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 3, Problem 3.9P
A particle of mass m is at rest at the end of a spring (force constant = k) hanging from a fixed support. At t = 0, a constant downward force F is applied to the mass and acts for a time t0. Show that, after the force is removed, the displacement of the mass from its equilibrium position (x = x0, where x is down) is
where
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
A mass hanging from a spring is set in motion and its ensuing velocity is given by v(t) = - 4x sin at for t20.
Assume that the positive direction is upward and s(0) = 4.
b) If a mass mis 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 + c, sin wt
(1)
where w is a constant that depends on spring and mass. Show that set of al
functions
in
(1)
is
a
vector
space.
A 0.175 kg mass on a table is attached to a horizontal spring with a spring constant of 155 N/m. Ignore all friction and air resistance in this problem (i.e. assume no non-conservative forces act on the system).
Assuming you call time t=0 the time when the mass is moving to the left (in the negative direction) through equilibrium, sketch a graph of the horizontal displacement of the mass vs. time for two full periods. Make sure to label your axes correctly and to mark the numerical values of your amplitude and period on the graph.
Determine whether the mass is located to the left or the right of equilibrium at t=0.240 s.
Determine if the mass is moving to the left or to the right at t=0.240 s.
At some time you make a measurement of the mass and find that it is moving at a speed of 0.815 m/s and is 0.03 m from equilibrium. Find the amplitude of the oscillation.
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
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
- An undamped spring-mass system with a mass weighing 9 lb and a spring constant of 6 lb/in is set in motion at t = 0 by an external force of 4 cos 9t lb. Find the position u (in feet) at any time t (in seconds). u(t) = Assume that acceleration due to gravity is 32ft/s².arrow_forwardIf 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_forwardA hollow steel ball weighing 24 pounds is suspended from a spring. This stretches the spring 4 inches. The ball is started in motion from a point 3 inches above the equilibrium position. Let u(t) be the displacement of the mass from equilibrium. Suppose that after t seconds the ball is u feet below its rest position. Find u (in feet) in terms of t. (Note that the positive direction is down.) Take as the gravitational acceleration 32 feet per second per second. u=arrow_forward
- A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by y = 1/6 sin(8t) + 1/8 cos(8t) where y is the displacement (in feet) from equilibrium of the weight and t is the time (in seconds). (a) Use the identity a sin(Bθ) + b cos(Bθ) = a2 + b2 sin(Bθ + C) where C = arctan(b/a), a > 0, to write the model in the form A) y = (b) Find the amplitude of the oscillations of the weight. (c) Find the frequency of the oscillations of the weightarrow_forwardAn object of mass 3 grams is attached to a vertical spring with spring constant 27 grams/secʻ. Neglect any friction with the air. (a) Find the differential equation y" = f(y, y') satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y" : -9y Σ (b) Find r1, r2, roots of the characteristic polynomial of the equation above. r1,r2 = Зі, - 3і Σ (b) Find a set of real-valued fundamental solutions to the differential equation above. Y1(t) = cos(3t) Σ Y2(t) = sin(3t) Σ (c) At t = 0 the object is pulled down 1 cm and the released with an initial velocity downwards of 3/3 cm/sec. Find the amplitude A > 0 and the phase shift o E (-1, 7| of the subsequent movement. A = Σφ Σarrow_forwardAn object of mass 2 grams is attached to a vertical spring with spring constant 32 grams/sec“. Neglect any friction with the air. (a) Find the differential equation y" = f(y, y') satisfied by the function y, the displacement of the object from its equilibrium position, positive downwards. Write y for y(t) and yp for y' (t). y" = Σ (b) Find r1, r2, roots of the characteristic polynomial of the equation above. r1, r2 = Σ (b) Find a set of real-valued fundamental solutions to the differential equation above. Y1 (t) = Σ Y2(t) = Σ (c) At t = 0 the object is pulled down v2 cm and the released with an initial velocity upwards of 4/2 cm/sec. Find the amplitude A > 0 and the phase shift o E (-1, 7] of the subsequent movement. A = ΣΦ Σarrow_forward
- A carriage runs along rails on a rigid beam. The carriage is attached to one end of a spring of equilibrium length r0 and force constant k, whose other end is fixed on the beam. On the carriage, another set of rails is perpendicular to the first along which a particle of mass m moves, held by a spring fixed on the beam, of force constant k and zero equilibrium length. Beam, rails, springs, and carriage are assumed to have zero mass. The whole system is forced to move in a plane about the point of attachment of the first spring, with a constant angular speed ω. The length of the second spring is at all times considered small compared to r0. Using generalized coordinates in the laboratory system, what is the Jacobi integral for the system? Is it conserved?arrow_forwardConsider a massless pendulum of length L and a bob of mass m at its end moving through oil. The massive bob undergoes small oscillations, but the oil regards the bob’s motion with a resistive force proportional to the speed with Fd=-b*θ. The bob is initially pulled back at t=0 with θ = αlpha with zero velocity. (a) Write down the differential equation governing the motion of the pendulum. (b) Find the angular displacement as a function of time by solving (a). Assume that b is smaller than the natural frequency (frequency in the absence of damping) of the pendulum. (c) Find the mechanical energy of the pendulum as a function of time. (d) Find the time when the mechanical energy decays to 1/e of its initial value.arrow_forwardA 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_forward
- A carriage runs along rails on a rigid beam. The carriage is attached to one end of a spring of equilibrium length r0 and force constant k, whose other end is fixed on the beam. On the carriage, another set of rails is perpendicular to the first along which a particle of mass m moves, held by a spring fixed on the beam, of force constant k and zero equilibrium length. Beam, rails, springs, and carriage are assumed to have zero mass. The whole system is forced to move in a plane about the point of attachment of the first spring, with a constant angular speed ω. The length of the second spring is at all times considered small compared to r0. What is the energy of the system? Is it conserved?arrow_forwardA cube, whose mass is 0.680 kg, is attached to a spring with a force constant of 122 N/m. The cube rests upon a frictionless, horizontal surface (shown in the figure below). m The cube is pulled to the right a distance A = 0.120 m from its equilibrium position (the vertical dashed line) and held motionless. The cube is then released from rest. (a) At the instant of release, what is the magnitude of the spring force (in N) acting upon the cube? N (b) At that very instant, what is the magnitude of the cube's acceleration (in m/s2)? m/s2 (c) In what direction does the acceleration vector point at the instant of release? Away from the equilibrium position (i.e., to the right in the figure). The direction is not defined (i.e., the acceleration is zero). Toward the equilibrium position (i.e., to the left in the figure). You cannot tell without more information.arrow_forwardA 1.8 kg mass attached to a horizontal spring oscillates at a frequency of 2.1 Hz. At t =0 s, the mass is at x= 2 m and has vx =− 3 m/s. What is the total energy (in Joules)?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
SIMPLE HARMONIC MOTION (Physics Animation); Author: EarthPen;https://www.youtube.com/watch?v=XjkUcJkGd3Y;License: Standard YouTube License, CC-BY