Question

2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10 points) write down the Lagrangian for this system in terms of rr.θ,9 and the corresponding Euler-Lagrange equations. Note: There are two types potential energies: potential energy of the spring and gravitation potential energy of the mass-Earth system. Choose your reference point for the gravitation potential energy of the mass-Earth system correctly. (b) (10 points) Write down the Hamiltonian in terms of p, .Por.0 and corresponding Hamilton equations. Is the Hamiltonian the total energy of the system? Is it conserved? (c) (5 points)Verify that the Hamilton equations are consistent with the Euler Lagrange equations? (d) (10 points) Find the speed of mass and the instantaneous length of the pendulum when it reaches the bottom (pendulum in vertical position), given: 0-60° ,m-1kg,L-1m,k-50N / m, g-10m / s2

Answer 1

2 ncr-cool ow, Langnagton of the system L= T-V ラ|L= mI(2 &2_m.gn (1-cos 0 )一1KL01-11ll) 2 2 -2L

and aL- e(Iv) becomes み 2 Now/ 见9.1(VJ bQCOMO// , b) Hamiltonton of勿2 system ingwenbu

we bnow, Pe- 2 +하나- 2 2. 2m几 2, hene, Yes, Homtlhoncan es the to+-a Sytam and ti cansenved

2 2 .tman (1-roroj t丄K.: 2 ー Pe M几 Po

at Langnagran quaon.
​​​​​