A \(25\,\mathrm{N}\) force acts perpendicular to a handle \(0.40\,\mathrm{m}\) from the axis. Find the torque magnitude.

A force's line of action passes through the rotation axis. What torque does it produce about that axis?

Counterclockwise torque is defined as positive. What is the sign of a clockwise torque?

A \(18\,\mathrm{N}\) force acts at the end of a \(0.50\,\mathrm{m}\) wrench at \(30^\circ\) to the wrench. Find the torque magnitude.

A force \(\vec F=12\hat\imath-4\hat\jmath\,\mathrm{N}\) acts at \(\vec r=0.20\hat\imath+0.30\hat\jmath\,\mathrm{m}\). Find \(\tau_z\).

A \(40\,\mathrm{N}\) force acts \(0.30\,\mathrm{m}\) to the right of a pivot and a \(20\,\mathrm{N}\) force acts \(0.50\,\mathrm{m}\) to the left. Both forces act downward. Find the net torque, taking counterclockwise as positive.

A \(60\,\mathrm{N}\) force acts \(0.25\,\mathrm{m}\) from a pivot at \(70^\circ\) to the radius. Find the perpendicular lever arm and the torque magnitude.

A \(12\,\mathrm{N\,m}\) clockwise torque is balanced by a perpendicular force applied \(0.30\,\mathrm{m}\) from the axis. Find the force magnitude and its torque direction.

A \(2.0\,\mathrm{m}\) light beam is pivoted at one end. A \(120\,\mathrm{N}\) load hangs at the far end. A cable at the far end pulls at \(35^\circ\) above the beam. Find the cable tension for zero net torque.

A force \(\vec F_1=20\hat\jmath\,\mathrm{N}\) acts at \(x=0.40\,\mathrm{m}\). A second force \(\vec F_2=F_y\hat\jmath\,\mathrm{N}\) acts at \(x=-0.25\,\mathrm{m}\). Find \(F_y\) for zero net torque about the origin, and interpret its sign.

Two coplanar forces act on a rigid plate: \(\vec F_1\) at \(\vec r_1\) and \(\vec F_2\) at \(\vec r_2\), with resultant \(\vec R=\vec F_1+\vec F_2\neq\vec0\). Derive the condition for replacing them by one force \(\vec R\), and find the line on which that force must act.

Two coplanar forces act on a rigid body. Derive the conditions under which their net effect is a pure couple independent of the choice of origin, find the couple torque, and explain why no single-force replacement exists in that case.