2 Consider a particle of mass m moving in the region x > 0 under the influence of a potential U (x) = Uo а xc + х а where Uo and a are positive constants. Sketch the potential, find the equilibrium point, and determine its sta- (a) bility. Find the frequency of small oscillations about the stable equilibrium (b) point.

2 Consider a particle of mass m moving in the region x > 0 under the influence of a potential U (x) = Uo а xc + х а where Uo and a are positive constants. Sketch the potential, find the equilibrium point, and determine its sta- (a) bility. Find the frequency of small oscillations about the stable equilibrium (b) point.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter3: Oscillations

Section: Chapter Questions

Problem 3.44P: Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped...

See similar textbooks

Similar questions

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.
C, N A uniform plank of length L and mass M is balanced on a fixed, semicircular bowl of radius R (Fig. P16.19). If the plank is tilted slightly from its equilibrium position and released, will it execute simple harmonic motion? If so, obtain the period of its oscillation.
Show that, if a driven oscillator is only lightly damped and driven near resonance, the Q of the system is approximately Q2(TotalenergyEnergylossduringoneperiod)
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.
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.
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.
Refer to the problem of the two coupled oscillators discussed in Section 12.2. Show that the total energy of the system is constant. (Calculate the kinetic energy of each of the particles and the potential energy stored in each of the three springs, and sum the results.) Notice that the kinetic and potential energy terms that have 12 as a coefficient depend on C1 and 2 but not on C2 or 2. Why is such a result to be expected?
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.
Consider a damped harmonic oscillator. After four cycles the amplitude of the oscillator has dropped to 1/e of its initial value. Find the ratio of the frequency of the damped oscillator to its natural frequency.
Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.
Check Your Understanding Identify one way you could decrease the maximum velocity of a simple harmonic oscillator.
Consider an undamped linear oscillator with a natural frequency ω0 = 0.5 rad/s and the step function a = 1 m/s2. Calculate and sketch the response function for an impulse forcing function acting for a time τ = 2π/ω0. Give a physical interpretation of the results.