The Lagrangian equation of motion for a system of spring and a mass m connected by a pulley(disk) of radius a as shown in the figure is given by:
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- What is Hamiltonian cycle?A cylindrical disc with a mass of 0.619 kg and radius of 0.575 m, is positioned such that it will oscillate as a physical pendulum as shown below. If the period of the small angle oscillations is to be 0.343 s, at what distance from the center of the disc should the axis of rotation be fixed? Assume that the position of the fixed axis is on the actual disc. The moment of inertia of a disc about its center is 1 = 0.5 M R²...Hint: Use the parallel axis theorem.Let G(u, v) = (3u + v, u - 2v). Use the Jacobian to determine the area of G(R) for: (a) R = [0, 3] x [0, 5] (b) R = [2, 5] x [1, 7]
- Brick with mass m that resting at the top of the inclined plane that has a height of 4.43 m and has an angle of theta = 15.2 degrees with respect to the horizontal. After begin released, is observed to be moving at v = 0.79 m/s a distance d after inclined plane. With coefficient of kinetic friction between brick and plane u_p = 0.1 and coefficient on horizontal surface is u_r = 0.2. Based in the image attached. What would the distance (meters) would be?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.(a) What does the quadrupole formula (P) = = = (Qij Q³ ³) compute? Reason the answer. (b) A point mass m undergoes a harmonic motion along the z-axis with frequency w and amplitude L, x(t) = y(t) = 0, z(t) = L cos(wt). Show that the only non-vanishing component of the quadrupole moment tensor is = Im L² cos² (wt). (c) Use the quadrupole formula to compute the power radiated by the emission of gravitational waves. (Hint: recall that (cos(t)) = (sin(t)) = 0 and (cos² (t)) = (sin² (t)) = ½½ for a given frequency 2.)