The solution to the differential equation of a damped oscillator, for the case in which the damping is small, is x = Age 2m cos(w't + 8) The phase constant o is determined by the
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- . Consider a particle with the an equation of motion I 26.xax = 0, where B anda are positive constants. The particle starts at rest from a position >0 Determine the natural frequency wo. (a) Write down the auxiliary equation corresponding to the above homoge- (b) neous linear differential equation and obtain the general solution æ(t) for strong damping B >wo. Sketch x(t) vs. t and the phase space diagram x(t) vs. (t) for the given (c) initial conditions.A mass of 2 kg on a spring with k = 6 N/m and a damping constant c= 4 Ns/m. Suppose Fo = v2 N. Using forcing function Fo cos(wt), find the w that causes practical resonance and find the amplitude.what if a damped harmonic oscillator has a damping constant of beta= 2w, is this overdamping, critical damping, or underdamping? please explain why?
- Given the function f(x) – - 1 sin(z) a. Equation of midline: b. Amplitude: c. Draw the graph. -2 -72 -1 37/2 27 -2A system with mass = 150g_and a spring constant of k=100N/m has a damping constant y=0.9. Assume the mass was pulled to the right 30 cm at t=0 and released. d) Estimate the time at which the amplitude has decayed to 4 of its initial value. e) Assume the system is connected to a forcing function given by (in Newtons) F(t)=2coswt f) Estimate the value of the amplitude at resonance.a particle of mass m makes simple harmonic motion with force constant k around the equilibrium position x = 0. Calculate the valu of the amplitude and the phase constant if the initial conditions are x(0) = 0 and v(0) = -Vi (vi > 0). Draw x-t, v - t and a - t figures for one complete cycle of oscillations.
- Please explain why beta = 2omega is an example of a critically damping motion for a damped harmonic oscillator?A system with 100g_mass and a spring constant of k=150N/m has a damping constant y=1.1. ASsume the mass was pulled to the right 30 cm at t=0and released. a)Estimate the time at which the amplitude has decayed to ¼ of its initial value. b)Assume the system is connected to a forcing function given by(in Newtons)F(t)=3coswtEstimate the value of the amplitude at resonance.need help with d
- IC-3 A 167 gram mass is vibrating about its equilibrium position on the end of a spring as shown in problem SHM-8. While vibrating, the mass is observed to have a maximum speed of 0.500m/s and a maximum acceleration of 6.00m/s. At t0 the mass is at the equilibrium position with a velocity to the left. a) Find the numerical values for the angular frequency o and the amplitude of the motion xm. Hint: think about how Vm and xm are related. b) Find the value of the spring constant of the spring. c) The position of the block is described by x Xmcos(@t+0,). Find all possible values of 0, and then explain how to determine the value of 0, that corresponds to the given conditions. 99+ hpDo it without any delayA simple pendulum with a length of 1.73 m and a mass of 6.74 kg is given an initial speed of 2.36 m/s at its equilibrium position. (a) Assuming it undergoes simple harmonic motion, determine its period (in s). (b) Determine its total energy (in J). (c) Determine its maximum angular displacement (in degrees). (For large v, and/or small /, the small angle approximation may not be good enough here.) (d) What If? Based on your answer to part (c), by what factor would the total energy of the pendulum have to be reduced for its motion to be described as simple harmonic motion using the small angle approximation where 0 ≤ 10°?