Whirligig Analysis

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?


Step 1: Diagram & coordinate system. Choose +x radially inward & +y vertically up.

Steps 2 & 3: Isolate. Draw Forces (see transparency).

Step 4: Take components.

Tx = T cos θ     Ty = T sinθ

Step 5: Apply Newton's Second Law for each component:

y-component   ΣFy = may = 0  gives     T sin θ = mg

x-component   ΣFx = max        gives     T cos θ = m v2/r

Important:  Here r is the radius of the circle! and not the string length.

Step 6: Solve.

y-equation: With T = 100 N and mg = 36.26 N,   θ = 21.26 degrees.

Neat! Divide the y-equation by the x-equation:   tan θ = g r/v2.

Either the x-equation or the neat relation then gives the speed.

Use r = 1.40 cos θ = 1.305 m, we obtain   v = 5.73 m/s.