How fast do you need to swing a 190-g ball at the end of a string in a horizontal circle of 0.50-m radius so that the string makes a 38 ∘ angle relative to the horizontal?

The speed you need to swing the ball at the end of the string in a horizontal circle is 2.504 m/s

Explanation:

As the ball swings, its motion creates tension (T) on the string which has both vertical component and horizontal component.

Vertical or Y-component: The tension on the string is equal to down ward force on the ball due to gravity.

Tsin38⁰ = mg

Tsin38⁰ = 0.19 x 9.8

Tsin38⁰ = 1.862

T = (1.862) / (0.6157)

T = 3.024 N

Horizontal or X-component: The tension on the string is equal to centripetal force on the ball which acts inward.

Tcos38⁰ = m (v²/r)

3.024 (0.78801) = 0.19 (v²/0.5)

2.3829 = 0.38*v²

v² = 2.3829 / 0.38

v² = 6.2708

v = √6.2708

v = 2.504 m/s

Therefore, the speed you need to swing the ball at the end of the string in a horizontal circle is 2.504 m/s