Show that for an underdamped harmonic oscillator, the position function can be expressed as x(t) = Beb/2m)tcos(wt-p) where w² = 4mk-b² and B is any constant.
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- A laminar boundary layer profile may be assumed to be approximately of the form u/U₁ = f(n) = f(y/6) i) Use an integral analysis with the following two-segment velocity profile, f(n)=(n/6)(10-3n-13), for 0≤17 ≤0.293 and ƒ (7) = sin (лn/2) for 0.293≤ n ≤1, to find expressions for the displacement thickness &, the momentum thickness e, the shape factor H, the skin-friction coefficient a, and the drag coefficient CD.Consider a periodic signal x(t) with period T defined as follows: T x(t) = (5t, -< t <0 (10, 0Sketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and also for y > 1. The phase line has upward-pointing arrows both below and above y = 1. Thus solutions below the equilibrium solution approach it, and those above it grow farther away. Therefore, ϕ(t) = 1 is semistable.By using hamiltonian equations. Find the solution of harmonic oscillator in : A-2 Dimensions B-3 dimensionsConsider a mass m attached to a spring with natural length 7, hanging vertically under the action of gravity mgk (where the unit vector k is pointing downwards) and a constant friction force F =-Fok. (a) Find the equilibrium point of the mass, write the equation of motion, and show that the motion of the particle is governed by the fundamental equation of simple harmonic motion. (b) Assume the particle is released from the spring when it has heighth above ground and initial velocity vo. Let y be the height above ground of the particle (note that the orientation of the axis is now opposite of z used in point (a)). Write the equation of motion (under the action of gravity and the friction force F). Solve them for the given initial condition and show that v(y)² = vz+2(g− ¹)(h—−y) m (c) Upon entering the ground (y=0) with velocity v₁, the particle is subject to a constant friction force F₁ where F₁ >0 is a constant. Calculate the distance d travelled by the particle into the ground in…A spring/mass/dashpot system has mass 5 kg, damping constant 70 kg/sec and spring constant 845 kg/sec/sec. Express the ODE for the system in the form a"+ 2px' + wr = 0 Identify the natural (undamped) frequency of the spring: wo 3= (square Hz) Identify the parameter p: (Hz) Now assume that the system has the oscillating forcing function cos(wod) with the same frequendy as the spring's natural frequency. + 14a'+ 169a = cos(wat) Find the general solution.A pendulum of length L=6 m and mass M=1 kg has a spring of force constant k=30 N/m connected to it at a distance h=0.3m below its point of suspension. Find the frequency of vibration of the system for small values of the amplitude ( for small angles, sine=0 , cose =1)). Assume the vertical suspension of length is rigid, but ignore its mass. State your answer in Hz to the nearest 0.01 (Use g=9.8m/s²) L www k M Your Answer:What is “fractional error” for the following formula? Z= 2/3 X2Y3 , X= 7m , Y= 4m, σx=0.2m , σy=0.1m σZ=?Suppose that you have a potential V (x) x2 + 6x – 8. Using a Taylor Series around Xo = 3, approximate the potential as a harmonic oscillator. O + (= – 3)? 7-2 (포-3)2 | (x – 3)? ||A 9-lb. weight suspended from a spring having spring constant k = 32 lb/ft, is pushed upwards with an initial velocity vo. The amplitude of the resulting vibrations is observed to be 4 inches. (a) What is the initial velocity? (b) What is the period of the vibration?Apply error propagation to determine the associated uncertainty in the average volume… PlzCalculate the energy, corrected to first order, of a harmonic oscillator with potential: