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Problem 1. (24 points) A particle of mass m moving in one dimension is subject to...
A particle of mass m moves in one dimension along the positive x axis. It is...
A particle of mass m moves in one dimension along the positive x axis. It is acted on by a constant force directed toward the origin with magnitude B, and an inverse-square law repulsive force with magnitude A/x^2. Find the potential energy function, U(x). Sketch the energy diagram for the system when the maximum kinetic energy is K_0 = 1/2 mv_0^2 Find the equilibrium position, x_0.
Consider a particle with a mass m subject to a force F(x) = ax - bx3...
Consider a particle with a mass m subject to a force F(x) = ax - bx3 where x is the displacement of the origin of the reference system and a and b are positive constants. a) Find an expression of the particle's total energy. Show that this total energy is constant. b) Find the equilibrium points and determine if they are stable or unstable.
The force acting on a particle constrained to move in 1-dimension is given by: F= ax(b-...
The force acting on a particle constrained to move in 1-dimension is given by: F= ax(b- cx?) [a= 3.25, b=2.05, c=7.33] Find the three equilibrium points, and enter them left-to-right in order of ascending x-coordinate. Then, label each equilibrium point as "stable" or "unstable". Number Number Number Equilibrium Points: Stability: stable unstable
A particle of mass m moves in one dimension. Its potential energy is given by U(x)...
A particle of mass m moves in one dimension. Its potential energy is given by U(x) = -Voe-22/22 where U, and a are constants. (a) Draw an energy diagram showing the potential energy U(). Choose some value for the total mechanical energy E such that -U, < E < 0. Mark the kinetic energy, the potential energy and the total energy for the particle at some point of your choosing. (b) Find the force on the particle as a function...
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...
1. A free particle of mass m moving from the left in one dimension scatters from...
1. A free particle of mass m moving from the left in one dimension scatters from the potential V(x) αδ(x). Suppose that the wave number of the particle is k and that α > 0. a. State the general form of the wave function including reflected and transmitted waves. b. Find the amplitude t of the transmitted wave in terms of α, k, m, and h. Find T
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
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 particle of mass m moves in one dimension. Let x(t) denote the position of the...
A particle of mass m moves in one dimension. Let x(t) denote the position of the particle at time t. The particle is subjected to a force which depends only on the position of the particle; when the particle is at position x, the force is -A sin(Bx), where A and B are some positive constants. Fill in the blank so that we end up with the differential equation that describes the motion: x" = Note that x = 0...
A particle is moving to the right with initial kinetic energy To, subject to a force...
A particle is moving to the right with initial kinetic energy To, subject to a force F(z)k function U(x) for this force ; (b) the kinetic energy and (c) the total energy of the particle as a function of its position; (d) find the turning points of the motion and the condition the total energy of the particle must satisfy if its motion is to exhibit turning points. (e) Sketch the potential, kinetic and total energy function (you can use...
Analytical mechanics seventh edition p.141 3.24 let a particle of unit mass be subject to a force...
analytical mechanics seventh edition p.141 3.24 let a particle of unit mass be subject to a force x-x^3 where x is its displacement from the coordinate origin (a) Find the equilibrium points, and tell whether they are stable or unstable (b) Calculate the total energy of the particle, and show that it is a conserved quantity (c) Calculate the trajectories of the particle in phase space
A particle of mass m moves in one dimension along the positive x axis. It is acted on by a constant force directed toward the origin with magnitude B, and an inverse-square law repulsive force with magnitude A/x^2. Find the potential energy function, U(x). Sketch the energy diagram for the system when the maximum kinetic energy is K_0 = 1/2 mv_0^2 Find the equilibrium position, x_0.
The force acting on a particle constrained to move in 1-dimension is given by: F= ax(b- cx?) [a= 3.25, b=2.05, c=7.33] Find the three equilibrium points, and enter them left-to-right in order of ascending x-coordinate. Then, label each equilibrium point as "stable" or "unstable". Number Number Number Equilibrium Points: Stability: stable unstable
A particle of mass m moves in one dimension. Its potential energy is given by U(x) = -Voe-22/22 where U, and a are constants. (a) Draw an energy diagram showing the potential energy U(). Choose some value for the total mechanical energy E such that -U, < E < 0. Mark the kinetic energy, the potential energy and the total energy for the particle at some point of your choosing. (b) Find the force on the particle as a function...
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...
1. A free particle of mass m moving from the left in one dimension scatters from the potential V(x) αδ(x). Suppose that the wave number of the particle is k and that α > 0. a. State the general form of the wave function including reflected and transmitted waves. b. Find the amplitude t of the transmitted wave in terms of α, k, m, and h. Find T
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 particle of mass m moves in one dimension. Let x(t) denote the position of the particle at time t. The particle is subjected to a force which depends only on the position of the particle; when the particle is at position x, the force is -A sin(Bx), where A and B are some positive constants. Fill in the blank so that we end up with the differential equation that describes the motion: x" = Note that x = 0...
A particle is moving to the right with initial kinetic energy To, subject to a force F(z)k function U(x) for this force ; (b) the kinetic energy and (c) the total energy of the particle as a function of its position; (d) find the turning points of the motion and the condition the total energy of the particle must satisfy if its motion is to exhibit turning points. (e) Sketch the potential, kinetic and total energy function (you can use...
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