Given a potential of the form: V[x] = V0(5 - 6 (x / a)^ 2 +...
Given a potential of the form: V[x] = V0(5 - 6 (x / a)^ 2 + 6 (x / a) ^4 (a) Sketch the potential. (b) Determine the equilibrium positions and specify whether they are stable or unstable, (c) Determine the effective spring constant for the stable equilibrium positions
Homework Answers
5. Consider a particle moving in the region x>0 under the influence of the potential U(x)...
5. Consider a particle moving in the region x>0 under the influence of the potential U(x) = C (a/x + x/a), where C=1J, and a=2m. (a) Find the equilibrium positions and determine whether they are stable or unstable. (b) Find U at those equilibrium positions. (c) Sketch U(x) without using a computer (explain how you get the sketch) and discuss the motion of the particle in details in all the different regions if its total energy E1 = 2 J,...
The potential energy of an object constrained to move only along the x-axis is given by...
The potential energy of an object constrained to move only along the x-axis is given by U=5x^2-4x where U is in Joules and x is in meters. a. Find the force associated with this potential. b. Assuming no other forces act on the object find the equilibrium positions of the object and determine whether each is stable, unstable, or neutral.
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
2. A particle of mass m is...
A particle of mass 5 kg is subject to a conservative force whose potential energy (in...
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...
4) Given the first order equation x'= [lx-x where x = x(t) and u is a...
4) Given the first order equation x'= [lx-x where x = x(t) and u is a real parameter. a) Find all fixed points for u0 and determine whether they are stable or unstable. b) Find all fixed points for u<0 and determine whether they are stable or unstable. c) Sketch the bifurcation diagram for this equation. Be sure to indicate the direction of flow on your diagram (with arrows). This is called a "pitchfork bifurcation" for obvious reasons.
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is...
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
The electric potential for a system is given by V(r) = V0e−ar/r, where V0 and a...
The electric potential for a system is given by V(r) = V0e−ar/r, where V0 and a (a > 0) are constants. Determine the magnitude of the electric field. (Use the following as necessary: V0, a, and r.)
The graph of the function f(x) is, and the the horizontal axis is x. Given the...
The graph of the function f(x) is, and the the
horizontal axis is x.
Given the differential equation
x?(t)=f(x(t))x?(t)=f(x(t)).
List the constant (or equilibrium) solutions to this differential
equation in increasing order and indicate whether or not these
equations are stable, semi-stable, or unstable. (The second line
after the comma has a drop menu that asks for stable, semi-stable,
or unstable.)
A.) ________ , ________
B.) ________ , ________
C.) ________ , ________
D.) ________ , ________
A particle is moving in a potential V ) given by the figure below. (b)Λ. Sketch...
A particle is moving in a potential V ) given by the figure below. (b)Λ. Sketch approximately the phase portrait 0 <q < 10. Label regions where (a) the particle has enough energy to make it over both local maxima. (b) the particle is confined to the left of both maxima. (c) the particle is confined to the right of both maxima. (d) the particle is confined bewteen the maxima. (e) the particle has enough enough energy to make it...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. ...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. (2 pts) In software, plot the potential U) Set your axis ranges: -2 SxS2 and 0s b. (5 pts) Find the equilibrium positions and determine whether they are stable or c. (8 pts) Describe the motion of the particle for total energy values E-О.0.05. 1.0, 2.0 unstable. Explain how you arrived at your answers. (all in Joules). What I am looking for here...
5. Consider a particle moving in the region x>0 under the influence of the potential U(x) = C (a/x + x/a), where C=1J, and a=2m. (a) Find the equilibrium positions and determine whether they are stable or unstable. (b) Find U at those equilibrium positions. (c) Sketch U(x) without using a computer (explain how you get the sketch) and discuss the motion of the particle in details in all the different regions if its total energy E1 = 2 J,...
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...
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...
4) Given the first order equation x'= [lx-x where x = x(t) and u is a real parameter. a) Find all fixed points for u0 and determine whether they are stable or unstable. b) Find all fixed points for u<0 and determine whether they are stable or unstable. c) Sketch the bifurcation diagram for this equation. Be sure to indicate the direction of flow on your diagram (with arrows). This is called a "pitchfork bifurcation" for obvious reasons.
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
Given the equation y' 9-16y, a) Find all Equilibrium solutions b) Determine whether each solution is stable, unstable or neither. c) Sketch the direction field.
A particle is moving in a potential V ) given by the figure below. (b)Λ. Sketch approximately the phase portrait 0 <q < 10. Label regions where (a) the particle has enough energy to make it over both local maxima. (b) the particle is confined to the left of both maxima. (c) the particle is confined to the right of both maxima. (d) the particle is confined bewteen the maxima. (e) the particle has enough enough energy to make it...
A particle is introduced to a region with a potential described by U(x)--2x2 +x*+1 Joules. 3. a. (2 pts) In software, plot the potential U) Set your axis ranges: -2 SxS2 and 0s b. (5 pts) Find the equilibrium positions and determine whether they are stable or c. (8 pts) Describe the motion of the particle for total energy values E-О.0.05. 1.0, 2.0 unstable. Explain how you arrived at your answers. (all in Joules). What I am looking for here...
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