Newtonian mechanics

Torque and rigid-body equilibrium

Torque measures the tendency to rotate about an axis; a rigid body is in equilibrium when net force and net torque both vanish.

A push at a door handle far from its hinge turns the door more readily than a push near the hinge. The effect depends on force, lever arm, and direction.

\vec\tau=\vec r\times\vec F, \qquad |\tau|=rF\sin\theta=F d_\perp, \qquad \sum\vec F=\vec0,\quad \sum\vec\tau_O=\vec0

Definition: Torque

$\vec r$ runs from the chosen origin/axis $O$ to the force application point; $d_\perp$ is the perpendicular distance from the axis to the line of action. Torque has units N·m and direction given by the right-hand rule. Rotational equilibrium depends on the origin, but when net force is zero all origins give the same condition.

Adjust forces and lever arms on both sides; observe net torque and beam tilt.

Rigid-body equilibrium

Static equilibrium requires no translational or angular acceleration: the vector sum of forces and the sum of moments about a point both vanish. In planar problems, choose a pivot that removes unknown forces from the torque equation.

Example: A balanced beam

A 30 N perpendicular force acts 0.40 m from a pivot. What perpendicular force on the other side, 0.60 m away, balances torque?

Solution

Torque balance gives $30(0.40)=F(0.60)$ , so $F=20\,\mathrm N$ in the opposite rotational sense.

Quick check

For a planar problem, choose one rotational sense as positive and keep each torque’s sign consistent. A force whose line of action passes through the pivot has zero torque, yet it still contributes to the net force. Torque balance alone does not ensure force balance: a bar may have no angular acceleration while translating. Check both conditions independently, and include the beam’s weight when it is not negligible.

Torque depends on the chosen origin. In equilibrium problems, choose a pivot through the application point of an unknown force to remove its torque from the equation. After using torque balance to find another force, return to the net-force equations to determine reaction forces. For an extended body, weight acts at its center of mass, not automatically at the end of a beam.

What torque about the pivot is produced by a force parallel to the rod from pivot to application point?

What is the condition for static equilibrium of a rigid body?

References

Young, Freedman (2019). University Physics