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Vertical circular motion
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Tension and speed vary around the loop; minimum speed at the top sets the loop condition.
v_top ≥ √(gr)In a vertical circle the speed and tension vary: at the top T + mg = mv²/r, at the bottom T − mg = mv²/r. The string stays taut only if v_top ≥ √(gr).
A ball on a 1 m string must have at least 3.13 m/s at the top.
The tension at the top is smaller than at the bottom.
In more detail
In vertical circular motion, the speed changes as the object moves up and down, and the tension or normal force changes with it. The critical point is the top of the circle, where the minimum speed is sqrt(gr).
Real-world uses
- Roller coasters are designed to keep riders on the track at the top of loops.
- Buckets of water can be swung in a vertical circle without spilling at the top.
- Roller coaster designers check that the cars stay on the track at the top of a loop by making sure the speed is high enough.
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Frequently asked questions
Why is the top the critical point?
Because at the top, gravity and the normal force or tension both point inward, so the required speed is smallest there.
What is the minimum speed at the top?
sqrt(gr), where g is gravitational acceleration and r is the radius.
Why does the bob need more speed at the top of a loop?
At the top, gravity and tension both point toward the centre, so the speed must be high enough for the required centripetal force.
Related topics in Newton's Laws & Forces
Banking of roads · Vehicle on flat circular turn · Conical pendulum · Loop-the-loop · Centrifugal force · Pseudo forces