3.) Find the equations of kinematics (? = ?0 + ?0? + 1/2 ??2 & ?...
3.) Find the equations of kinematics (? = ?0 + ?0? + 1/2 ??2 & ? = ?0 + ??) by first setting up the corresponding Hamiltonian and finding Hamilton’s equations (?̇ = − ??/?? & ?̇ = ??/?? ) for a particle of mass m that is only acted upon by the force of gravity and moving in a vertical line. Show all your work for credit.
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EXAM PAPER #6 MECHANICS II 1. Theory. Plane motion of a rigid body, Equations, resolution of...
EXAM PAPER #6 MECHANICS II 1. Theory. Plane motion of a rigid body, Equations, resolution of motion into translation and rotation Problem. The motion of a particle is defined by the equations 2. x0.01 .y-200-10t. Find the acceler on of the particle when it is on the axis Ox 3. Theory. The law of conservation of angular momentum (point Problem. A particle M of mass m initially at rest A) moves down on the inner surface of a cylinder of...
please consider my low understanding of the basic principles Problem 13.39. The sphere al A is...
please consider my low understanding of the basic
principles
Problem 13.39. The sphere al A is given a downward velocity v, of magnitude 5 m/s and swings in a vertical plane at the end of a rope of length / 2 m attached to a support at O. Determine the angle at which the rope will break, knowing that it can withstand a maximum tension equal to twice the weight of the sphere. b.) using the equations Shown how do...
A 4.0-kg particle is moving horizontally with a speed of 5.0 m/s when it strikes a...
A 4.0-kg particle is moving horizontally with a speed of 5.0 m/s when it strikes a vertical wall. The particle rebounds with a speed of 3.0 m/s. The collision occurs in 2 ms. (a) what is the magnitude of the impulse delivered to the particle? (b) What the average impulsive force acted to the particle? Answers: (a) 24 Ns and (b) 1.6 x 104 N. Show steps and equations used.
3. Consider a particle of mass m moving in a potential given by: W (2, y,...
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless ...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
PROBLEM 2 (10 POINTS) A particle of mass m moving along a straight line is acted...
PROBLEM 2 (10 POINTS) A particle of mass m moving along a straight line is acted on by a retarding force (one directed against the motion) F = beau, where b and a are constants and u is the velocity. At に0, it is moving with velocity 10. Find the velocity at later times always
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,...
Problem 4*: (Motion along a spiral) A particle of mass m moves in a gravitational field...
Problem 4*: (Motion along a spiral) A particle of mass m moves in a gravitational field along the spiral z = k0, r = constant, where k is a constant, and z is the vertical direction. Find the Hamiltonian H(z, p) for the particle motion. Find and solve Hamilton's equations of motion. Show in the limit r = 0, 2 = -g.
please solve with explanations 3. (20 pts) A particle of mass m and charge q is in a one dimensional harmonic oscillato...
please solve with explanations
3. (20 pts) A particle of mass m and charge q is in a one dimensional harmonic oscillator potential ()1ma'. A time dependent uniform electric field E, ()E, os eris 2 applied in the x direction. The particle is in the harmonic oscillator ground state at time a) What is the time dependent perturbation Hamiltonian H'(t) - the potential enegy of the charge in this electric field? b) Find the amplitude ci(t) of finding the particle...
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...
EXAM PAPER #6 MECHANICS II 1. Theory. Plane motion of a rigid body, Equations, resolution of motion into translation and rotation Problem. The motion of a particle is defined by the equations 2. x0.01 .y-200-10t. Find the acceler on of the particle when it is on the axis Ox 3. Theory. The law of conservation of angular momentum (point Problem. A particle M of mass m initially at rest A) moves down on the inner surface of a cylinder of...
please consider my low understanding of the basic
principles
Problem 13.39. The sphere al A is given a downward velocity v, of magnitude 5 m/s and swings in a vertical plane at the end of a rope of length / 2 m attached to a support at O. Determine the angle at which the rope will break, knowing that it can withstand a maximum tension equal to twice the weight of the sphere. b.) using the equations Shown how do...
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
PROBLEM 2 (10 POINTS) A particle of mass m moving along a straight line is acted on by a retarding force (one directed against the motion) F = beau, where b and a are constants and u is the velocity. At に0, it is moving with velocity 10. Find the velocity at later times always
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,...
Problem 4*: (Motion along a spiral) A particle of mass m moves in a gravitational field along the spiral z = k0, r = constant, where k is a constant, and z is the vertical direction. Find the Hamiltonian H(z, p) for the particle motion. Find and solve Hamilton's equations of motion. Show in the limit r = 0, 2 = -g.
please solve with explanations
3. (20 pts) A particle of mass m and charge q is in a one dimensional harmonic oscillator potential ()1ma'. A time dependent uniform electric field E, ()E, os eris 2 applied in the x direction. The particle is in the harmonic oscillator ground state at time a) What is the time dependent perturbation Hamiltonian H'(t) - the potential enegy of the charge in this electric field? b) Find the amplitude ci(t) of finding the particle...
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...
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