A ball on a 1.4meter long string is being whirled in midair in a horizontal circle at a
constant speed v. The tension in the string is 100 N. The mass of the ball is 3.70 kg.
What is v?
As usual, begin with a freebody diagram.
Follow this up with an appropriate choice of coordinate system.
Let's go with a coordinate system with +x towards the center of the circle and +y vertically up.
The tension needs to be broken into components:
T_{x} = T cos(θ)
T_{y} = T sin(θ)
Apply Newton's Second Law in each direction:
y direction    x direction  

ΣF_{y} = ma_{y} = 0   
 
T sin(θ) = mg   

If we divide one equation by the other, we get a neat relationship for the angle of the string:
tan(θ)  = 

In this situation we still need to use the yequation to find the angle. With T = 100 N and mg = 36.26 N, the angle works out to:
θ = 21.26 degrees
Either the xequation or the neat relationship we derived above will then get us the speed.
Be careful with r, which is NOT the length of the string.
r = 1.40 cos(θ) = 1.305 m
This gives v = 5.73 m/s