Radian Measurement, Angular Displacement, and Angular Velocity

Radian Measurement and Degree Conversion

• Definition of Radian: One radian ($1\,\text{rad}$) is defined as the angle subtended at the center of a circle by an arc with a length equal to the radius of the circle.

• Conversion Between Radians and Degrees:

  • For one complete revolution, the number of radians must be equal to the number of degrees in a full circle ($360^\circ$).

  • Expressed mathematically:     $2\pi\,\text{rad} = 360^\circ$

  • Solving for one radian:     $1\,\text{rad} = \frac{360^\circ}{2\pi} = \frac{360^\circ}{2 \times 3.14} = 57.3^\circ$

Direction of Angular Displacement and Right Hand Rule

• Vector Nature: Angular displacement is a vector quantity, possessing both magnitude and direction.

• Right Hand Rule:

  • The Right Hand Rule is used to specify the direction of angular displacement.

  • Procedure: Hold the axis of rotation in the right hand with fingers curling in the direction of rotation; the thumb then gives the direction of angular displacement.

Angular Velocity

• Definition: The time rate of change of angular displacement of a body is called angular velocity.

• Mathematical Formulation:

  • If $\Delta \theta$ is the small angular displacement occurring in a time interval $\Delta t$, then the angular velocity $\omega$ (omega) is:     $\omega = \frac{\Delta \theta}{\Delta t}$

• Units of Angular Velocity:

  • SI Unit: Radian per second ($\text{rad}\,\text{s}^{-1}$).

  • Other Units:

  • Degrees per second ($\text{deg/s}$)

  • Revolutions per second ($\text{rev/s}$)

  • Revolutions per minute ($\text{rev/min}$ or $\text{rpm}$)

• Direction of Angular Velocity:

  • The direction of angular velocity is identical to that of angular displacement.

  • When rotation occurs counterclockwise, $\omega$ points in the positive $z$--direction.

  • When rotation occurs clockwise, $\omega$ points in the negative $z$--direction.