}$ is applied at point $P$. The torques at point $P$ with respect to point $O$ and point $Q$ are:

The center of an equilateral triangle $ABC$ is $O$. Three forces $F_1$,$F_2$,and $F_3$ are applied along the sides $AB$,$BC$,and $AC$ respectively. If the total torque about $O$ is zero,what is the relationship between $F_1$,$F_2$,and $F_3$?