Concept of Radian in a stupidly simple way

Radian is not a difficult concept, rather a very clever mathematical idea for thinking of rotation.


Starting with very simple calculations.

2πr/r  = 2π  (no unit, just a number)

2πr/2π = r  (length, unit in m (meter))

Please also note the following basic relation.

  θ          s
-----  =  -----
 2π        2πr

where  θ = an angle in radian,  s = the corresponding arc length along the circle of the radius r. Or

θ = angular displacement, s =  displacement on the circle of the radius r.

Then it becomes simply

              s
  θ   =  -----  
              r


When considering velocity

that the angular velocity ω is

where v is a linear velocity along the circle of the radius r. This r is crucially important.

The left hand side. The unit of ω is radian / sec
The right hand side. The unit of v in meter / sec and r is in meter. Then v/r is in a Number (divided by r) / sec. Please recall that  <radian> is not a physics unit but just a number. A number divided <r>. This can be applied to the above

              s
  θ   =  -----  
              r

 θ = an angle in radian.

and similarly or analogously the angular acceleration ⍺ is 



where at is a liner Tangential Acceleration (m / sec square) corresponding to the linear velocity v.  Again ⍺ is < divided by r > so the unit is radian / sec square. at / r is in <number / sec square> and devided by r.


Please remember how Radius r works. Radius r leads you by dividing either linear displacement or velocity or acceleration to the Angle Rotation world in Radian.


sptt