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A particle of mass m moves in one dimension. Let x(t) denote the position of the...
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
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 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.
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....
32. A Brownian particle is placed in a very long narrow channel. The particle is trapped...
32. A Brownian particle is placed in a very long narrow channel. The particle is trapped in a harmonic potential V (x) = {kx2 (where k is a constant and x is the distance from the origin) and is subject to random thermal noise. In other words, we have a one-dimensional harmonic oscillator coupled to a thermal bath. At time t = 0, the particle is as the position x (0) = 20. Assuming the inertia can be neglected (the...
A mass of m kilograams (kg) is mounted on top of a vertical spring. The spring is L metres long when disengaged and the end not attached to the mass is fixced to the ground. The mass moves vertic...
A mass of m kilograams (kg) is mounted on top of a vertical spring. The spring is L metres long when disengaged and the end not attached to the mass is fixced to the ground. The mass moves vertically up and down, acted on by gravity, the restoring force T of the spring and the damping force R due to friction: see the diagram below The gravitational force is mg dowswards, where g- 9.8m is acceleration due to gravity, measured...
2. Let x(t) be the coordinate operator for a free particle in one dimension in the...
2. Let x(t) be the coordinate operator for a free particle in one dimension in the Heisenberg picture. Evaluate [z(t), x(0)]. Comment on the uncertainty in the position over the course of time.
solve the following question For the system shown in the figure below x and y denote,...
solve the following question
For the system shown in the figure below x and y denote, respectively, the absolute displacements of the mass m and the end Q of the damper c1 (1) Derive the equation of motion of the mass m (2) Find the steady state displacement of the mass m (3) Find the force transmitted to the support at P when the end Q is subjected to harmonic motion y (t)-y cos wt x(t) y(t) cos ω t
A simple harmonic oscillator at the position x=0 generates a wave on a string. The oscillator...
A simple harmonic oscillator at the position x=0 generates a
wave on a string. The oscillator moves up and down at a frequency
of 40.0 Hz and with an amplitude of 3.00 cm. At time t =
0, the oscillator is passing through the origin and moving down.
The string has a linear mass density of 50.0 g/m and is stretched
with a tension of 5.00 N.
A simple harmonic oscillator at the position x = 0 generates a wave...
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at ti...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
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 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.
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....
32. A Brownian particle is placed in a very long narrow channel. The particle is trapped in a harmonic potential V (x) = {kx2 (where k is a constant and x is the distance from the origin) and is subject to random thermal noise. In other words, we have a one-dimensional harmonic oscillator coupled to a thermal bath. At time t = 0, the particle is as the position x (0) = 20. Assuming the inertia can be neglected (the...
A mass of m kilograams (kg) is mounted on top of a vertical spring. The spring is L metres long when disengaged and the end not attached to the mass is fixced to the ground. The mass moves vertically up and down, acted on by gravity, the restoring force T of the spring and the damping force R due to friction: see the diagram below The gravitational force is mg dowswards, where g- 9.8m is acceleration due to gravity, measured...
2. Let x(t) be the coordinate operator for a free particle in one dimension in the Heisenberg picture. Evaluate [z(t), x(0)]. Comment on the uncertainty in the position over the course of time.
solve the following question
For the system shown in the figure below x and y denote, respectively, the absolute displacements of the mass m and the end Q of the damper c1 (1) Derive the equation of motion of the mass m (2) Find the steady state displacement of the mass m (3) Find the force transmitted to the support at P when the end Q is subjected to harmonic motion y (t)-y cos wt x(t) y(t) cos ω t
A simple harmonic oscillator at the position x=0 generates a
wave on a string. The oscillator moves up and down at a frequency
of 40.0 Hz and with an amplitude of 3.00 cm. At time t =
0, the oscillator is passing through the origin and moving down.
The string has a linear mass density of 50.0 g/m and is stretched
with a tension of 5.00 N.
A simple harmonic oscillator at the position x = 0 generates a wave...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
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