drag the bottom-speed slider — watch tension flip sign at the top
TAUT
Live Data
Speed
0.0 m/s
Tension
0.0 N
θ from bottom
0°
Height gained
0.00 m
Time period
measuring…
Kinetic energy
0.00 J
Potential energy
0.00 J
Total energy
0.00 J
Parameters
v_min for full loop = √(5gr)
Minimum speed to just complete the loop, at r = 1.2 m:
Bottom (√5gr): — m/s
Right 3 o'clock (√3gr): — m/s
Top (√gr): — m/s
Left 9 o'clock (√3gr): — m/s
slow motion — physics unchanged, just playback speed
Live Equations
At this instant (angle θ from bottom)
mg cosθ − T = − mv²/r → solving for T
Bottom (θ=0)
v_B = — m/s
T_B = mv_B²/r + mg = — N
Side (θ=90° / 270°, horizontal)
v_side = √(v_B² − 2gr) = — m/s
T_side = mv_side²/r = — N
Top (θ=180°)
v_T = √(v_B² − 4gr) = — m/s
T_T = mv_T²/r − mg = — N
Energy at 4 Key Points — fixed by conservation of energy
Bottom (θ=0°)
v = — m/s
T = — N
KE = — J
PE = — J
E = — J
Right — 3 o'clock (θ=90°)
v = — m/s
T = — N
KE = — J
PE = — J
E = — J
Top (θ=180°)
v = — m/s
T = — N
KE = — J
PE = — J
E = — J
Left — 9 o'clock (θ=270°)
v = — m/s
T = — N
KE = — J
PE = — J
E = — J
Live — current position (θ = 0° from bottom)
v = — m/s | T = — N
KE = — J + PE = — J = E = — J
Key Facts
Tension is maximum at bottom, minimum at top.
Speed is maximum at bottom, minimum at top.
String goes slack the instant T would go negative — object becomes a projectile.
v_top,min = √(gr), and by energy conservation v_bottom,min = √(5gr).
v_side,min = √(3gr) — this is a dual-purpose number: it's the minimum speed needed if the object starts at the side (horizontal point), AND it's the speed the object has when it passes the side, if it started at the bottom with v_bottom,min = √(5gr).