Solution Found!
The method of Lagrange multipliers works perfectly well
Chapter 7, Problem 7.52(choose chapter or problem)
The method of Lagrange multipliers works perfectly well with non-Cartesian coordinates. Consider a mass m that hangs from a string, the other end of which is wound several times around a wheel (radius R, moment of inertia I) mounted on a frictionless horizontal axle. Use as coordinates for the mass and the wheel x, the distance fallen by the mass, and 4, the angle through which the wheel has turned (both measured from some convenient reference position). Write down the modified Lagrange equations for these two variables and solve them (together with the constraint equation) for .3j and and the Lagrange multiplier. Write down Newton's second law for the mass and wheel, and use them to check your answers for x and 4i. Show that X of/ax is indeed the tension force on the mass. Comment on the quantity X afoo.
Questions & Answers
QUESTION:
The method of Lagrange multipliers works perfectly well with non-Cartesian coordinates. Consider a mass m that hangs from a string, the other end of which is wound several times around a wheel (radius R, moment of inertia I) mounted on a frictionless horizontal axle. Use as coordinates for the mass and the wheel x, the distance fallen by the mass, and 4, the angle through which the wheel has turned (both measured from some convenient reference position). Write down the modified Lagrange equations for these two variables and solve them (together with the constraint equation) for .3j and and the Lagrange multiplier. Write down Newton's second law for the mass and wheel, and use them to check your answers for x and 4i. Show that X of/ax is indeed the tension force on the mass. Comment on the quantity X afoo.
ANSWER:Step 1 of 6
The constraint equation for mass can be represented as,
The potential energy of the mass hanged from a string can be given as,
The kinetic energy of the system can be represented as,
