Problem 1
A small particle of mass m starts from rest on a smooth approach track and then enters the inside of a vertical circular loop of radius R at its lowest point. The approach joins the loop smoothly, and the initial height H is measured above the loop bottom. Take g=9.8 m/s² and neglect all friction. (a) Find speed at position angle θ from the upward vertical, taking the bottom as zero potential. (b) Write the normal force and contact condition at the top. (c) Find minimum H to maintain contact throughout, then find the bottom normal force in this limiting case. Use SI units throughout and treat the track and constraints as ideal as specified. State sign conventions, retain units, and check the contact condition at every point of the loop.
problems.proof.stepOf
problems.proof.analysis
The equation follows by applying the physical law to the chosen system; ideal constraints link the variables. Signs must remain consistent with the defined directions.
problems.proof.pitfall. Pitfall: do not change sign conventions between equations; a negative sign may encode direction rather than an invalid result.