A \(15\,\mathrm{N}\) force acts perpendicular to a \(0.40\,\mathrm{m}\) spanner. Find the torque magnitude about the pivot at the end of the spanner.

State the two vector conditions required for a rigid body to be in static equilibrium.

A downward \(80\,\mathrm{N}\) force acts \(0.30\,\mathrm{m}\) to the right of a pivot. Taking counterclockwise as positive, find the torque.

A \(10\,\mathrm{N}\) force acts at \(25^\circ\) to a \(0.60\,\mathrm{m}\) lever arm. Find the torque magnitude.

A gate hinge experiences a clockwise torque of \(18\,\mathrm{N\,m}\). What counterclockwise torque is required for rotational equilibrium?

A force \(F\) is applied perpendicular to a \(0.80\,\mathrm{m}\) lever arm and must balance a \(32\,\mathrm{N\,m}\) torque. Find \(F\).

Explain why \(\sum\vec{F}=0\) alone is not enough to guarantee a rigid body remains at rest.

A \(4.0\,\mathrm{m}\) beam is supported at both ends. The beam weighs \(120\,\mathrm{N}\) and a \(260\,\mathrm{N}\) crate is \(1.0\,\mathrm{m}\) from the left support. Find both support reactions and state which support carries more load.

A uniform horizontal beam of length \(L\) and weight \(W_b\) is pinned at the left end. A cable attached to the right end makes angle \(\theta\) above the beam. Build the equilibrium equations, solve for the cable tension and hinge reactions, state your assumptions, and interpret the limit \(\theta\to0\).

A beam of length \(L\) and weight \(W_b\) rests on supports at \(x=0\) and \(x=L\). A load \(P\) may be placed at coordinate \(x\), even outside the supports. Derive the two support reactions and the symbolic condition on \(x\) for both supports to remain in contact. State the modeling assumptions.