Different units while calculating Acceleration of Rotating Objects

[magnetismman]

Question on Acceleration of Rotating Objects:

The Physics Textbook I have says that the Acceleration of a spinning object is the Pythagorean-theorum result of sides for Centripetal-Acceleration (toward the rotation center) and Tangential-Acceleration (perpendicular to Centripetal-Acceleration in the direction of movement). It seemed pretty straight-forward, until I got to calculating each. According to the book:

Centripetal-Acceleration – a_c :
Is either:
(1) Tangential-Velocity squared, over radius OR
(2) Angular-Velocity squared, times radius.

Tangential Acceleration – a_t :
Radius * Angular-Acceleration.

The problem I think I've found is in the units for each.

For Centripetal-Acceleration:
Tangential-Velocity = radius * angular-velocity (meter-radians-per-second)
Radius is a base unit (meters)
...so Tangential-Velocity squared, over radius has the unit:
(radian-radian-meters-per-second-per-second)
Please note, I did get a “Domain of result may be larger” warning from my calculator.

For Tangential-Acceleration:
Angular-Acceleration = Angular-Velocity-Change per second (Radians-per-second-per-second)
Radius is a base unit (meters)
...so Radius times Angular-Acceleration has the unit:
(radian-meters-per-second-per-second)

How can the two be combined for a total rotational-acceleration if the units are different?

Centripetal-Acceleration (a_c): (radians²*meters / second²)
Tangential-Acceleration (a_t): (radians*meters / second²)

Since the units are different, the combined acceleration would be in neither unit.
Is this a flaw in the theory?
Is this some flaw in my reasoning?
Thank you for your advice.

Information-Source: Cutnell&Johnson - 'Physics' 6th Edition (Chapter 8) – ISBN:0-471-15183-1

 

[dauto]

radians aren't physical units. Strictly speaking an angle is defined as an arch (measured in meters) divided by a radius (measured in meters), giving you the identity 1 radian = 1 meter/meter = 1. That means radians are adimensional. That's why the units (meter-radians-per-second) that you obtained for the tangential velocity may also be expressed as the more familiar (meters-per-second).

 

[magnetismman]

radians are adimensional.

Thank you. So am I to understand that since a radian is a meter-per-meter, the units cancel evenly? If so, is there any difference implied by the fact that the radians (meters-per-meter) cancel out different numbers of times for 'equivalent' units? Thanks again.

 

[dauto]

Thank you. So am I to understand that since a radian is a meter-per-meter, the units cancel evenly?

Yes.

If so, is there any difference implied by the fact that the radians (meters-per-meter) cancel out different numbers of times for 'equivalent' units? Thanks again.

No, no difference.

 

[AlephZero]

The important thing about radians here is that they are lexactly "meters per meter". If you measured the angular velocity in degrees per second, or revolutions per minute, you would have an extra factor in the formulas, just like if you measured velocity in miles per hour but distance in meters.

In real life angular velocities are often measured in RPM etc, so you have to convert them to radians/second before using the formulas.