A small block of mass slides without friction on the inside of a rigid circular loop…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Kinematics and Newtonian dynamics of particles · Lagrangian mechanics

Action and Symmetry, Ch Multipliers, non-holonomic constraints and forces outside the potential

Problem

A small block of mass m=0.25 kgm=0.25\,\mathrm{kg} slides without friction on the inside of a rigid circular loop of radius R=0.60 mR=0.60\,\mathrm m fixed in a vertical plane, under gravity (g=9.8 m s−2g=9.8\,\mathrm{m\,s^{-2}}). Let θ\theta be the angle measured from the bottom of the loop, and let vbv_{\rm b} be the block's speed at the bottom. ((a)) Using r=Rr=R as the constraint, express the normal force N(θ)N(\theta) the track exerts on the block (directed towards the centre) as a multiplier, in terms of the block's speed v(θ)v(\theta). ((b)) Using energy conservation, find the minimum speed vbv_{\rm b} at the bottom for which the block keeps contact with the track all the way round. ((c)) For vb=6.0 m s−1v_{\rm b}=6.0\,\mathrm{m\,s^{-1}}, find NN at the top and at the bottom of the loop.

Related theory

Browse the problem library