Problem 1
A small mass slides without friction inside a circular loop of radius . It is released from rest at height above the loop's bottom. Find its speed at the top and the minimum for which it maintains contact throughout the loop. Model each stage with the appropriate conservation law or equation of motion; quantities stated as constant remain fixed during that stage. (a) Establish the governing physical relation for each stage. (b) Determine the requested observables from the stated initial conditions and constraints. (c) Check signs, dimensions, and the physical limiting behavior of the results.
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problems.proof.analysis
At the top the centripetal direction is downward. Gravity and the normal force provide centripetal acceleration; contact requires . This follows from the chosen physical model and the stated initial conditions; the sign convention must remain consistent throughout. A dimensional check catches algebraic errors before the result is interpreted.