A ball on a 1.4-meter long string is being whirled in mid-air 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 free-body 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
ΣFy = may = 0	|	ΣFx = max = m v2 r
T sin(θ) = mg	|	T cos(θ) = m v2 r

If we divide one equation by the other, we get a neat relationship for the angle of the string:

tan(θ)

=

g r v2

In this situation we still need to use the y-equation to find the angle.
With T = 100 N and mg = 36.26 N, the angle works out to:

θ = 21.26 degrees

Either the x-equation 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