Problem 1

A body slides without friction from height

h

above the bottom into a vertical loop of radius

R

. Find the minimum height to maintain contact at the top. (a) State the assumptions and establish the governing equation for the requested quantity. (b) Solve it in terms of the given symbols, identifying the direction, sign, or physical nature of the result. (c) Check a simple limiting case and state when the model remains valid. Use SI units consistently, retain the symbols in the setup, and enforce all geometric constraints and initial conditions.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{mg+N=mv^2/R,\quad N=0\Rightarrow v^2=gR,\quad mgh=2mgR+mv^2/2,\quad h_{min}=5R/2}

problems.proof.analysis

The results are listed in the order requested and use one consistent sign convention. Their dimensions, initial conditions, and model constraints must all agree.