A particle of mass moves in one dimension under a spring force whose fixed…

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

Action and Symmetry, Ch Symmetry, Noether's theorem and what is conserved

Problem

A particle of mass

m=0.50\ \mathrm{kg}

moves in one dimension under a spring force whose fixed equilibrium is at the ground-frame origin, so its ground-frame potential energy is

V(x)=\tfrac12 kx^2

with

k=20\ \mathrm{N\,m^{-1}}

. Its motion is described instead using the coordinate

q=x-ut

, the position measured from an origin moving at a steady

u=1.5\ \mathrm{m\,s^{-1}}

, giving

L(q,\dot q,t)=\tfrac12 m(\dot q+u)^2-\tfrac12 k(q+ut)^2.

t=1.0\ \mathrm{s}

the particle is at

q=0.20\ \mathrm{m}

. Find

\mathrm{d}h/\mathrm{d}t

at that instant, where

h

is the Jacobi integral built from

q

, and state what the sign and size of this rate tell you about whether

h

equals the ground-frame energy

E=\tfrac12 m\dot x^2+\tfrac12 kx^2