Question

A block of mass m is attached to a top of a spring (spring constant k). The bottom of the spring is attached to a car of mass M that is free to move on a horizontal track. The spring is rigid enough that it is only able to move up and down, not side to side. See below picture. Illlllll car a. Write the Lagrangian in terms of x, y, i, and y. b. Write the Hamiltonian in terms of x, y, px, and Py. c. Find the differential equations of motion from the Lagrangian. d. Find the differential equations of motion from the Hamiltonian.

Answer 1

Answers Given that The Bottom A block of mass = m spring is attached to a car of mass spring is rigid enough that of is only able to move up and dow. The following Diagram show Below The m If we Cav lo ogginal length of Spring м m 1 os 4 potential energy Ve may & I k ( y-lo)

T = & met 1 m ( x² + y2) Kineatic energy Langrangain I TV - hot an imtraum Ť. Car-h) ) 7 - home The Hamiltonism in termy of aty Pay som t the Du du Py my Jy Hamiltonism H = Epi Qi-2 - Pe x +Pyy-L + ma²+ mn²+ my ²-1/2 mm2 +1 m (42.42) - may - 1 (le (g-lo) - 12 m² + m (Muty 2) + may + 1 k ( g-lo) 2 + Py2 m Ť # = (htm) Paz (M+m)? ingy +1 k(y-lo] m2

+ Pm2 + Py? - z (or h ) + to the (mthite lagraniam equations of mention d dc du ju (umrun) dt dy Poly) a (my) + mg +k( g-lo) to my at my & Cyle) =o & Handtons equention of motion dre dy fic hap Jy de Jeu He It dpu dey opy my f(y-lo) dt dt dy dt P4 m withi