A mass on a string traces a horizontal circle while the string makes angle theta with the vertical.

Force balance:

• Vertical: T*cos(theta) = mg
• Horizontal: Tsin(theta) = m$omega^2$r = m$omega^2$Lsin(theta)

Results:

• omega = sqrt($\frac{g}{L*cos(theta}$))
• T (tension) = $\frac{mg}{cos}$(theta) > mg
• Period: $T_p$ = 2pisqrt$\frac{L*cos(theta}{g}$) = 2pisqrt$\frac{h}{g}$, where h = L*cos(theta)

Key insight: The period depends only on the vertical height h below the support point, not independently on L and theta. A longer string at the same angle gives a longer period (proportional to sqrt(L)).

The string can never be perfectly horizontal (theta = 90 would require infinite tension to support mg).