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Suppose a mass M is attached to a rod of length L (also mass M) and free to pivot around a frictionless pin. The rod hangs vertically in gravity 9 as a (massless) horizontal spring with spring constant is connected to the midpoint of the rod, and connected to the wall at the opposite end. a) What is the Lagrangian L (0,0) of this system? Use the angular displacement to characterize the rod's orientation. b) Apply the Euler-Lagrange equation to get a 2nd-order differential equation in . c) Now assume is small. Show that the equation reduces to the harmonic oscillator equation + ²0 = 0 d) If you release the mass from rest after displacing it a small amount, at what frequency would the system oscillate?

Suppose a mass M is attached to a rod of length L (also mass M) and free to pivot around a frictionless pin. The rod hangs vertically in gravity 9 as a (massless) horizontal spring with spring constant is connected to the midpoint of the rod, and connected to the wall at the opposite end. a) What is the Lagrangian L (0,0) of this system? Use the angular displacement to characterize the rod's orientation. b) Apply the Euler-Lagrange equation to get a 2nd-order differential equation in . c) Now assume is small. Show that the equation reduces to the harmonic oscillator equation + ²0 = 0 d) If you release the mass from rest after displacing it a small amount, at what frequency would the system oscillate?

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Suppose a mass M is attached to a rod of length L (also mass M) and free to pivot around a frictionless pin. The rod hangs vertically in gravity 9 as a (massless) horizontal spring with spring constant is connected to the midpoint of the rod, and connected to the wall at the opposite end. a) What is the Lagrangian L (0,0) of this system? Use the angular displacement to characterize the rod's orientation. b) Apply the Euler-Lagrange equation to get a 2nd-order differential equation in 8. c) Now assume is small. Show that the equation reduces to the harmonic oscillator equation + w²0=0 d) If you release the mass from rest after displacing it a small amount, at what frequency would the system oscillate?
Transcribed Image Text:Suppose a mass M is attached to a rod of length L (also mass M) and free to pivot around a frictionless pin. The rod hangs vertically in gravity 9 as a (massless) horizontal spring with spring constant is connected to the midpoint of the rod, and connected to the wall at the opposite end. a) What is the Lagrangian L (0,0) of this system? Use the angular displacement to characterize the rod's orientation. b) Apply the Euler-Lagrange equation to get a 2nd-order differential equation in 8. c) Now assume is small. Show that the equation reduces to the harmonic oscillator equation + w²0=0 d) If you release the mass from rest after displacing it a small amount, at what frequency would the system oscillate?
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