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 12, Problem 12.4P
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 κ12 as a coefficient depend on C1 and ω2 but not on C2 or ω2. Why is such a result to be expected?
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
A wheel is free to rotate about its fixed axle. A spring is attached to one of its spokes at distance r from the axle. Assuming that the wheel is a hoop of mass m and radius R, what is the angular frequency w of small oscillations of this system in terms of m, R, r, and the srping constant k? What is w if r =R and if r = 0?
Thank you for the help. I think I'm tripping myself up with using r in the equations.
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?
Consider the simple pendulum: a ball hanging at the end of a string. Derive the expression for the period of this physical pendulum, taking into account the finite size
ball (i.e. the ball is not a point mass). Assume that the string is massless. Start with the expression for the period T'of a physical pendulum with small amplitude oscillati
T
= 2π
The moment of inertia of the ball about an axis through the center of the ball is
Here, I, is the moment of inertia about an axis through the pivot (fixed point at the top of the string, m is the mass of the ball, g is the Earth's gravitational constant of
acceleration, and h is the distance from the pivot at the top of the string to the center of mass of the ball.
Note, this pre-lab asks you to do some algebra, and may be a bit tricky.
I
mgh
Iball = / mr²
T
Chapter 12 Solutions
Classical Dynamics of Particles and Systems
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
- 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_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 potential given by U(x) = -9 cos(15 x). This potential has a stable equilibrium at x = 0. Calculate the period of small oscillation, in s, of a 0.9 kg mass about that equilibrium. (Please answer to the fourth decimal place - i.e 14.3225)arrow_forward
- Consider a simple pendulum of length L with a mass m. Derive the angular frequency of oscillation for the pendulum when it is displaced by a small angle. Assume it is in a gravitational field with the magnitude of acceleration due to gravity equal to g. (That is, you need to find MOI and d for a simple pendulum and then use the equation for ω for a simple pendulum.)arrow_forwardScientists have developed a clever way to measure a mass of virus using a spring. A cantilever beam in the scanning electron microscope image below is like a diving board, except that it is extremely small (a couple of micrometer). The cantilever beam with mass m can oscillate (imagine a vibrating diving board) and it can be modeled as a spring with a spring constant k. What you can measure experimentally is the frequency of oscillation of the cantilever first without the virus (f1) and after the virus had attached itself to the cantilever (f2). (a) Find the mass of virus from f1 and f2 (assume that we don’t know the spring constant k) (b) Suppose the mass of cantilever is 10.0 * 10^-16 g and a frequency of 2.00 * 10^15 Hz without the virus and 2.87 * 10^14 Hz with the virus. What is the mass of the virus?arrow_forwardWhat 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.arrow_forward
- Consider a non-linear oscillator governed by the differential equation x''(t) - ϵx[x'(t)] + x = 0, where the initial conditions are given by x(0)=xo and v(0)=0. Employ the Lindstedt method to expand up to the second order. Specifically, determine the three sets until the order of ϵ^2. Next, calculate the values of ω1 and ω2, and use them for x(t) up to the 2nd order.arrow_forwardTwo blocks of masses m1=1.0 kg and m2=3 kg are connected by an ideal spring of force constant k=4 N/m and relaxed length L. If we make them oscillate horizontally on a frictionless surface, releasing them from rest after stretching the spring, what will be the angular frequency ω of the oscillation? Choose the closest option. Hint: Find the differential equation for spring deformation.arrow_forwardThe figure depicts the displacement of an oscillator x[m], as a function of time t[s] 2 1 -1 -2 -3 0 0.5 1 1.5 Mass of the oscillator is 1.1 kg. What is the oscillator's amplitude (with more than 2 significant digits)? A = 30*10^-1 m 2 Your last answer was interpreted as follows: 30-10-¹ What is the oscillator's angular velocity (with more than 2 significant digits)? w = 9.4*10^0 rad 2.5 3 3.5 4 Your last answer was interpreted as follows: 9.4. 10⁰ Write the equation for oscillator's position in the following format: x = A sin (wt + 6)? Insert the answer as follows: x = (A)*sin((omega)*t + (phi)), where (A), (omega). (phi) are replaced with numerical values of amplitude, angular velocity and smallest positive phase angle rounded to one decimal after the decimal separator.arrow_forward
- A solid cylinder of mass M, radius R, and length L, rolls around a light, thin axle at its center. The axle is attached to a spring of stiffness K on either side. The springs are attached to same wall. The cylinder is brought back a distance A, parallel to the wall, and allowed to oscillate. What is the maximum speed of the oscillations of the cylinder? Answer: ((3*k*A^2)/(2*M))^1/2arrow_forwardFor the damped oscillator system, withm = 250 g, k = 85 N/m, and b = 70 g/s, what is the ratio of the oscillationamplitude at the end of 20 cycles to the initial oscillationamplitude?arrow_forwardanswer both Q3:An object undergoes simple harmonic motion. When it stops atits turnaround points, which of the following statements is trueabout its potential energy U and kinetic energy K?minimum U, minium Kmaximum U, maximum Kmaximum U, minimum Kminimum U, maximum KNone of the above 4. And which of the following statements is true about the force(s)magnitude(s) acting on the object at the turnaround points?maximum forceminimum forceInsufficient informationarrow_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
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning
SIMPLE HARMONIC MOTION (Physics Animation); Author: EarthPen;https://www.youtube.com/watch?v=XjkUcJkGd3Y;License: Standard YouTube License, CC-BY