3.1 Angular Measurements

3.1 ANGULAR MEASUREMENTS

• Definition of radian: If the length of arc AB equals the radius r, the subtended angle at the center O is 1 radian; rad is the SI unit of angular measurement.

• Angular displacement (AO): For a particle P on a circle of radius r attached to a massless rod OP, the angle swept during a time interval corresponds to the angular displacement AO.

• Vector nature for small angles: For very small angular changes Δθ, the angular displacement is treated as a vector.

• Sign convention: AO is positive for counterclockwise (CCW) rotation.

• Axis of rotation: The axis is normal to the plane of rotation; the z-axis is taken along this axis with O as the origin; x and y lie in the plane of rotation.

• Direction of angular displacement (right-hand rule): Grasp the axis with the right hand; curl fingers in the direction of rotation; the thumb points in the direction of angular displacement.

• Units of angular displacement: degrees (deg), revolutions, and radians (rad).

• Arc length–angle relation (in radians): For an arc of length s on a circle of radius r subtending angle θ at the center,
  \theta = \frac{s}{r}
  and equivalently
  s = r\,\theta
  When the arc length equals the radius (s = r), the angle is 1 radian.

• Key definitions: The angular displacement AO is defined by the rotation of OP about the axis; its magnitude relates to the angle swept, and its direction is given by the right-hand rule.