The ghost rad unit in angular motion

[stefano77]

unit of measurement '' rad '' appears and disappears as in calculating angular frequency as square root of k/m in spring-mass model.how can l explain to students?not as a dogma? you can say unit of measuremnt is the inverse of second, but what about the final formula of rad/ s ? rad appears in final result as a ghost

 

[PeroK]

A radian is the ratio of two lengths, so is dimensionless. Like the ratio of two masses, or the ratio of two velocities.

 

[Orodruin]

To put it slightly differently: The unit of rad is equal to 1, dimensionless. Angles are dimensionless and therefore must be measured in dimensionless units. Another unit of angle is the degree, which is $\pi/180$ - still dimensionless.

We write out these units for convenience and for showing that we are dealing with an angle.

 

[Gavran]

unit of measurement '' rad '' appears and disappears as in calculating angular frequency as square root of k/m in spring-mass model.how can l explain to students?not as a dogma? you can say unit of measuremnt is the inverse of second, but what about the final formula of rad/ s ? rad appears in final result as a ghost

You can omit specifying radians, but you can not omit specifying degrees.

$\sin(2\pi\,rad)$ has the same value as $\sin(2\pi)$ while $\sin60\,^\circ$ does not have the same value as $\sin60$

for angular velocity
$2\pi\,rad/s$ has the same value as $2\pi\,1/s$ while $60\,^\circ/s$ does not have the same value as $60\,1/s$

 

[Herman Trivilino]

The radian is a unit, but it's not a dimension. Compare this to something like the meter, which is both a unit and a dimension.

The same issue that arises with the radian also arises with all other dimensionless units. Take for example the revolution. A disc of circumference 2.0 m spins at a rate of 3.0 rotations per second. How fast is a point on its rim moving? You multiply 2.0 m by 3.0 rotations per second and get a result of 6.0 m/s. How did the unit rotation disappear?

 

[A.T.]

A disc of circumference 2.0 m spins at a rate of 3.0 rotations per second.

Confusingly the dimensionless something affects what unit is used for something/second: Hertz vs. Becquerel.

 

[Herman Trivilino]

Confusingly the dimensionless something affects what unit is used for something/second: Hertz vs. Becquerel.

I never thought about it before, but perhaps that confusion is another layer that conflates the confusion over the use of the radian.