Two particles connected by a spring form an isolated system (no external force)…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Lagrangian mechanics · Lagrangian mechanics

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

Problem

Two particles connected by a spring form an isolated system (no external force) moving along a line. Their masses are

m_1=3.0\ \mathrm{kg}

and

m_2=5.0\ \mathrm{kg}

; the spring force depends only on

x_1-x_2

. At

t=0

the positions are

x_1=0

x_2=2.0

m and the velocities are

\dot x_1=4.0\ \mathrm{m\,s^{-1}}

\dot x_2=-1.0\ \mathrm{m\,s^{-1}}

. ((a)) Derive, from the boost

x_i\to x_i+\epsilon t

applied to both particles, the conserved quantity

K=Pt-MR_{\rm cm}

, where

P=m_1\dot x_1+m_2\dot x_2

M=m_1+m_2

and

R_{\rm cm}=(m_1x_1+m_2x_2)/M

. ((b)) Use

K

together with the constancy of

P

to find

R_{\rm cm}

t=3.0\ \mathrm{s}