Problem 1

A small mass

m

slides without friction inside a circular loop of radius

R

. It is released from rest at height

h

above the loop's bottom. Find its speed at the top and the minimum

h

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.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{v_{\rm top}^2\ge gR,\qquad h_{\min}=2R+\frac R2=\frac52R}

problems.proof.analysis

Apply the model consistently to each part, then check dimensions and physical limits.