Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
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Chapter 3, Problem 3.14P
To determine
The displacement
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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. P
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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- Let the initial position and speed of an overdamped, nondriven oscillator be x0 and v0, respectively. (a) Show that the values of the amplitudes A1 and A2 in Equation 3.44 have the values A1=2x0+v021 and A2=1x0+v021 where 1 = 2 and 2 = + 2. (b) Show that when A1 = 0, the phase paths of Figure 3-11 must be along the dashed curve given by x=2x, otherwise the asymptotic paths are along the other dashed curve given by x=1x. Hint: Note that 2 1 and find the asymptotic paths when t .arrow_forwardSimple Harmonic Motion is defined as a periodic motion of a point along a straight line, such that its acceleration is always towards a fixed point in that line and is proportional to its distance from that point. In other words, its acceleration is proportional to the displacement but acts in an opposite direction. Prove algebraically that the motion of a particle defined by x=cos2t + 2sin2t is simple harmonic.arrow_forwardSimple Harmonic Motion is defined as a periodic motion of a point along a straight line, such that its acceleration is always towards a fixed point in that line and is proportional to its distance from that point. In other words, its acceleration is proportional to the displacement but acts in an opposite direction. Prove algebraically that the motion of a particle defined by x = cos2t + 2sin2t is simple harmonic.arrow_forward
- 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_forwardEx. 16 : A particle performing S.H.M. has a velocity of 10 m/s, when it crosses the mean position. If the amplitude of oscillation is 2 m, find the velocity when it is midway between mean and extreme position.arrow_forwardthe general solution to a harmonic oscillator are related. There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t: 1. x(t) = A cos (wt + p) and 2. x(t) = C cos (wt) + S sin (wt). Either of these equations is a general solution of a second-order differential equation (F= mā); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.) Part D Find analytic expressions for the arbitrary constants A and in Equation 1 (found in Part A) in terms of the constants C and Sin Equation 2 (found in Part B), which are now considered as given parameters. Express the amplitude A and phase (separated by a comma) in terms of C and S. ► View Available Hint(s) Α, φ = V ΑΣΦ ?arrow_forward
- A certain oscillator satisfies the equation of motion: ä + 4x = 0. Initially the particle is at the point x = V3 when it is projected towards the origin with speed 2. 2.1. Show that the position, x, of the particle at any given time, t, is given by: x = V3 cos 2t – sin 2t. (Note: the general solution of the equation of motion is given by: x = A Cos 2t + B Sin 2t, where A and B are arbitrary constants)arrow_forwardFind the undamped position too plsarrow_forwardSuppose that a mass is initially at X = Xo with an initial velocity Vo. Show that the resulting motion is the sum of two oscillations, one corresponding to the mass initially at rest at X = Xo and the other corresponding to the mass initially at the equilibrium position with velocity Vo. What is the amplitude of the total oscillation?arrow_forward
- A horizontal spring mass system oscillates on a frictionless plane. At time t=0, it is moving left at position x=9 cm. It has velocity v=0, at positions x=0 cm and 12 cm, and completes one full cycle in 2 seconds. Write the position and velocity kinematic equations for this oscillating system, including the phase constant.arrow_forwardAn over damped harmonic oscillator satisfies the equation, x+10x +16x=0. At time t =0 the particle is projectecd from the point x = I towards the origin with sneed u. Find x(t) in the subsequent motion.arrow_forwardSolve it correctly.arrow_forward
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