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

v_{\rm top}^2=2g(h-2R)

problems.proof.analysis

Solve the energy equation for speed squared; reaching the top requires $h\ge2R$ . 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.

problems.proof.pitfall. Do not mix sign conventions or omit a system constraint; check the sign and units before interpreting the result.