TBR CH2 #9 Centripetal force/angular velocity

[LuminousTruth]

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Three identical bugs (marked A, B, C) are standing on a turntable as it begins to spin. As the spinning increases, which of the bugs is most likely to slip first?

(The figure is a circular turntable with A, B, C in the same line with A closest to center, B in the middle, and C next to the edge of the table)

A) Bug A
B) Bug B
C) Bug C
D) All three of the bugs are equally likely to slip.

Considering conservation of angular momentum, Bug A, with smaller inertia, I (since it is closest to the center), would exhibit a greater angular speed in order to conserve angular momentum. A greater speed in the center will cause Bug A to slip.

But from TBR, it shows Bug C slipping because "Bug C has a larger centripetal acceleration and greater speed" since it is closer to the rim. I thought centripetal acceleration, v^2/r would imply that a bigger r, leads to smaller acceleration (so the further out from the center, the smaller the acceleration).

In that case, I can't seem to see why Bug C has a larger centripetal acceleration/speed. Can someone show me why it is Bug C that slips instead of Bug A?

 

[Saint Fu]

You are right that bigger r alone means lower centripetal acceleration. However, what happens to v as r increases? They all experience the same angular velocity, w, so v increases proportionally with r as well (v = r*w). Since v is squared, an increase in r actually increases acceleration.

You can modify the expression for centripetal acceleration to illustrate this:

v^2/r = (wr)^2/r = w^2 r

 

[anon4895]

Considering conservation of angular momentum, Bug A, with smaller inertia, I (since it is closest to the center), would exhibit a greater angular speed in order to conserve angular momentum. A greater speed in the center will cause Bug A to slip.

Conservation of angular momentum doesn't mean all these bugs have the same momentum.

But from TBR, it shows Bug C slipping because "Bug C has a larger centripetal acceleration and greater speed" since it is closer to the rim. I thought centripetal acceleration, v^2/r would imply that a bigger r, leads to smaller acceleration (so the further out from the center, the smaller the acceleration).

In that case, I can't seem to see why Bug C has a larger centripetal acceleration/speed. Can someone show me why it is Bug C that slips instead of Bug A?

All the bugs have the same angular velocity, w. Their linear velocity, v, depends how far they are from the center.

(w^2)r is probably the best formula to use here. w is the same for all the bugs so the biggest r requires the biggest force to stay on the table.

If you use v^2/r just realize that v also depends on r, and since it's squared the r in the numerator has more influence. so an r increase is an overall increase even though there's an r in the denominator.