$A$ couple produces

$A$ 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:

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

$A$ 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 $\tau_z$?

$A$ force $\vec{F}=(2 \hat{i}+3 \hat{j}-5 \hat{k}) \, N$ acts at a point $\vec{r}_1=(2 \hat{i}+4 \hat{j}+7 \hat{k}) \, m$. The torque of the force about the point $\vec{r}_2=(\hat{i}+2 \hat{j}+3 \hat{k}) \, m$ is ............. $N m$.

$A$ force $F_1 = A \hat{j}$ is applied to a point whose radius vector is $r_1 = a \hat{i}$, while a force $F_2 = B \hat{i}$ is applied to the point whose radius vector is $r_2 = b \hat{j}$. Both the radius vectors are determined relative to the origin of the coordinate axes $O$. The moment of the force relative to $O$ is