Radians and the unit of rotational energy

In summary, the conversation discusses the unit of radians and rotational energy. Radian is a dimensionless unit and the correct unit for rotational energy is joules or N.m. This is because the units of length cancel out in the ratio of arc length to radius length.
  • #1
dara bayat
8
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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
 
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  • #2
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."
 

FAQ: Radians and the unit of rotational energy

1. What is a radian?

A radian is a unit of measurement for angles in a circle. It is defined as the angle formed at the center of a circle by an arc that is equal in length to the radius of the circle. One radian is approximately 57.3 degrees.

2. How is rotational energy measured?

Rotational energy is measured in joules (J), which is the standard unit of energy in the International System of Units (SI). It can also be measured in other units such as kilojoules (kJ) or electronvolts (eV).

3. What is the relationship between radians and degrees?

Radians and degrees are two different units for measuring angles. One radian is equal to 57.3 degrees, or approximately 180 divided by pi. This means that 360 degrees, a full circle, is equal to 2π radians.

4. How is rotational energy related to linear energy?

Rotational energy and linear energy are both forms of kinetic energy, which is the energy of an object in motion. Rotational energy is the energy of an object that is rotating, while linear energy is the energy of an object that is moving in a straight line. The relationship between the two is that rotational energy is equal to the moment of inertia multiplied by the square of the angular velocity.

5. How does rotational energy affect the motion of an object?

Rotational energy plays a crucial role in determining the motion of an object. The higher the rotational energy of an object, the faster it will rotate. This means that objects with higher rotational energy will have a greater angular velocity and will rotate more quickly. Rotational energy also affects the stability and balance of an object, as well as its ability to resist changes in its rotational motion.

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