Suppose the potential energy function for a particlemoving in 1D is given by U(x) below. Find alequilibrium points and determine whether they arestable or unstable. Hint: there are 3 equilibrium points.U (x) = xª – x²

Suppose the potential energy function for a particle moving in 1D is given by U(x) below. Find all equilibrium points and determine whether they are stable or unstable. Hint: there are 3 equilibrium points.

Suppose the potential energy function for a particle moving in 1D is given by U(x) below. Find all equilibrium points and determine whether they are stable or unstable. Hint: there are 3 equilibrium points.

Classical Dynamics of Particles and Systems

5th Edition

ISBN:9780534408961

Author:Stephen T. Thornton, Jerry B. Marion

Publisher:Stephen T. Thornton, Jerry B. Marion

Chapter2: Newtonian Mechanics-single Particle

Section: Chapter Questions

Problem 2.47P: Consider a particle moving in the region x > 0 under the influence of the potential where U0 = 1 J...

See similar textbooks

Related questions

Q: A mass m = 2kg is placed at the end of a horizontal spring that has a spring constant k = 500N/m and…

Q: Find the force corresponding to the potential energy U(x) = (Use the following as necessary: a, b,…

Q: Below is a side-view picture, where a box is brought up to a height at point A and moves a distance…

A: The gravity’s work done on the box can be given by the work-energy theorem as, Here, U represents…

Q: Given F1 = 2xzi + yj + x2k and F2 = yi − xj:(a) Which F, if either, is conservative?(b) If one of…

A: Given: Forces:∫F→.dr→=∬∇→×F→.n^ dA F1=2xzi+yj+x2kF2=yi-xj

Q: The 1 kg collar shown is moving freely in the horizontal plane with a spring whose constant is 100 N…

A: Given: The 1 kg collar shown is moving freely in the horizontal plane with a spring whose constant…

Q: The force F= (yzi + zxj + xyz)/xyz acts on the particle P(x,y,z) which moves in space. (a) Using the…

A: Given data: The force on a particle is: F The potential function associated with force F is: V The…

Q: ) A box of mass m= 38 kg which is sliding on a horizontal rough surface and momentarily comes to st…

A: We know that We have given that the mass of 38 kg block moving on a horizontal plane with the…

Q: The light rod is pivoted at O and carries the 5- and 10-lb particles. If the rod is released from…

Q: A small sphere is attached to one end of a string of length 1.67 mm. The other end of the string is…

Q: Find the force corresponding to the potential energy U(x) (Use the following as necessary: a, b, and…

A: Given : potential energy is U(x) = - a/x2 + b/x3

Q: A particle is suspended from a point A by an inextensible string of length L. It is projected from B…

Q: A particle that can move only along the positive x-axis is part of a system with potential energy…

Q: A block of mass M, capable of sliding with negligible friction over a horizontal air rail, is…

Q: A ball of unknown mass mm is tossed straight up with initial speed vv. At the moment it is released,…

A: Energy conservation law,

Q: Given the potential energy function U(x) =−x^2 + 4x+ 2 calculate the force F(x) associated with…

Q: Find the force corresponding to the potential energy U(x) = (Use the following as necessary: a, b,…

A: The given potential energy is, Ux=-ax2+bx3 The corresponding force is, Fx=-dUdx

Q: The force vector is conservative or non conservative

A: Given that:- F→=(x2yz)i^+(-xyz2)k^ For a conservative force, its curl is zero.

Q: A 0.4 Gram mass is required to stretch a soap film confined in arectangular frame by 2mm downward…

A: Mass of the object m=0.4 g so force applied by the weight of the object F=mg=0.4*9.8*10-3=3.92*10-3N…

Q: A ball of unknown mass m is tossed straight up with initial speed v. At the moment it is released,…

A: according to the conservation of energy, KEinitial+PEinitial=PEfinal

Q: A particle is constrained to move in one dimension along the z axis and is acted upon by a force…

A: According to question we need to find expression for potential energy.

Q: An object undergoes simple harmonic motion. When it momentarily passes through the equilibrium…

Q: Confirm that the force field F(x, y) = ye"Y i+xe"Yjis conservative in some open connected region…

A: In this question we have to find the work done by the force field. Please give positive feedback if…

Q: tendons obey Hooke's Law. In laboratory tests on a particular tendon, it was found that, when b)…

Q: Particle of mass m slides without friction on a wedge of angle alpha and mass M that can move…

A: here we have to take the cartesian coordinate system with y=0 at the base of the wedge and x= 0 to…

Q: A ball of unknown mass m is tossed straight up with initial speed v. At the moment it is released,…

A: Solution:- Step-1

Q: a) Find the two equilibrium positions of the particle. b) For each of the two equilibrium…

A: Given data: The potential energy is: U The conservative force is: F (a)The potential energy for the…

Q: An ideal sprng Of Go J when Stretched by 5.0cm from epuribrim position. What Š the fariel spring…

A: Energy stored in the spring, Deflection in the spring,

Q: In any problem, if the number of unknowns is greater than the number of equilibrium equations, the…

A: The 1st question is unclear. Please repost with necessary information. We are solving the 2nd…

Q: A ball of unknown mass m is tossed straight up with initial speed v. At the moment it is released,…

A: Consider the motion of the ball when mass is going down at the same location where it is in…

Q: A frictionless simple pendulum, with a small but dense 4.4-kg mass at the end and a length of 75 cm,…

Q: A weight is connected to a spring that is suspended verticallyfrom the ceiling. If the weight is…

A: Part A:Yes ,If there is no friction total energy of the system will be conserved.When the block is…

Q: Calculate the maximum angle 0, that the block can remain stationary on the slope. M Ms

Q: A particle is constrained to move in one dimension along the a axis and is acted upon by a force…

Q: A particle moves in one dimension (along the x-axis), where the potential energy is given by the…

A: Given data: Potential function, U(x)=Ax2+Bx A=2 J/m2 & B=4 J/m2 Total energy of the system, E=8…

Q: . .A frictionless spring with a 5-kg mass can be held stretched 1.2 meters beyond its natural length…

A: Given data: Mass attached with spring is, m=5 kg. Stretching distance is, x=1.2 m. Magnitude of…

Q: (Use the following as necessary: a, b, and x. Express your answer in vector form.) Find the force…

Q: Q.n.3 A central force is defined to be a force that points radially, and whose magnitude depends on…

A: The vector differential operator of the spherical coordinate system is known as,…

Q: A particle moves in a force field described by the Yukowa potential V (r) = −k/r exp(−r/a) where k…

Q: Half ring with internal radius a and external b, with surface mass distribution σ = σ0/r, where σ0…

A: Given:inner radius, aouter radius, bmass density, σ = σor (σo is a constant)height of point P ,…

Q: please answer all with steps

Q: A particle that can move along the x-axis is part of a system with potential energy U(x) = A x 2 − B…

A: (a) Consider the given function of the potential energy, take the values for the constants A and B…

Q: In 1953, the US Army field-tested the M65 Atomic Cannon, capable of firing a W9 warhead (364 kg) at…

Q: A mass attached to a spring and allowed to oscillate horizontally on a tabletop is subject to both…

Q: Which of the following defines the relationship between potential energy (U) and work (W) for…

A: The work done by the conservative force is independent of the path. The work done is equal to the…

Question

Suppose the potential energy function for a particle
moving in 1D is given by U(x) below. Find al
equilibrium points and determine whether they are
stable or unstable. Hint: there are 3 equilibrium points.
U (x) = xª – x²

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1 Objective

VIEW

Step 2 Formula used and calculation

VIEW

Step 3 Conclusion

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

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

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.
Verify that v=8kBTm. Make the same scaling transformation as in the preceding problem.`
If the modulus of the sum of two vectors is less than the modules of the two vectors, we can conclude that a) The vectors point in the same direction, in opposite directions. b) The vectors point in the same direction and in the same direction. c) The scalar product of the two vectors is positive. d) The scalar producer of the two vectors is negative. e) None of the above.
Let's consider a simple pendulum, consisting of a point mass m, fixed to the end of a massless rod of length l, whose other end is fixed so that the mass can swing freely in a vertical plane. The pendulum's position can be specified by its angle Φ from the equilbrium position. Prove that the pendulum's potential energy is U(ϕ)=mgl(1−cosϕ). Write down the total energy E as a function of Φ and ϕ˙. Show that by differentiating E with respect to t you can get the equation of motion for Φ. Solve for Φ(t). If you solve properly, you should find periodic motion. What is the period of the motion?
A 0.4 Gram mass is required to stretch a soap film confined in arectangular frame by 2mm downward before the film breaks. If the length of the movable bar is 2.5 cm, what will be the surface tension and change in surface free energy of the soap film?
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.
Show,by integration that the moment of inertia of a uniform thin rod AB of mass 3m and length 4a about an axis through one end and perpendicular to the plane of the rod, is 16ma2 The rod is free to rotate in a vertical plane about a smooth horizontal axis through its end A Find i)The raduis of gyration of the rod about an axis through A ii)The period of small oscillations of the rod about its position of stable equilibrum
A ladder of length L and mass M is leaning against a wall. Assuming thewall and the floor are friction-less, the ladder will slide down the wall andalong the floor until the left end loses contact with the wall. Before the ladder loses contact with the wall there is one degree of freedom (θ). Using the formula for Euler-Lagrange EOM to find the Lagrangian and then the EOM for the ladder. Assume that gravity is present.
A slingshot, consisting of a leather pouch attached to two rubber bands which are tied to the prongs of a wooden Y- shaped frame, is used to shoot a pebble horizontally. When the slingshot is stretched by a distance d, it gives the pebble speed v. What speed does it give the same pebble when it is stretched to distance 3d?
The Question is as follows, "A particle with a mass of .7 kg is attached to a horizonal spring with a force constant of 40 N/m. At the moment t=0, the particle has its maximum speed of 30 m/s and is moving to the left, (assume right is positive direction). A) Determine the particles equation of motion, specifying its position as a function of time (X in meters and T is seconds, give angles in radians) B) where in the motion is the potential energy three times the kinetic energy? C) Find the length of a simple pendulum with the same period." Thanks for the help! If possible could you write out the formulas when you use em? Thanks
A mysterious force acts on all particles along a particular line and always points towards a particular point P on the line. The magnitude of the force on a particle increases as the cube of the distance from that point; that is Fr3 , if the distance from P to the position of the particle is r. Let b be the proportionality constant, and write the magnitude of the force as F=br3. Find the potential energy of a particle subjected to this force when the particle is at a distance D from P, assuming the potential energy to be zero when the particle is at P.
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(ghmax