Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 3, Problem 3.5P
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.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
What is the smallest positive phase constant (ø) for the harmonic oscillator with the position function x(t) given in the figure below if the position function has the form x = Acos(ωt+ø)? The vertical axis scale is set by xs = 6.0 cm. Answer in radians. Hint: Evaluate at t=0 and solve for phi.
In the case b =6, find the physical form of the solution, the damping factor, envelope curves and quasiperiod.
Sketch the graph for b=6, 8 and 10.
Problem 5: Consider a 1D simple harmonic oscillator (without damping). (a) Compute
the time averages of the kinetic and potential energies over one cycle, and show that they
are equal. Why does this make sense? (b) Show that the space averages of the kinetic and
potential energies are
(T)₂ = k1²
KA² and (U),= KA².
Why is this a reasonable result?
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...
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- 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_forwardC, 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_forwardAllow 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_forward
- 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.arrow_forwardConsider the function ƒ(x) = 4 sin(( 3)) + 6. State the amplitude A. period P 2 A, a midline. State the phase shift and vertical translation. In the full period [0, P], state the maximum a N minimum y-values and their corresponding a-values. Enter the exact answers. Amplitude: A = 4 Period: P = Bù Midline: y = Number The phase shift is Click for List The vertical translation is Click for List 1 Hints for the maximum and minimum values of f(x): π and = 2 프 The maximum value of y = sin(x) is y = 1 and the corresponding a values are x multiples of 2 7 less than and more than this a value. You may want to solve ( x − 3) = 2 3T and • The minimum value of u = sin (r) is 71 sin (r) is = -1 and the corresponding r values are Submit Assignment Quit & Savearrow_forwardFind the amplitude, period and phase shift of the graph of y = 2sin+) – 1, and sketch its graph over one period. Find the range of the functionarrow_forward
- Here are several solutions curves in the phase plane. Which of these could be solutions of a harmonic oscillatorarrow_forwardFor a simple harmonic oscillator with x = A sin ot write down an expression for the velocity. Please use II* II for products (e.g. B*A), "/" for ratios (e.g. B/A) and the usual "+" and "-" signs as appropriate (without the quotes). For trigonometric functions use the usual sin and cos, while for Greek letters such as w, use omega. Please use the "Display response" button to check you entered the answer you expect.arrow_forwardFor a simple harmonic oscillator with x = A sin wt write down an expression for the magnitude of acceleration. Please use II * II for products (e.g. B*A), "/" for ratios (e.g. B/A) and the usual "+" and "-" signs as appropriate (without the quotes). For power (e.g. A) use A^2 or A*A. For trigonometric functions use the usual sin and cos, while for Greek letters such as w, use omega. Please use the "Display response" button to check you entered the answer you expect.arrow_forward
- For a simple harmonic oscillator with x = A sin ot write down an expression for the velocity. Please use "*" for products (e.g. B*A), "/" for ratios (e.g. B/A) and the usual "+" and "-" signs as appropriate (without the quotes). For trigonometric functions use the usual sin and cos, while for Greek letters such as w, use omega. Please use the "Display response" button to check you entered the answer you expect. U= Display responsearrow_forwardSolve the following questions relating to the Harmonic Oscillator.arrow_forwardExplain the term amplitude as it relates to the graph of asinusoidal function.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
SIMPLE HARMONIC MOTION (Physics Animation); Author: EarthPen;https://www.youtube.com/watch?v=XjkUcJkGd3Y;License: Standard YouTube License, CC-BY