3.3.4—Vertical circular motion

In a vertical circle, speed changes with height. Use conservation of energy between points, then apply radial Newton’s law to find tension or normal reaction at that point.

At the top and bottom, define inward separately: tension or reaction may add to weight at the bottom but oppose it at the top. Complete contact requires the limiting reaction to remain non-negative.

For a particle on a string, the minimum speed at the top occurs when tension is zero, giving mv²/r=mg there; energy then determines the required bottom speed.

The condition T=0 is a limiting contact condition, not “no gravity”, and using one fixed speed around the circle violates energy conservation.