A mass m on a string of length L making angle theta with the vertical, tracing a horizontal circle.

Force equations:

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

From horizontal: T = m*$omega^2$L From vertical: m$omega^2$Lcos(theta) = mg => $omega^2$ = $\frac{g}{L*cos(theta}$)

Time period: $T_p$ = 2pi/omega = 2pi*sqrt$\frac{L*cos(theta}{g}$)

Tension: T = $\frac{mg}{cos}$(theta) (always greater than mg)

Radius of circle: r = L*sin(theta)

Height below support: h = Lcos(theta), so $T_p$ = 2pi*sqrt$\frac{h}{g}$ — depends only on the height, not L or theta independently.