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

Force equations:

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

From horizontal: T = momega^2L From vertical: momega^2Lcos(theta) = mg => omega^2 = g/(Lcos(theta))

Time period: T_p = 2pi/omega = 2pisqrt(Lcos(theta)/g)

Tension: T = 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(h/g) — depends only on the height, not L or theta independently.