### 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?

Use DID TASC

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

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

Step 4: Take components.

T_{x} = T cos θ
T_{y} = T sinθ

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

*y-component*
Σ**F**_{y} = ma_{y} = 0
gives
T sin θ = mg

*x-component*
Σ**F**_{x} = ma_{x}
gives
T cos θ = m v^{2}/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/v^{2}.

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.