The displacement x(t) of a damped harmonic oscillator satisfies the equation x(t) - 2Bx(t) + wzx(t) = 0 Let the initial condition be dx x(0) = Xại v(0) = = 0 dt=0 Find x (t), if wo = 4, and (a) ß = 5, (b) B = 4, and (c) B = 1.
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- A mass of 0.38 kg is attached to a spring and set into oscillation on a horizontal frictionless surface. The simple harmonic motion of the mass is described by x(t) = (0.26 m)cos[(16 rad/s)t]. Determine the following. %3D (a) amplitude of oscillation for the oscillating mass How does the amplitude of oscillation compare to the magnitude of the maximum displacement from equilibrium? m (b) force constant for the spring N/m (c) position of the mass after it has been oscillating for one half a period m (d) position of the mass one-third of a period after it has been released (e) time it takes the mass to get to the position x = -0.10 m after it has been releasedA mass-spring motion is governed by the ordinary differential equation d²x dx +b + k(t)x= F(t), dt² dt m where m is the mass, b is the damping constant, k is the spring constant, and F(t) is the external force. We consider the initial conditions x(0) = 1 and x/(0) = 0. Assume the following numerical values for this part of the project: m = 1, k = 1/5, b= 1/5, and F(t) = cos yt. 1 (a) Read section 4.10. Explain what is the resonance frequency, and then compute the resonance frequency for this mass-spring system. (b) The ODE45-solver can be used to obtain the solution curves in MATLAB. Use the script Project2_Q2.m to plot the solutions and estimate the amplitude A of the steady response for y = 0.2, 0.42, 0.6, and 0.8. (c) The script also provide you with the graph of A versus y. For what frequency 7 is the amplitude the greatest? Is it equal to that you obtained in (a)?Part 2) An organ pipe of length L = 4.6 m is open at both ends. It is driven to oscillate with a standing wave that has two nodes within the pipe. (a) What is the wavelength of the standing wave? m (b) If the speed of sound in air is 330m/s, what is the frequency that the organ pipe is oscillating at in this mode? Hz
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- Asap plzA simple harmonic oscillator consists of a mass m attached to a spring with spring constant k, with displacement given by x = A sin(wt + ¢) . Which one of the following is NOT true? O The frequency is independent of the amplitude O The potential energy is a maximum when (wt + 4) = 0 The kinetic energy is a maximum when (wt + ¢) = 0 O The restoring force must be proportional to the negative of the displacement O Increasing the mass will decrease the frequencyA driving force of the form F(t) = (0.345 N) sin (27 ft) acts on a weakly damped spring oscillator with mass 5.35 kg, spring constant 363 N/m, and damping constant 0.159 kg/s. What frequency fo of the driving force will maximize the response of the oscillator? fo= Find the amplitude Ao of the oscillator's steady-state motion when the driving force has this frequency. Ao = Hz m