Question

Two blocks of equal mass m are connected by a flexible cord. One block is placed on a smooth horizontal table, the other block hangs over tire edge. Find the acceleration of the blocks and cord assuming (a) die mass of die cord is negligible and (b) the cord is heavy, of mass m'.

Find the Hamiltonian and Hamilton’s canonical equations of motion for the systems

described in both parts of the above problem.

Answer 1

mass of negligible the cord is here . Since the cord has a fixed length ' so there is only one degree of freedom of the system. y length of the cord hangs downward from the edge & length of the cord connected to block placed on the table. Here, atyal 00, ya lox cie consider, y generalised coordinate. Total Kit of the system T = 2 mg + km2 zmir +mige myr and P.EV = to-mgy -- mga Lagrangian L-T-V = myt+ ingy Corresponding momentum. ty - zmi . La Pu+mgy Hamiltonian of the system. H = ypy - 2 mga

Hamiltonians canonical eque, kg - g + mg by Considering the mass of the cord, m', we have the KiE of the cord T = km gr Here, the block of mass m is hung by a height y downward. So the centre of mass of the cord is at a height / downward. So, P.E of the word is V = - my zeg mass of the cord handing downward my New Lagrangian , L'- ' y* (2m + m') + med + mo yu - Þý - noviny - (2m +m' } ý 09, ý nemen om L ebentar tunggu belangor New Hamiltonian. H'=ýpi Canonical eque ja Py = - 34 oma mg + myy

ay cord, for negligible mass of the - L = my+mgy. Eqn of motion, b> or, amy - mg = 0 or, ý = 8/ acceleration of the blocks - 82. and also you do Considering the mass of the cord, L = 2 (2m + n) + ingy + mg z Fons of motion, (2m+ming -may-ng- me my 2mtim' e (antul) Soli y = Ye + Up ABA + Bekt te where _mg l(amen) or the ty and came y(t) = Aert & Bert - Me acceleration, ý = V(Art & Bett)