How fast do you need to swing a 250-g ball at the end of a string in a horizontal circle of 0.6-m radius so that the string makes a 39 degree angle relative to the horizontal?

2.7 m/s

Explanation:

Draw a free body diagram of the ball. There are two forces:

Weight force mg pulling down

Tension force T pulling 39° above the horizontal

Sum of the forces in the y direction:

∑F = ma

T sin θ - mg = 0

T = mg / sin θ

Sum of the forces in the radial (+x) direction:

∑F = ma

T cos θ = m v² / r

Substitute:

(mg / sin θ) cos θ = m v² / r

mg / tan θ = m v² / r

g / tan θ = v² / r

v = √ (gr / tan θ)

Given that r = 0.6 m and θ = 39°:

v = √ (9.8 m/s² * 0.6 m / tan 39°)

v ≈ 2.7 m/s