Live Tension:
Tension at Top:
Tension at Bottom:
Physics Behind Circular Motion
Horizontal Circular Motion:
When an object moves in a horizontal circle:
- The tension in the string provides the centripetal force: T = mv²/r
- Speed remains constant throughout the motion
- Angular velocity ω = v/r remains constant
- Kinetic energy KE = ½mv² remains constant
- Potential energy PE = 0 (assuming no vertical displacement)
Vertical Circular Motion:
When an object moves in a vertical circle, both speed and tension vary:
- At the bottom (θ = π/2):
- Maximum speed: v = v₀
- Maximum tension: T = mv₀²/r + mg
- At the top (θ = 3π/2):
- Minimum speed: v = √(v₀² - 4gr)
- Minimum tension: T = mv²/r - mg
- General position (angle θ):
- Speed: v = √(v₀² - 2gr(sinθ - 1))
- Tension: T = mv²/r + mgsinθ
- Kinetic energy: KE = ½mv²
- Potential energy: PE = mgr(1 - cosθ)
Critical Conditions:
- String goes slack when T ≤ 0 at the top (v₀ ≤ √(5gr))
- String breaks when T exceeds the breaking tension
- Minimum speed at top: v ≥ √(gr) to complete the circle
Energy Conservation:
The total mechanical energy (KE + PE) remains constant throughout the motion:
½mv₀² + mgr = ½mv² + mgr(1 - cosθ)