Radial vs Tangential acceleration

[MathewsMD]

I ma having a little bit of trouble distinguishing radial and tangential acceleration.

For example:

The magnitude of the acceleration of a point on a spinning wheel is increased by a factor of 4 if:

A. the magnitudes of the angular velocity and the angular acceleration are each multiplied by a factor of 4
B. the magnitude of the angular velocity is multiplied by a factor of 4 and the angular accel- eration is not changed
C. the magnitudes of the angular velocity and the angular acceleration are each multiplied by a factor of 2
D. the magnitude of the angular velocity is multiplied by a factor of 2 and the angular accel- eration is not changed
E. the magnitude of the angular velocity is multiplied by a factor of 2 and the magnitude of the angular acceleration is multiplied by a factor of 4
ans: E

But if a_r = v²/r = ω²r so if angular velocity is multiplied by a factor of 2, this works. But, doesn't αt = ω? So a = α²t²r is also valid, right? Therefore, 2 is also the factor the angular acceleration should be multiplied.

I realize a = αr, but isn't this tangential acceleration and isn't the question assessing radial acceleration?
Any help in differentiating the two types of acceleration would great! Thank you :)

 

[TSny]

I realize a = αr, but isn't this tangential acceleration

Yes

and isn't the question assessing radial acceleration?

The question is asking about the magnitude of the total (or net) acceleration (with contribution from both the centripetal and tangential acceleration).

[EDIT]

But, doesn't αt = ω? So a = α²t²r is also valid, right? Therefore, 2 is also the factor the angular acceleration should be multiplied.

ω = αt assumes constant angular acceleration. This is not assumed in this problem.

 

[MathewsMD]

Yes



The question is asking about the magnitude of the total (or net) acceleration (with contribution from both the centripetal and tangential acceleration).

[EDIT]
ω = αt assumes constant angular acceleration. This is not assumed in this problem.

Oh, thank you! So, then using components, the end result would a_f/a_i = (32/2)^1/2 and this gives a factor of 4. Exactly what I was looking for!

 

[TSny]

Oh, thank you! So, then using components, the end result would a_f/a_i = (32/2)^1/2 and this gives a factor of 4. Exactly what I was looking for!

Since I don't know what your reasoning was that led you to a_f/a_i = (32/2)^1/2, I can't say if you worked it correctly. But maybe it's fine.

 

[HalfLostAndSearching]

Radial acceleration is the component of acceleration that is directed towards or away from the center of rotation. It is perpendicular to the tangential direction and is caused by changes in the direction of motion. This type of acceleration is responsible for the change in speed of the rotating object.

Tangential acceleration, on the other hand, is the component of acceleration that is parallel to the tangential direction. It is caused by changes in the magnitude of the angular velocity and is responsible for the change in direction of the rotating object.

In the given scenario, if the magnitude of the angular velocity is multiplied by a factor of 2 and the magnitude of the angular acceleration is multiplied by a factor of 4, the resulting acceleration will have both radial and tangential components. This is because the change in angular velocity affects both the direction of motion (radial) and the magnitude of motion (tangential).

It is important to differentiate between the two types of acceleration in order to accurately analyze the motion of rotating objects. In this case, the correct answer is E because it takes into account both the changes in radial and tangential acceleration.