Uniform circular motion proportionality question

[Sean1218]

Homework Statement

David spins a sling in a horizontal circle above his head. What would happen to the period of rotation if he applied the same force and the length was a) doubled, b) halved

Homework Equations

a_c = (4π²r) / T²

The Attempt at a Solution

Just not sure what to do in general, I tried a few different things like this:

r $$\alpha$$ T²

2r $$\alpha$$ T²
r $$\alpha$$ T²/2

Then compared T² and T²/2
square rooted both of them

T and T/sqrt(2)

1/sqrt(2) is 0.7, which is the answer to b), even though I was trying to do a).

edit: think I figured it out

I can just equate T and T/sqrt(2) can't I? The first T is T_a, the second is T_b, and that equation is saying that T_a is .7x smaller than T_b, in other words, T_b is 1.4x larger, right?

 

[Delphi51]

My approach would be to solve the formula for T:
T1 = 2π*sqrt(mR/F)
I wrote that T1 to indicate it is the original period.
When R is doubled, you get
T2 = 2π*sqrt(m2R/F) and I wish I could make that 2R a red 2 to make it easier to follow. That 2 needs to go out of the sqrt where it becomes a root 2 and move to the front so you can see
T2 = sqrt(2)*2π*sqrt(mR/F) = sqrt(2)*T1
It is an excellent technique; works every time!