# Radians and the unit of rotational energy

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1. May 26, 2015

### 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

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 ?

Dara

2. May 26, 2015

### 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."