Moment arm

The moment arm is the perpendicular distance from an axis of rotation to the line along which a force acts. Torque — the rotational analogue of force — is equal to force times moment arm. A wrench works because applying a small force far from the bolt produces a large torque; a door handle is mounted opposite the hinge because moving the handle in close would demand a stronger push to swing the door.

Geometrically, the moment arm is what is left of a force's geometric relationship to the axis after you strip away any part of the force pointing along the line connecting them. Only the perpendicular component rotates; the parallel component just tries to translate the rigid body's center of mass (a push that does no rotational work). This is why a bolt cannot be loosened by pulling directly along the wrench handle toward the bolt — there is no moment arm, and no torque, no matter how hard you pull.

For a rolling body on an incline, the moment arm appears in a subtle way: static friction at the contact point provides the torque that spins the body up to match its accelerating translation. The contact point sits at radius R from the wheel's center, so friction's moment arm about the axle is R. This is the geometric reason rolling KE depends on the shape factor I/(mR²) rather than on R alone — R sets both the moment arm and the lever length on both sides of Newton's second law for rotation, and the ratio is what matters.