Show that for an underdamped harmonic oscillator (i.e., b < 2/sqrt(mk)), the position function can be expressed as x(t) = Be^(-(b/2m)t) cos(wt – phi) where w^2 = (4mk–b^2)/4m^2 and B is any constant.
Q: Question 10 If the function is given as U(x) = x4, is the equilibrium is stable at x=0? check your…
A:
Q: A 0.25-slug mass is attached to a spring and spring is stretched 1.28 ft from its natural length.…
A:
Q: Consider the function f(x) = 4 sin((x − 3)) + 6. State the amplitude A, period P, a - midline. State…
A:
Q: which the Lagrangian is I = mc² (1-√√1-B²)-kx² where ß = == a) Obtain the Lagrange equation of…
A: Required to find the equation of motion.
Q: Find the expected ualue of the Harmonic oscillator function Un lx) =Nexp ( *z) Hn Lx)
A: it is required to find the expected value of the given harmonic oscillator function. Here the given…
Q: (a) Show that the transformation Q = p + iaq, P = (p − iaq) / (2ia) is canonical and find a…
A:
Q: There is a pendulum in an elevator going down with constant velocity v. Assuming the mass of the…
A:
Q: 12.1 Consider the nonlinear first-order equation x = 2√x - 1. (a) By separating solution x₁(1). (b)…
A:
Q: In the physical system of unit particles in a one-dimensional harmonic oscillator described at time…
A:
Q: If atoms of the same mass form a linear chain in the normal mode with the force constants between…
A: The dispersion relation gives the relation between the angular frequency and wavevector of the wave…
Q: Simple Harmonic Motion is defined as a periodic motion of a point along a straight line, such that…
A: Simple harmonic motion is defined as an oscillatory motion in which the acceleration of the particle…
Q: Example 1: A particle is executing simple harmonic motion of period Tabout a centre O andit passes…
A: Let the time measured from A, then X=acos(muot)
Q: Given an infinite well of length 0 to L, and an initial wavefunction which is a tent shaped…
A:
Q: O A Simple Harmonic oscillator of mass m and natural freguency Wo, driving 4t)- maloscot .…
A: Given condition at t=o, The given differential equation The solution of this equation is sum of…
Q: The energy E of a system of three independent harmonic oscillators is given by 1 1 E = (nx + ½ )ħw +…
A:
Q: 2) Consider a particle in a three-dimensional harmonic oscillator potential V (x, y, 2) = ;mw (x² +…
A: Let the given particle is the electron. Then the transition probability (W) be given as, Where, ω…
Q: Find the Fourier series of a periodic wave f(x+ f(x) = 2c if -L < x< 0, f(x) = 3c if 0 < x < L. 2L)…
A: Given: The function is given by: fx=2c if -L≤x<0fx=3c if 0≤x<L Introduction: A…
Q: Consider the one-dimensional linear chain in which the force constants neighbor atoms are…
A: The dispersion relation for the frequency is ω2=1MC1+C2±C12+C22+2C1C2coska (a) The value of ω2 for…
Q: 3. A simple harmonic oscillator makes 25 complete vibrations after 5.0 seconds. Determine its (a)…
A:
Q: Assume that the given function is periodically extended outside the original range. Find the…
A: The Fourier series of a periodic function f(x) with period 2L can be written as the sum of an…
Q: Using the matrix expression for A and of the harmonic oscillator, find the matrix representations of…
A:
Q: Prob.1 (1) State the required conditions of simple harmonic motion (SHH).
A: We need to find the necessary condition for simple harmonic motion.
Q: Show that the minimum energy of a simple harmonic oscillator is hω/2. What is the minimum energy in…
A: The mean value of x2av is the mean square devation ∆x2By substituting the mean square deviation…
Q: State the required conditions of simple harmonic motion (SHH). Consider the torsional pendulum with…
A: (i) There are two main conditions for a motion to be classified as SHM: The magnitude of the…
Q: f(x, y) = (cos(x) + cos(yV3/2 – a/2) + cos(yv3/2+x/2))² Plot the contours of this function for -3…
A: A contour plot is a graphical method to visualize the 3-D surface by plotting constant Z slices…
Q: How to solve %diff in t just for on secti
A: Solution: 1. The mathematical expression for the time period is given by, T=2πMTK From the above…
Q: The Longrongion of 1D harmonic is, ()- write Euler-Congrange equation of system? ().write…
A:
Q: Write the vector function for the harmonic oscillator using the generator function, then find the…
A: The generator function for the harmonic oscillator is given by: G(u) = (mω^2/2u^2 - iu/2)^(-1/2)…
Q: mass of 12 slugs is hanging at rest on a frictionless spring whose constant is k = 1/3 . Beginning…
A: Given: The spring constant k=1/3. The external force is Ft=20 cosωt. To find: (a) The…
Q: Show that the function x(t) = A cos ω1t oscillates with a frequency ν = ω1/2π. What is the frequency…
A:
Q: A laminar boundary layer profile may be assumed to be approximately of the form u/U₁ = f(n) = f(y/6)…
A: Step 1: Understanding the Velocity ProfileThe velocity profile is given in two segments:where…
Q: Show that the harmonic oscillator Hamilonin Can : be writteu as: 2.
A:
Q: Exercise b. A 24 lb weight stretches a spring 12 inches. If the weight is pulled 10 inches below the…
A:
Q: Coupled Flarmonic Oscillators X2 : 0 ) Write down the 2nd law for each of the masses. Use…
A: We will only answer the first question since the exact question to be answered was not specified.…
Q: Sketch f(y) versus y. Show that y is increasing as a function of t for y < 1 and also for y >…
A:
Q: A mass of 55 grams stretches a spring by 8 cm. (Note that this means the forces balance, and thus mg…
A: m = 55 gram g = 981 m/s2 x= 8cm Initial velocity v= 23 cm/s = 0.23 m/s We can see that the particle…
Q: A double pendulum consists of two equal masses 'm' suspended by two strings of length L. What is the…
A: Solution: The free-body diagram is
Q: The Longrongion of 1D harmonic oscilator 1 = ² ² ²x² ах 2 m. 2 () write Euler-Longronge equation of…
A: Lagrangian is difference of kinetic and potential energy.…
Q: By using hamiltonian equations. Find the solution of harmonic oscillator in : A-2 Dimensions B-3…
A: For a Harmonic oscillator, the Kinetic energy T and Potential Energy V are given by, Considering…
Q: Simple Harmonic Motion is defined as a periodic motion of a point along a straight line, such that…
A: Motion of particle: x(t) = cos2t + 2sin2t
Q: What is the curl of the linear restoring force for an isotropic harmonic oscillator?
A: The linear restoring force for an isotropic harmonic oscillator is given by Where k = Oscillator…
Show that for an underdamped harmonic oscillator (i.e., b < 2/sqrt(mk)), the position function can be expressed as x(t) = Be^(-(b/2m)t) cos(wt – phi) where w^2 = (4mk–b^2)/4m^2 and B is any constant.
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
- Consider 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…Find the volume of the solid cut from the thick-walled cylinder 4≤x^2+y^2≤7 by the cones z=±(4x^2+4y^2)^1/2.Suppose function fhas the graph as shown below
- 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?Find the Fourier series representation of the periodic function below if m = 5 and k = 7. Then, evaluate the first few terms (up to n = 7) of the series at x = 0.615.A spring/mass/dashpot system has mass 9 kg, damping constant 288 kg/sec and spring constant 3249 kg/sec/sec. Express the ODE for the system in the form x"+2px'+wx = 0 Identify the natural (undamped) frequency of the spring: (square Hz) wo = Identify the parameter p: p= (Hz) Now assume that the system has the oscillating forcing function cos(wot) with the same frequency as the spring's natural frequency. Complexify the ODE and use the real part as a particular solution: x"+32x+361x = cos(wot) Xp= (meters)