Rotational Mechanics

Torque of a Force About the Axis of Rotation

Consider a force F acting on a particle P. Choose an origin O and let r be the position vector of the particle experiencing the force. The torque of the force F about O is defined as:

G = r × F

This is a vector quantity with direction perpendicular to both r and F.

When a rigid body rotates about an axis AB, the torque of a force F acting on a particle P of the body about O (somewhere on the axis of rotation) is G = r × F. The component of G along OA (a part of the rotation axis AB) is called the torque of F about OA. It is calculated as |r × F| cos(θ), where θ is the angle between r × F and OA.

The torque of a force about a line is independent of the choice of the origin as long as it is chosen on the line. Let O₁ be another point on the line AB. Then:

OP × F = (OO₁ + O₁P) × F = OO₁ × F + O₁P × F

Since OO₁ × F is perpendicular to OO₁, its component along AB is zero. Therefore, the component of OP × F is equal to the component of O₁P × F.

Special Cases:

• Case I: F || AB

  r × F is perpendicular to both r and F. Since F is parallel to AB, r × F is perpendicular to AB. Hence, the component of r × F along AB is zero.

• Case II: F intersects AB

  If F intersects AB at point O, then taking O as the origin, r and F are collinear. Therefore, r × F = 0, and the component along OA is zero.

• Case III: F ⊥ AB but F and AB do not intersect

  Consider the plane through P that is perpendicular to the axis of rotation AB, intersecting AB at point O. If we take O as the origin, then G = r × F = OP × F.