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
Question
Chapter 3, Problem 3.4P
To determine
The time averages of the kinetic and potential energies over one cycle and show that these quantities are equal. And determine the space averages of the kinetic and potential energies.
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
True or False and please explain why:
In a simple harmonic oscillator without dissipation (friction, drag, resistance, etc.) the energy is conserved.
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?
A particle of mass m is in a potential given by
aro
a
V (r) =
3r3
where a and
are positive constants. When disturbed slightly from its stable
equilibrium position it undergoes a sımple harmonic oscillation. The time period of
oscillation is
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
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.arrow_forwardShow 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_forward
- Check Your Understanding Identify one way you could decrease the maximum velocity of a simple harmonic oscillator.arrow_forwardConsider a particle moving in the region x > 0 under the influence of the potential where U0 = 1 J and α = 2 m. Plot the potential, find the equilibrium points, and determine whether they are maxima or minima.arrow_forwardConsider 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.arrow_forward
- 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.arrow_forwardCalculate the energy, corrected to first order, of a harmonic oscillator with potential:arrow_forwardCalculate the velocity of a simple harmonic oscillator with amplitude of 29 cm and frequency of 5 Hz at a point located 5 cm away from the equilibrium position. Give your answer in SI units. Answer: Choose...arrow_forward
- A harmonic oscillator has angular frequency w and amplitudeA. (a) what are the magnitudes of the displacement and velocity when the elastic potential energy is equal to the kinetic energy? (Assume that Uet = 0 at equilibrium.) (b) How often does this occur in each cycle? what is the time between occurrences? (c) At an instant when the displacement is equal to, what fraction of the total energy of the system is kinetic and what fraction is potential?arrow_forwardHere are several solutions curves in the phase plane. Which of these could be solutions of a harmonic oscillatorarrow_forward2 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.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
University Physics Volume 1PhysicsISBN:9781938168277Author:William Moebs, Samuel J. Ling, Jeff SannyPublisher:OpenStax - Rice University
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning
University Physics Volume 1
Physics
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:OpenStax - Rice University