# Different units while calculating Acceleration of Rotating Objects

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 – ac :
Is either:
(1) Tangential-Velocity squared, over radius OR

Tangential Acceleration – at :

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

For Centripetal-Acceleration:
Radius is a base unit (meters)
...so Tangential-Velocity squared, over radius has the unit:
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:

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

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?

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

Last edited:

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).

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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.

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.

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AlephZero
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