Small block on a rotating table

[Legaldose]

Imagine a surface that rotates with frequency f about its center, if we set a small block (or a coin, or any flat object for that matter) on the table, I wanted to calculate the maximum radius that you can place this block from the center before it starts to move outward from the center. This is how I did it:

I figured the maximum centripetal force on block had to equal the maximum frictional force keeping it in place.

F_c = F_f

and since:

F_c = mv²/r = 4π²rf²m
F_f = μ_smg

where v² = 4π²r²f²

where f is the frequency of rotation in Hz

Now I set

4π²rf²m = μ_smg

4π²rf² = μ_sg

The mass variables m cancel, this is why we can add any object, as long as it's center of mass rests at a distance r from the center.

Now it's easy to see that

r = μ_sg/4π²f²



I think it's interesting that the distance is only dependent on how fast the table is rotating, and that there are no other factors(other than what planet you are on :p) that determine it.

Would this be considered a correct derivation? Basically I just want to know if I missed anything or if I have the correct basic intuition behind this type of situation.

Thanks PF! :)

 

[abrewmaster]

Well technically the object does matter since you will get different frictional coefficients depending on the materials you end up putting on the table. But yea, if you can get the same frictional coefficients then they would be the same.

 

[Legaldose]

Yea I forgot to include the friction coefficient in that sentence, oh well D: