An Atwood machine has masses and joined by a light inextensible string of fixed…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Kinematics and Newtonian dynamics of particles · Lagrangian mechanics

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

Problem

An Atwood machine has masses

m_1 = 4.0\,\mathrm{kg}

and

m_2 = 1.0\,\mathrm{kg}

joined by a light inextensible string of fixed length

\ell_{\rm s}

over a light, frictionless pulley. Let

x_1

and

x_2

be the distances each mass has fallen below the pulley, so the constraint is

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

. ((a)) Using

x_1

and

x_2

as coordinates with the multiplier

\lambda_1

for

f_1

, write and solve Lagrange's equations to find the acceleration and the multiplier. Take

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

and add the constraint to the Lagrangian as

L+\lambda_1f_1

. ((b)) State the tension in the string.

Related theory

Lagrangian mechanics

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