Two blocks of masses and move along a straight frictionless line and are joined by a…

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

Action and Symmetry, Ch The action, the first variation and the boundary term

Problem

Two blocks of masses

m_1 = 1.0\,\mathrm{kg}

and

m_2 = 2.0\,\mathrm{kg}

move along a straight frictionless line and are joined by a spring of stiffness

k = 50\,\mathrm{N\,m^{-1}}

that exerts a force depending only on their separation

q_1 - q_2

L = \tfrac12 m_1\dot q_1^2 + \tfrac12 m_2\dot q_2^2 - \tfrac12 k(q_1 - q_2)^2.

Both

q_1(t)

and

q_2(t)

are held fixed at

t_1

and

t_2

in the variation. ((a)) Derive the two Euler-Lagrange equations, treating

\delta q_1

and

\delta q_2

as independent. ((b)) At an instant when

q_1 - q_2 = 0.020\,\mathrm{m}

, evaluate the acceleration of each block.