A double pendulum consists of two simple pendula, with one pendulum suspended from the bob of the other. If the two pendula have equal length and have a bob of equal mass and if both pendula are confined to move in the same plane, Find Lagrange’s equations of motion for the system. Do not assume small angles
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A double pendulum consists of two simple pendula, with one pendulum suspended from
the bob of the other. If the two pendula have equal length and have a bob of equal mass and
if both pendula are confined to move in the same plane, Find Lagrange’s equations of motion
for the system. Do not assume small angles.
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- Quartic oscillations Consider a point particle of mass m (e.g., marble whose radius is insignificant com- pared to any other length in the system) located at the equilibrium points of a curve whose shape is described by the quartic function: x4 y(x) = A ¹ Bx² + B² B²), (1) Where x represents the distance along the horizontal axis and y the height in the vertical direction. The direction of Earth's constant gravitational field in this system of coordinates is g = −gŷ, with ŷ a unit vector along the y direction. This is just a precise way to say with math that gravity points downwards and greater values of y point upwards. A, B > 0. (a) Find the local extrema of y(x). Which ones are minima and which ones are maxima? (b) Sketch the function y(x). (c) What are the units of A and B? Provide the answer either in terms of L(ength) or in SI units. (d) If we put the point particle at any of the stationary points found in (a) and we displace it by a small quantity³. Which stationary locations…A 0.62-kg block attached to a spring with force constant 154 N/m is free to move on a frictionless, horizontal surface as in the figure below. The block is released from rest after the spring is stretched 0.13 m. At that instant, find the force on the block (magnitude and direction). At that instant, find its acceleration (magnitude and direction).Consider the schematic of the single pendulum. M The kinetic energy T and potential energy V may be written as: T = ²m²²8² V = -gml cos (0) аас dt 80 The Lagrangian L is given by L=T-V, and the Euler-Lagrange equations for the motion of the pendulum are given by the following second order differential equation in : ас 80 = 11 = 0 Write down the second order ODE using the specific T and V defined above. Please write this ODE in the form = f(0,0). Notice that this ODE is not linear! Now you may assume that l = m = g = 1 for the remainder of the problem. You may still suspend variables to get a system of two first order (nonlinear) ODEs by writing the ODE as: w = f(0,w) What are the fixed points of this system where all derivatives are zero? Write down the linearized equations in a neighborhood of each fixed point and determine the linear stability. You may formally linearize the nonlinear ODE or you may use a small angle approximation for sin(0); the two approaches are equivalent.
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