Radians and the unit of rotational energy

[dara bayat]

Hello everyone

I have a question regarding radians and the unit of rotational energy (which has been probably asked several times elsewhere but is still confusing for me :-) ).

As I understand radian (rad) is a UNIT that is dimensionless (thus cannot be omitted), correct ?

Now if I want to look at the unit of Energy I think it goes as follows :

Er = T*theta --> unit=N.m.rad
Er : rotational energy
T : torque
theta : angle

but also

Er = 1/2*I*w² --> unit=kg.m².rad².s⁻²=N.m.rad²
I : moment of inertia
w : angular velocity

however since w=sqrt(k/I) , k is the torsion spring constant :
Er = 1/2*k*theta² = 1/2*I*w²*theta² (I saw this formula in a textbook) --> unit=N.m.rad⁴

I also read in other posts that the unit of rotational energy is sometimes N.m/rad

which one is correct ? Am I making a mistake ?

can I just omit radians and say that rotational energy is N.m? if yes, why ? If no, why ? :-)

is this really confusing ? Or have I not understood something ?

Thanks in advance for your help

Dara

 

[Overt]

Radians are dimensionless. So as your intuition would tell you the correct units for rotational energy, which is an energy, are joules or N^.m.

Taken from Graham Kemp's response in https://math.stackexchange.com/questions/803955/why-radian-is-dimensionless

"Thus the radian measure of angle as the ratio of arc length to radius length is one where the units of length cancel out."