PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Rotation, rigid bodies and accelerated frames · Rotation, rigid bodies and accelerated frames
Action and Symmetry, Ch Non-inertial frames from one Lagrangian
Problem
A puck of mass m=0.20 kg rests on a horizontal, frictionless turntable, held by a peg at radius r0=0.50 m while the turntable turns at a constant angular velocity Ω=1.0 rad/s about the vertical axis through its centre; the puck co-rotates with the turntable up to the moment of release. At t=0 the peg is removed. The table is frictionless and horizontal, so afterwards no horizontal force acts on the puck in the inertial (ground) frame, and gravity does no work on the horizontal motion. By t=3.0 s the puck's distance from the axis has grown, in either frame, to r′=1.58 m. An observer riding the turntable, who has learned that "the" energy of a frictionless particle is conserved, claims that the kinetic energy she measures in her own (rotating) frame, T′=21m∣r˙′∣2, must be the same at t=3.0 s as it was at release, since nothing pushes the puck once the peg is gone.
((a)) Write down the Jacobi integral h for this motion (there is no real potential for the horizontal motion, so V=0), evaluate it using the state at release, and explain why it must stay constant afterwards.
((b)) Use h and the given value of r′ at t=3.0 s to find T′ at that instant, and decide whether the observer's claim is correct.
((c)) Explain in one or two sentences, in terms of the forces the rotating observer would draw, why her instinct fails even though the puck genuinely feels no friction.