Lever arm

Where a force acts matters as much as how hard it pushes, and the lever arm is the piece of that geometry the physics actually counts. EK 5.3.A.2 defines it in a single line: the lever arm is the perpendicular distance from the axis of rotation to the line of action of the exerted force.

The line of action is the infinite straight line you get by extending the force arrow in both directions. Drop a perpendicular from the axis onto that line, and its length is the lever arm, written $r_{\perp}$.

It is the geometry hiding inside the sine. With $r$ the distance from the axis to the point of application and $\theta$ the angle between the force and that position vector, $r_{\perp} = r\sin\theta$, which turns the equation into the familiar $\tau = rF\sin\theta$. The two forms are one statement seen from either side: resolve the distance, or resolve the force.

Three consequences worth taking into an exam:

• A force whose line of action passes through the axis has zero lever arm and exerts zero torque, however large the force is. Pushing a door at its hinge does nothing.
• The lever arm is largest when the force is perpendicular to the position vector, so a push at 90 degrees is the most effective one.
• Sliding a force along its own line of action leaves the torque unchanged, because the perpendicular distance has not moved.

The routine for finding every lever arm in a diagram and balancing the torques is in how to calculate torque.