Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
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Question
Chapter 3, Problem 3.38P
To determine
The response of a damped oscillator to a forcing function of the form
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Chapter 3 Solutions
Classical Dynamics of Particles and Systems
Ch. 3 - Prob. 3.1PCh. 3 - Allow the motion in the preceding problem to take...Ch. 3 - Prob. 3.3PCh. 3 - Prob. 3.4PCh. 3 - Obtain an expression for the fraction of a...Ch. 3 - Two masses m1 = 100 g and m2 = 200 g slide freely...Ch. 3 - Prob. 3.7PCh. 3 - Prob. 3.8PCh. 3 - A particle of mass m is at rest at the end of a...Ch. 3 - If the amplitude of a damped oscillator decreases...
Ch. 3 - Prob. 3.11PCh. 3 - Prob. 3.12PCh. 3 - Prob. 3.13PCh. 3 - Prob. 3.14PCh. 3 - Reproduce Figures 3-10b and c for the same values...Ch. 3 - Prob. 3.16PCh. 3 - For a damped, driven oscillator, show that the...Ch. 3 - Show that, if a driven oscillator is only lightly...Ch. 3 - Prob. 3.19PCh. 3 - Plot a velocity resonance curve for a driven,...Ch. 3 - Let the initial position and speed of an...Ch. 3 - Prob. 3.26PCh. 3 - Prob. 3.27PCh. 3 - Prob. 3.28PCh. 3 - Prob. 3.29PCh. 3 - Prob. 3.30PCh. 3 - Prob. 3.31PCh. 3 - Obtain the response of a linear oscillator to a...Ch. 3 - Calculate the maximum values of the amplitudes of...Ch. 3 - Consider an undamped linear oscillator with a...Ch. 3 - Prob. 3.35PCh. 3 - Prob. 3.36PCh. 3 - Prob. 3.37PCh. 3 - Prob. 3.38PCh. 3 - Prob. 3.39PCh. 3 - An automobile with a mass of 1000 kg, including...Ch. 3 - Prob. 3.41PCh. 3 - An undamped driven harmonic oscillator satisfies...Ch. 3 - Consider a damped harmonic oscillator. After four...Ch. 3 - A grandfather clock has a pendulum length of 0.7 m...
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- Show that, if a driven oscillator is only lightly damped and driven near resonance, the Q of the system is approximately Q2(TotalenergyEnergylossduringoneperiod)arrow_forwardConsider 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.arrow_forwardA 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.arrow_forward
- 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.arrow_forwardIf the amplitude of a damped oscillator decreases to 1/e of its initial value after n periods, show that the frequency of the oscillator must be approximately [1 − (8π2n2)−1] times the frequency of the corresponding undamped oscillator.arrow_forwardThe total energy of a simple harmonic oscillator with amplitude 3.00 cm is 0.500 J. a. What is the kinetic energy of the system when the position of the oscillator is 0.750 cm? b. What is the potential energy of the system at this position? c. What is the position for which the potential energy of the system is equal to its kinetic energy? d. For a simple harmonic oscillator, what, if any, are the positions for which the kinetic energy of the system exceeds the maximum potential energy of the system? Explain your answer. FIGURE P16.73arrow_forward
- 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.arrow_forwardFor a damped, driven oscillator, show that the average kinetic energy is the same at a frequency of a given number of octaves* above the kinetic energy resonance as at a frequency of the same number of octaves below resonance.arrow_forwardCalculate the maximum values of the amplitudes of the response functions shown in Figures 3-22 and 3-24. Obtain numerical values for β = 0.2ω0 when a = 2 m/s2, ω0 = 1 rad/s, and t0 = 0.arrow_forward
- 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.arrow_forwardAn object of mass m on a spring of stiffness k oscillates with an amplitude A about its equilibrium position. Suppose that m = 300 g, k = 10 N/m, and A = 10 cm. (a) Find the total energy. (b) Find the mechanical frequency of vibration of the mass. (c) Calculate the change in amplitude when the system loses one quantum of energy.arrow_forwardShow that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt = bv2 and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, E=12mv2+12kx2, and use Equation 12.28.arrow_forward
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