$A$ particle of mass $m$ is moving along a trajectory given by $x = x_0 + a \cos \omega_1 t$ and $y = y_0 + b \sin \omega_2 t$. The torque acting on the particle about the origin at $t = 0$ is:

The rotational motion of a body is caused by:

Let $\overrightarrow{F}$ be the force acting on a particle having position vector $\overrightarrow{r}$,and let $\overrightarrow{\tau}$ be the torque of this force about the origin. Then:

$A$ door $1.2 \,m$ wide requires a force of $1 \,N$ to be applied perpendicularly at the free end to open or close it. The perpendicular force required at a point $0.2 \,m$ distant from the hinges for opening or closing the door is: (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 $2 \hat{k} \text{ m}$.

$A$ wheel of radius $R$ with an axle of radius $R/2$ is as shown in the figure. It is free to rotate about a frictionless axis through its centre and perpendicular to the page. Three forces are exerted as shown in the figure: a force $F$ at an angle of $45^{\circ}$ to the tangent of the outer rim,a force $F$ tangential to the inner axle,and a force $2F$ tangential to the outer rim. The magnitude of the net torque acting on the system is nearly ............. $FR$.