a): Using Hamiltonian equation of motion, Show that the Hamiltonian, p2 H =e-rt mw? +. 2m Leads to the equation of motion of a damped harmonic oscillator, * + ri + w?x = 0.
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- 4.26. Maximum speed * A critically damped oscillator with natural frequency w starts out at position xo > 0. What is the maximum initial speed (directed toward the origin) it can have and not cross the origin?please solveConsider a for which Ko is 5o N/m: state and sketch the nature damped harmonic osaillater and is the oscillately the of yotion when damping Constat 04N-5/m i0 1.0 N - is
- Question 1. Find the steady state solution of the forced Mass-Spring-Damper with the following parameters undergoing forcing function F(t). 3 kg, c = 22 Ns/m, k = 493 N/m, F(t) = 21 cos(6t) in the form of (t) = A cos(6t – 8) Enter your answers for A and to four decimal places in the appropriate boxes below: m = A: d:A small block with mass m slides along a frictionless horizontal surface with a speed v = with and sticks to the end of a uniform rod (mass M = m, length L = 0.75 m) that can rotate around a frictionless axle through its upper end as shown. :0.4 m/s. It collides %3D a) Use Newton's 2nd law to find the angular frequency w of small angle oscillations for the combined system. b) What is the amplitude (maximum displacement 0max) from the vertical, expressed in degrees?Solve b and c
- The vibration modes and the motion pattern of a bridge system can be foreseen through the application of eigenvalue and eigenvector. The dynamic system of the bridge can be expressed in the matrix form as: а 1 3] A = |1 0 l2 1 1 al where a is the last digit of your matrix number. If the last digit of your number is zero or one then take a = 2. Use try value v = [1 0 1]T and calculate until |mk+1 – mx| < 0.005 or FIVE (5) iterations whichever comes first. Do the calculations in 3 decimal places. Estimate the mode shape of the vibration by finding the dominant (in absolute value) eigenvalue and its motion pattern (corresponding eigenvector). (а)Problem 2 (Estimating the Damping Constant). Recall that we can experimentally mea- sure a spring constant using Hooke's law-we measure the force F required to stretch the spring by a certain y from its natural length, and then we solve the equation F = ky for the spring constant k. Presumably we would have to determine the damping coefficient of a dashpot empirically as well, but how would we do so? As a warm-up, suppose we have a underdamped, unforced spring-mass system with mass 0.8 kg, spring constant 18 N/m, and damping coefficient 5 kg/s. We pull the mass 0.3 m from its rest position and let it go while imparting an initial velocity of 0.7 m/s. %3D (a) Set up and solve the initial value problem for this spring-mass system. (b) Write your answer from part (a) in phase-amplitude form, i.e. as y(t) = Aeºt sin(ßt – 4) and graph the result. Compare with a graph of your answer from (a) to check that you have the correct amplitude and phase shift. (c) Find the values of t at which y(t)…1
- We can model a molecular bond as a spring between two atoms that vibrate with simple harmonic motion.The figure below shows an simple harmonic motion approximation for the potential energy of an HCl molecule.This is a good approximation when E < 4 ×10^−19. Since mH << mCl, we assume that the hydrogen atomoscillates back and forth while the chlorine atom remains at rest. Estimate the oscillation frequency of theHCl molecule using information in the figure below.A mass of 458 g stretches a spring by 7.2 cm. The damping constant is c = 0.34. External vibrations create a force of F(t)= 0.4 sin 5t Newtons, setting the spring in motion from its equilibrium position with zero velocity. What is the imaginary part v, of the complex root of the homogeneous equation? Use g-9.8. Express your answer in two decimal places.Can you solve it plz An object with a mass of 0.5kg and potential energyexpressed as the imagesimple harmonic motion aroundequlibrium point x=0. What is theangular frequency of this object?