Level 1: Frictionless Loop of Radius R: Is a 2R Start Enough?
A rolling ball instead of a cart.
A small solid ball rolls without slipping down a track into a vertical loop of radius R (the ball's own radius is much smaller than R). The least starting height that lets it get round the loop is about:
At the top the ball still needs v² ≥ gR. But a rolling solid sphere also carries rotational kinetic energy, equal to 2/5 of its translational kinetic energy, so its total kinetic energy is 0.7mv².
Rolling needs a start of 2.7R, higher than the 2.5R a sliding cart needs, because part of the energy goes into spinning. That energy is not lost; it is simply not available to carry the ball upward.
Every Level 2 starts from a quick question, Level 1, and changes one thing. Seeing what that change does is the skill JEE Main tests. Try it here, free and with no signup, then read the worked answer.
