Find the Lagrangian and Lagrange's equations for a simple pendulum if the cord is replaced by a spring constant k. If unstretched the spring length is r0, and the polar coordinates of the mass m are (r, theta), the potential energy of the spring is 1/2(k)(r-r0)^2
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Find the Lagrangian and Lagrange's equations for a simple pendulum if the cord is replaced by a spring constant k. If unstretched the spring length is r0, and the polar coordinates of the mass m are (r, theta), the potential energy of the spring is 1/2(k)(r-r0)^2
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- Express the Lagrangian for a free particle moving in a plane in a plane polar coordinates. From this proves that, in terms of radial and tangential components, the acceleration inpolar coordinates isa = (¨r − rθ˙2) er + (rθ¨ + 2 r˙ θ˙) eθ(where er and eθ are unit vectors in the positive radial and tangential directions).Problem 2 Consider the block of mass m, connected to a spring of spring constant k and placed on a inclined plane of angle a. Let la be the length of the spring at equili brium, and r be the elongation. The block oscillates and at the same time is rotating around origin 0, in the plane of the inclined, by a variable angular velocity . 1. Calculate the degrees of freedom of the block 2. What is the kinetic energy of the block 3. What is the potential energy of the block 4. Write the Lagrangian function (don't derive the Euler Lagrange equa- tions) k reference plane m o'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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