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?
Related questions
Question
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?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 5 images
