A particle of mass m is moving in the potential
.
1) Determine the force F(x) acting on the particle. Sketch the
force and the potential in a single diagram, as functions of
position, with
. Find the physical dimension of constant A.
2) Find all equilibria of the particle on the interval
. Determine whether these equilibria are stable or not.
3) If the initial position of x0 = a/2, find all possible values of initial velocity for which the particle will escape to infinity.
Homework Answers
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A particle of mass m is moving in the potential
.
1) Determine the force F(x)...
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A particle of mass 5 kg is subject to a conservative force whose potential energy (in joules) as a function of position (in meters) is given by the equation U(x) =-100x5e-1x [where x > 0] (a) Determine the position xo where the particle experiences stable equilibrium (b) Find the potential energy Uo of the particle at the position x 2106 The particle is displaced slightly from position x = xo and released (c) Determine the effective value of the spring...
Find the law of motion of a particle mass m and zero energy in one dimension...
Find the law of motion of a particle mass m and zero energy in
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constant. Given the inital position x0, compute how much time does
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3....
A For a particle with mass m moving under a one dimensional potential V(x), one solution...
A For a particle with mass m moving under a one dimensional potential V(x), one solution to the Schrödinger equation for the region 0<x< oo is x) =2 (a>0), where A is the normalization constant. The energy of the particle in the given state is 0, Show that this function is a solution, and find the corresponding potential V(x)?
2. A particle of mass m is moving in a plane under a force whose potential energy is given by V(r...
2. A particle of mass m is moving in a plane under a force whose potential energy is given by V(r) -kin r + cr + gr cos θ with k,c,g positive constants. (a) Write down the force in polar coordinates. (b) Find the positions of equilibrium (1) if c>g and (2) if c<g. (c) By considering the direction of the force near these points, determine whether the equilibrium is stable or not
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The Lagrangian for a particle of mass m moving in a vertical plane and experiencing the constant ...
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A particle of mass m is in a potential energy field described by, V(x, y) =...
A particle of mass m is in a potential energy field described by, V(x, y) = 18kx² +8ky? where k is a positive constant. Initially the particle is resting at the origin (0,0). At time t = 0 the particle receives a kick that imparts to it an initial velocity (vo, 2vo). (a) Find the position of the particle as a function of time, x(t) and y(t). (b) Plot the trajectory for this motion (Lissajous figure) using Vo = 1,...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a state given by the normalised wave function (x, 0) =3/2eAl2| | -ao x << 0, realU>0. Find the potential energy V = the energy eigenvalue E. Fix zero energy according to the convention V(x) » 0 for ao. Is there a delta function singularity at x0? V (x) for which this is an energy eigenstate and determine [6]
At time t = 0, a...
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Consider a particle of mass m moving in a one-dimensional potential of the form V. for...
Consider a particle of mass m moving in a one-dimensional potential of the form V. for 0<x<b, V(a) = 0 for Islal<e, for 1212, with V., b and c positive constants and c>b. a Explain why the wave function of the particle can be assumed to be cither an even function or an odd function of a. b For the case that the energy E of the particle is in the range 0<ESV., find the (unnormalized) even cigenfunctions and give...
A particle of mass 5 kg is subject to a conservative force whose potential energy (in joules) as a function of position (in meters) is given by the equation U(x) =-100x5e-1x [where x > 0] (a) Determine the position xo where the particle experiences stable equilibrium (b) Find the potential energy Uo of the particle at the position x 2106 The particle is displaced slightly from position x = xo and released (c) Determine the effective value of the spring...
Find the law of motion of a particle mass m and zero energy in
one dimension in the field U(x) = -Ax^(4) where A is a positive
constant. Given the inital position x0, compute how much time does
it take for the particle to escape to infinity if the vector of
initial velocity of the particle is pointing away from the origin
x=0. Describe the motion when the vector of inital velocity of the
particle is pointing toward x=0.
3....
A For a particle with mass m moving under a one dimensional potential V(x), one solution to the Schrödinger equation for the region 0<x< oo is x) =2 (a>0), where A is the normalization constant. The energy of the particle in the given state is 0, Show that this function is a solution, and find the corresponding potential V(x)?
2. A particle of mass m is moving in a plane under a force whose potential energy is given by V(r) -kin r + cr + gr cos θ with k,c,g positive constants. (a) Write down the force in polar coordinates. (b) Find the positions of equilibrium (1) if c>g and (2) if c<g. (c) By considering the direction of the force near these points, determine whether the equilibrium is stable or not
2. A particle of mass m is...
The Lagrangian for a particle of mass m moving in a vertical plane and experiencing the constant gravitational force mg is 2 Find the Hamiltonian and so the Hamilton-Jacobi equation Using the separable ansatz s- S(a)+Sy(v)-at ciple function i constants a and ay . Taking the separation constants a and ay as the new momenta find the new constant coordinates ßz and ßy. Find the particle's trajectory as a function of the constants Oz, αψ β, and β . Find...
A particle of mass m is in a potential energy field described by, V(x, y) = 18kx² +8ky? where k is a positive constant. Initially the particle is resting at the origin (0,0). At time t = 0 the particle receives a kick that imparts to it an initial velocity (vo, 2vo). (a) Find the position of the particle as a function of time, x(t) and y(t). (b) Plot the trajectory for this motion (Lissajous figure) using Vo = 1,...
At time t = 0, a mass-m particle in a one-dimensional potential well is in a state given by the normalised wave function (x, 0) =3/2eAl2| | -ao x << 0, realU>0. Find the potential energy V = the energy eigenvalue E. Fix zero energy according to the convention V(x) » 0 for ao. Is there a delta function singularity at x0? V (x) for which this is an energy eigenstate and determine [6]
At time t = 0, a...
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