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 3.8P
To determine
Show that the oscillations are exactly isochronous with a frequency
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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.
Problems 5 & 6 refer to the mass-spring oscillator
depicted in the figure on the right. The block has
a mass of 350 g, and is attached to a spring with a
spring constant of k
pushed 20 cm to left from C, its equilibrium position,
before being released and allowed to move horizon-
tally on a frictionless surface. .
Ax = 20 cm
45 N. The mass is initially
m
A B
DE
5. What is the frequency of oscillation for the block?
A. 0.13 Hz
В. 0.35 Hz
С. 1.8 Hz
D. 7.8 Hz
5.
6. What is the speed of the block as it passes point B?
A. 0.50 m/s
В. 1.0 m/s
C. 1.5 m/s
D. 2.0 m/s
6.
It is known that a general solution for the position of a
harmonic oscillator is
x(t) = C cos (wt) + S sin (wt),
where C, S, and ware constants. (Figure 2)
Your task, therefore, is to determine the values of C, S,
and w in terms of k, m,and init and then use the
connection between x (t) and a (t) to find the
acceleration.
Figure
L
min
x=0
Xinit
1 of 2
>
Part A
Combine Newton's 2nd law and Hooke's law for a spring to find the acceleration of the block a (t) as a function of time.
Express your answer in terms of k, m, and the coordinate of the block x (t).
► View Available Hint(s)
a (t) =
Submit
17 ΑΣΦ
Part B Complete previous part(s)
Review | Constants
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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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