Two blocks of mass and are joined by a light inextensible string of fixed length that…

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

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

Problem

Two blocks of mass

m_1 = 5.0\,\mathrm{kg}

and

m_2 = 3.0\,\mathrm{kg}

are joined by a light inextensible string of fixed length

\ell_{\rm s}

that passes over a small frictionless, massless pulley fixed to a support (an Atwood machine). Let

x_1

and

x_2

be the distances of

m_1

and

m_2

below the pulley, so that the string's fixed length gives the constraint

x_1+x_2 = \ell_{\rm s}

. Both blocks move only vertically and the string does not stretch. Take

g = 9.8\,\mathrm{m\,s^{-2}}

. ((a)) Using

x_1

as the single generalised coordinate (eliminate

x_2 = \ell_{\rm s}-x_1

via the constraint, as in the ordinary Lagrangian treatment of the Atwood machine), find

\ddot x_1

from Lagrange's equation, then apply Newton's second law to

m_2

alone to find the string's tension. ((b)) Now keep

x_1

and

x_2

as two independent coordinates. Add a Lagrange multiplier

\lambda_1

for

f_1 \equiv x_1+x_2-\ell_{\rm s} = 0

to the two equations of motion, solve for

\lambda_1

, and show that

-\lambda_1

equals the same tension. ((c)) State the numerical value of the acceleration and of the tension found by each route.