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 2, Problem 2.52P
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 ω 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.
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
A particle that can move along the x-axis is part of a system with potential energy U(x) = A x 2 − B x , where A and B are positive constants. (a) Make a sketch of U(x) vs x.
(b) Where are the particle’s equilibrium positions? (You may neglect the “point” where x is very large)
(c) Describe how a particle would move if given small displacements from the equilibrium positions. For each of the points you identified in Part A, identify if it is a point of stable or unstable equilibrium.
A box (mass = 100 g) is initially connected to a compressed (x = 80 cm) spring (? = 100 N/m) at point A. It was released and started moving along the horizontal surface (?? = 0.2) until it moves up along the inclined surface (?? = 0.3). The box then stops at point D alone the incline. Consider that ? = 30 m and θ = 30.(d) What is the length of the incline, ?? (Ans. 34.88 m) (e) What is the change in gravitational potential energy from point A to point D? (Ans. 17.10 J) Kindly answer only d and e
Suppose that the potential energy of a particle moving along the x axis isU(x) = b/x^2 - 2c/x
where b and c are positive constants.(a) Plot U(x) as a function of x; assume b = c = 1 for this purpose.Where is the equilibrium point?
(b) Suppose the energy of the particle is E = −c^2/2b. Find the turning points of the motion.
(c) Suppose that the energy of the particle is E = c^2/2b . Find the turning points of the motion. How many turning points are there in this case?
Chapter 2 Solutions
Classical Dynamics of Particles and Systems
Ch. 2 - Prob. 2.1PCh. 2 - Prob. 2.2PCh. 2 - If a projectile is fired from the origin of the...Ch. 2 - A clown is juggling four balls simultaneously....Ch. 2 - A jet fighter pilot knows he is able to withstand...Ch. 2 -
In the blizzard of ’88, a rancher was forced to...Ch. 2 - Prob. 2.7PCh. 2 - A projectile is fired with a velocity 0 such that...Ch. 2 - Consider a projectile fired vertically in a...Ch. 2 - Prob. 2.11P
Ch. 2 - A particle is projected vertically upward in a...Ch. 2 -
A particle moves in a medium under the influence...Ch. 2 - A projectile is fired with initial speed 0 at an...Ch. 2 -
A particle of mass m slides down an inclined...Ch. 2 - A particle is projected with an initial velocity 0...Ch. 2 - A strong softball player smacks the ball at a...Ch. 2 - Prob. 2.19PCh. 2 - A gun fires a projectile of mass 10 kg of the type...Ch. 2 - Prob. 2.21PCh. 2 - Prob. 2.22PCh. 2 - A skier weighing 90 kg starts from rest down a...Ch. 2 - A block of mass m = 1.62 kg slides down a...Ch. 2 - A child slides a block of mass 2 kg along a slick...Ch. 2 - A rope having a total mass of 0.4 kg and total...Ch. 2 - A superball of mass M and a marble of mass m are...Ch. 2 - An automobile driver traveling down an 8% grade...Ch. 2 - A student drops a water-filled balloon from the...Ch. 2 - Prob. 2.31PCh. 2 - Two blocks of unequal mass are connected by a...Ch. 2 - A particle is released from rest (y = 0) and falls...Ch. 2 - Perform the numerical calculations of Example 2.7...Ch. 2 - Prob. 2.36PCh. 2 - A particle of mass m has speed υ = α/x, where x is...Ch. 2 - The speed of a particle of mass m varies with the...Ch. 2 - A boat with initial speed υ0 is launched on a...Ch. 2 - A train moves along the tracks at a constant speed...Ch. 2 - Prob. 2.42PCh. 2 - Prob. 2.45PCh. 2 - Prob. 2.46PCh. 2 - Consider a particle moving in the region x > 0...Ch. 2 - Prob. 2.48PCh. 2 - Prob. 2.49PCh. 2 - According to special relativity, a particle of...Ch. 2 - Let us make the (unrealistic) assumption that a...Ch. 2 - A particle of mass m moving in one dimension has...Ch. 2 - A potato of mass 0.5 kg moves under Earth’s...Ch. 2 - Prob. 2.55P
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
- Consider a particle moving in the region x > 0 under the influence of the potential where U0 = 1 J and α = 2 m. Plot the potential, find the equilibrium points, and determine whether they are maxima or minima.arrow_forwardA weight is connected to a spring that is suspended vertically from the ceiling. If the weight is displaced downward from its equilibrium position and released, it will oscillate up and down. (a) If air resistance is neglected, will the total mechanical energy of the system (weight plus Earth plus spring) be conserved? (b) How many forms of potential energy are there for this situation?arrow_forwardConsider the data for a block of mass m = 0.250 kg given in Table P16.59. Friction is negligible. a. What is the mechanical energy of the blockspring system? b. Write expressions for the kinetic and potential energies as functions of time. c. Plot the kinetic energy, potential energy, and mechanical energy as functions of time on the same set of axes. Problems 5965 are grouped. 59. G Table P16.59 gives the position of a block connected to a horizontal spring at several times. Sketch a motion diagram for the block. Table P16.59arrow_forward
- A block of mass m = 2.00 kg is attached to a spring of force constant k = 500 N/m as shown in Figure P7.15. The block is pulled to a position xi = 5.00 cm to the right of equilibrium and released from rest. Find the speed the block has as it passes through equilibrium if (a) the horizontal surface is frictionless and (b) the coefficient of friction between block and surface is k = 0.350. Figure P7.15arrow_forwardA mass of 1kg is attached to a spring with a spring constant of k=10 N/m. you stretch the spring to distance A from equilibrium and let go. The mass oscillates with a certain period, frequency, and total energy. Now stretch the spring to a distance of 3A. compared to the original total energy E, what is the new total energy of the system now?arrow_forwardI'm doing calc1 by myself right now and not sure where to start with this problem: Here are the questions: An object attached to a spring swings vertically around its equilibrium position. The height of the object relative to its position of equilibrium is given by the following function: h(t)=(e^-(x/10)*100sin(x)-10cos(x))/101 *(See photo i have provided)* a) Give the object movement for the first 10 seconds. I know i can find the speed of an object if i have it's acceleration. But how do i find it's movement fom it's height? I have done the integral of the height's equation and it's -(100e^-(x/10)(20sin(x)+99cos(x))/10201+Constant b) Give the total distance traveled by the object during the first 10 seconds I would assume this is something along the lines of finding the final positition of the object and the initial position and substracting them? c) Give the total distance that will be traveled by the object from time t = 0. at this point i am clueless to what i should find I…arrow_forward
- A simple pendulum of length LL is set to oscillate in simple harmonic motion. The bob gravitationalpotential energy is zero at its lowest vertical point. When the bob’s height is at half its maximum, h =hmax/2, then its velocity is:(a) vv = rdical(ghmax/2)(b)vv = rdical(ghmax(c) vv = rdical(2ghmax(d)vv = 2rdical(ghmaxarrow_forwardA spring with spring constant k = 80 N/m has an equilibrium length of 1.00 m. The spring is compressed to a length of 0.5 m and a mass of m = 1.8 kg is placed at its free end on a frictionless slope which makes an angle of θ = 37° with respect to the horizontal. The spring is then released. [Note: you may use the approximations sin 37° = 0.6 and cos 37° = 0.8 for simplicity] a) If the mass is not attached to the spring, how far up the slope will the mass move before coming to rest? b) If the mass is attached to the spring, how far up the slope will the mass move before coming to rest? c) Now the incline has a coefficient of kinetic friction μk. If the block, attached to the spring, is observed to stop just as it reaches the spring’s equilibrium position, what is the coefficient of friction?arrow_forwardA heavy crate (m = 23 kg) is resting on a frictionless inclined plane while suspended from a wall by a spring.The plane is inclined at an angle of 30◦ from the horizontal, and the spring is stretched from its equilibriumposition by 25 cm. What is the spring constant of the spring? If the frictionless inclined plane is now replacedwith an otherwise identical plane except with a rough surface, the crate now stretches the spring by 12 cmfrom its equilibrium position. What is the coefficient of static friction? You can assume that the staticfriction is at its maximum value.arrow_forward
- Prove that when the average is taken with respect to position over one cycle, the average potential energy equals kA^2/6 and the average kinetic energy equals kA^2/3.arrow_forwardA 50.0 cm long spring with spring constant 237 N/m has a mass 1.8 kg attached to it, and it can oscillate on a horizontal table without any friction. The spring is pulled by a distance 5 cm from the resting position and released. What is the kinetic energy (in joules) of the mass at the instant when the length of the spring is 49 cm.arrow_forwardHelp me pleasearrow_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
Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
College PhysicsPhysicsISBN:9781285737027Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
College Physics
Physics
ISBN:9781285737027
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
SIMPLE HARMONIC MOTION (Physics Animation); Author: EarthPen;https://www.youtube.com/watch?v=XjkUcJkGd3Y;License: Standard YouTube License, CC-BY