No ai please. 11.4** (a) Find the normal frequencies for the system of two carts and three springs shown in Figure11.1, for the case that m₁ = m² and k₁ = k3, (but k₂ may be different). Check that your answer is correctfor the case that k₁ = k₂ as well. (b) Find and describe the motion in each of the two normal modes inturn. Compare with the motion found for the case that k₁ k2 in Section 11.2. Explain any similarities.-k₁k2k3MomFigure 11.1equilibriumx2equilibriumTwo carts attached to fixed walls by the springs labeled k₁ andk3, and to each other by k₂. The carts' positions x₁ and x2 are measured fromtheir respective equilibrium positions.

Answered: Image /qna-images/answer/f73e513b-5778-4f5f-9cea-6702b2aabc22.jpg

Answered: 11.4** (a) Find the normal frequencies…

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter12: Coupled Oscillations

Section: Chapter Questions

Problem 12.16P

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Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction s=0.100 . (a) How far can the spring be stretched without moving the mass? (b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coefficient of friction is k=0.0850 , what total distance does it travel before stopping? Assume it starts at the maximum amplitude.
Plot a velocity resonance curve for a driven, damped oscillator with Q = 6, and show that the full width of the curve between the points corresponding to is approximately equal to ω0/6.
Allow the motion in the preceding problem to take place in a resisting medium. After oscillating for 10 s, the maximum amplitude decreases to half the initial value. Calculate (a) the damping parameter β, (b) the frequency υ1 (compare with the undamped frequency υ0), and (c) the decrement of the motion.
Obtain the response of a linear oscillator to a step function and to an impulse function (in the limit τ → 0) for overdamping. Sketch the response functions.
Obtain an expression for the fraction of a complete period that a simple harmonic oscillator spends within a small interval Δx at a position x. Sketch curves of this function versus x for several different amplitudes. Discuss the physical significance of the results. Comment on the areas under the various curves.
Prove that using x(t)=Asin(t+) will produce the same results for the period for the oscillations of a mass and a spring. Why do you think the cosine function was chosen?
An undamped driven harmonic oscillator satisfies the equation of motion m(d2x/dt2+ 02x) = F(t). The driving force F(t) = F0 sin(t) is switched on at t= 0. (a) Find x(t) for t 0 for the initial conditions x = 0 and v = 0 at t = 0. (b) Find x(t) for = 0 by taking the limit 0 in your result for part (a). Sketch your result for x(t). Hint: In part (a) look for a particular solution of the differential equation of the form x = A sin(t) and determine A. Add the solution of the homogeneous equation to this to obtain the general solution of the inhomogeneous equation.
The expression x = 8.50 cos (2.40 t + /2) describes the position of an object as a function of time, with x in centimeters and t in seconds. What are the a. frequency, b. period, c. amplitude, and d. initial phase of the objects motion? e. What is the position of the particle at t = 1.45 s?
(a) If frequency is not constant for some oscillation, can the oscillation be SHM? (b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?
A particle of mass m moving in one dimension has potential energy U(x) = U0[2(x/a)2 (x/a)4], where U0 and a are positive constants. (a) Find the force F(x), which acts on the particle. (b) Sketch U(x). Find the positions of stable and unstable equilibrium. (c) What is the angular frequency of oscillations about the point of stable equilibrium? (d) What is the minimum speed the particle must have at the origin to escape to infinity? (e) At t = 0 the particle is at the origin and its velocity is positive and equal in magnitude to the escape speed of part (d). Find x(t) and sketch the result.
A simple harmonic oscillator has amplitude A and period T. Find the minimum time required for its position to change from x = A to x = A/2 in terms of the period T.
Show that, if a driven oscillator is only lightly damped and driven near resonance, the Q of the system is approximately Q2(TotalenergyEnergylossduringoneperiod)

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