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 3.36P
To determine
The expression for the displacement of a linear oscillator analogous to Equation 3.110 for the initial conditions
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
The θ integration fails if α is non-constant. There is also a type of situation where the step before the integration fails, involving ω.
Question 3: derive θ = θ0 + (ω + ω0) t/2.
You probably need to use prior derived results.
Our unforced spring mass model is mx00 + βx0 + kx = 0 with m, β, k >0. We know physically that our spring will eventually come to rest nomatter the initial conditions or the values of m, β, or k. If our modelis a good model, all solutions x(t) should approach 0 as t → ∞. Foreach of the three cases below, explain how we know that both rootsr1,2 =−β ± Sqrt(β^2 − 4km)/2mwill lead to solutions that exhibit exponentialdecay.(a) β^2 − 4km > 0.
(b) β^2 − 4km =0.
(c) β^2 − 4km >= 0.
Consider the solution tothe harmonic oscillator given above by x(t)=Ccos(wt−v)
Prove tha
tx(t0)=x(t0+2piw)
In other words, the solution has the same value at time:t0 and at time:t0+2piw
regardless of what value we have for ?0. The value 2piw is then the period T of the harmonic oscillator.
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
- A body of mass m is suspended by a rod of length L that pivots without friction (as shown). The mass is slowly lifted along a circular arc to a height h. a. Assuming the only force acting on the mass is the gravitational force, show that the component of this force acting along the arc of motion is F = mg sin u. b. Noting that an element of length along the path of the pendulum is ds = L du, evaluate an integral in u to show that the work done in lifting the mass to a height h is mgh.arrow_forwardA hinged rigid bar of length l is connected by two springs of stiffnesses K1 and K2 and is subjected to a force F as shown in Fig. 1.33(a). Assuming that the angular displacement of the bar (θ) is small, find the equivalent spring constant of the system that relates the applied force F to the resulting displacement x.arrow_forwardNote: Because the argument of the trigonometric functions in this problem will be unitless, your calculator must be in radian mode if you use it to evaluate any trigonometric functions. You will likely need to switch your calculator back into degree mode after this problem.A massless spring is hanging vertically. With no load on the spring, it has a length of 0.24 m. When a mass of 0.59 kg is hung on it, the equilibrium length is 0.98 m. At t=0, the mass (which is at the equilibrium point) is given a velocity of 4.84 m/s downward. At t=0.32s, what is the acceleration of the mass? (Positive for upward acceleration, negative for downward)arrow_forward
- Anisotropic Oscillator Consider a two-dimensional anisotropic oscillator is rational (that is, ws/wy = p/q 4. Prove that if the ratio of the frequencies w and wy where p and q are integers with gcd(p, q) = 1, where gcd stands for greatest common divisor) then the motion is periodic. Determine the period. Parrow_forwardGenerate a function f(t)=y depending on a single input parameter time (t), in which the result (y) will have a declining acceleration for the first 10 years.arrow_forwardWrite the equations that describe the simple harmonic motion of a particle moving uniformly around a circle of radius8units, with linear speed 3units per second.arrow_forward
- The θ integration fails if α is non-constant. There is also a type of situation where the step before the integration fails, involving ω. Question 1: derive ω = αt + ω0. Start with the derivatives I began with, and derive this equation. Question 2: derive θ = 1/2αt2 + ω0t + θ0. You may use prior derived results. Question 3: derive θ = θ0 + (ω + ω0) t/2. You probably need to use prior derived results.arrow_forwardThe Duffing oscillator with mass m is described by the non-linear second order DE d? 3 + ax + Bx° = y sin (wt), dt? where [x] = L and [t] = T. Calculate the dimensions of , B, y and w. Answer: a [B] = [w] = ||arrow_forwardThe diagram below shows an infinitely long straight wire with a current I that is directed in the +x direction. The ultimate goal of this problem is to use the Biot Savart Law to derive the magnetic field by the wire at point P which is a distance d from the wire. P +y I ds, 1 dB 1₂ - [H-1² 4T +z ds 2 The Biot Savart Law gives an expression of the magnetic field by a infinitesimal segment of the wire. Hods x f p2 +x Notice that the direction of the field corresponding to a current segment (dB) is given by the direction of the cross product ds x î.arrow_forward
- If a mass m is placed at the end of a spring, and if the mass is pulled downward and released, the mass-spring system will begin to oscillate. The displacement y of the mass from its resting position is given by a function of the form y = c,cos wt + c2 sin wt (1) where w is a constant that depends on spring and mass. Show that set of all functions in (1) is a vector space.arrow_forwardFind the gradient of w = x2y3z at (1, 2, −1).arrow_forwardThe motion of the point K moving along the plane is given by the equation x = 3sin (t), y = 2cos (t). Find the equation of the track where the punk is moving.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
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning