Question

A) Write the Lagrangian for a simple pendulum consisting of a point mass m suspended at the end of a massless string of length l. Derive the equation of motion from the Euler-Lagrange equation, and solve for the motion in the small angle approximation. B) Assume the massless string can stretch with a restoring force F = -k (r-r0), where r0 is the unstretched length. Write the new Lagrangian and find the equations of motion. C) Can you re-write the resulting two coupled 2 nd -order differential equations as four coupled 1 st - order differential equations (Hint: Let v=dr/dt and ω = dϑ/dt).

Answer 1

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LAqrangian Ahe Sysdem L T-Va Fuler- LagTane equntion From m O = 6us TBu (9 Equation motior Sine o then 9 + Ss1ution his eauntion is gi The ven by Constomt asbitiry found ond fonboundary where are which Can be Condi Hons

(B) Resbring prce, f -x(r-) 2 sbrinyr-1j Patentied ener f the Y . Total Potential energy of the SyStem V= О1bu- (r-7) + length he strirg where ris the

bab Position fthe Sine . kinefic ene ry T- Laganginn L T-v Nea Equationsfotion are a Amimmglno - Klr-n,) Cos - K -0)

and 4mr marine Sineo 2m. Sine 2 mo - (2) e) Lef dr (3) = 1 = (A) 2 belomes Then 2 mrwma os0 - (T-Y) Equation (2) becomes Sino= C6 four (3), 41,(5), ) diffecentil equtons Equahion are Coupled A St-order