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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

Is either:

(1) Tangential-Velocity squared, over radius OR

(2) Angular-Velocity squared, times radius.

Tangential Acceleration – a

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?

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

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**

Tangential-Acceleration (a_{c}): (radians^{2}*meters / second^{2})Tangential-Acceleration (a

_{t}): (radians*meters / second^{2})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

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