A massless inextensible string passes over a pulley which is a fixed distance above the floor. A bunch of bananas of mass m is attached to one end of the string. A monkey of mass M is initially at the other end B. The monkey climbs the string, and her displacement d(t) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are d(0) = 0, d/dt(0) = 0.a) write the transformation equations, and calculate the lagrangian of the system in terms of these coordinates. Show that the equation governing the height z(t) of the monkey above the flooris (M + m)z¨ − m d¨ = (m − M)g.b) Find the Hamiltonian H and Hamilton’s equations. Rederive the equation for z¨ from Hamilton’s equationsc) Is H = E = T + V ? Is H conserved? Is E conserved? Justify briefly.(d) Integrate the equation to find z(t) as a function of t. In the special case that m = M, show that the bananas and the monkey rise through equal distances so that the vertical separation between them is constant. 2. A massless inextensible string passes over a pulley which is a fixed distance above the floor. A bunch of bananas of mass mis attached to one end of the string. A monkey of mass M is initially at the other end B. The monkey climbs the string, andher displacement d(t) with respect to the end B is a given function of time. The system is initially at rest, so that the initialconditions are d(0) = 0, d(0) = 0.(a) Introduce suitable generalized coordinates, write the transformation equations, and calculate the lagrangian of thesystem in terms of these coordinates. Show that the equation governing the height 2(t) of the monkey above the flooris (M + m) – md = (m − M)g.(M+m) mä-(b) Find the hamiltonian H and Hamilton's equations. Rederive the equation for from Hamilton's equations.(c) Is H E T+V? Is H conserved? Is E conserved? Justify briefly.(d) Integrate the equation to find z(t) as a function of t. In the special case that m = M, show that the bananas and themonkey rise through equal distances so that the vertical separation between them is constant.

(a) The equation governing the height z(t) of the monkey is (M+m)z¨−md¨=(m−M)g..(b) The…

Answered: 2. A massless inextensible string…

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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A massless inextensible string passes over a pulley which is a fixed distance above the floor. A bunch of bananas of mass m is attached to one end of the string. A monkey of mass M is initially at the other end B. The monkey climbs the string, and her displacement d(t) with respect to the end B is a given function of time. The system is initially at rest, so that the initial conditions are d(0) = 0, d/dt(0) = 0.

a) write the transformation equations, and calculate the lagrangian of the system in terms of these coordinates. Show that the equation governing the height z(t) of the monkey above the floor
is (M + m)z¨ − m d¨ = (m − M)g.

b) Find the Hamiltonian H and Hamilton’s equations. Rederive the equation for z¨ from Hamilton’s equations

c) Is H = E = T + V ? Is H conserved? Is E conserved? Justify briefly.
(d) Integrate the equation to find z(t) as a function of t. In the special case that m = M, show that the bananas and the monkey rise through equal distances so that the vertical separation between them is constant.