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The Lagrangian for a particle of mass m moving in a vertical plane and experiencing the constant ...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
A particle of mass m moves in the vertical plane (xOz) under the influence of two...
A particle of mass m moves in the vertical plane (xOz) under the influence of two forces: gravity: G-m g (g oriented in the -z direction), and the frictional force: F--k V, where V is the instantaneous velocity. The initial conditions are: a) Find x(t) and zit) b) Find the equation z(x) of the trajectory. Under what conditions is this trajectory a parabola? c) Assuming that Xo > 0, and ї0, find the coordinates (xA,-A) of the highest point on...
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,...
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,...
Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave fu...
Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave function is given by o)-A. Find the normalization constant A function in this basis. Solve for the coeffici You may find useful the integrals in the front of the (6 pt) d) Now consider the finite potential spherical well with V(r)- ing only the radial part...
A block of mass m = 3.5 kg is on an inclined plane with a coefficient...
A block of mass m = 3.5 kg is
on an inclined plane with a coefficient of friction
μ1 = 0.31, at an
initial height h = 0.53 m above
the ground. The plane is inclined at an angle θ =
54°. The block is then compressed against
a spring a distance Δx = 0.11 m
from its equilibrium point (the spring has a spring constant of
k1 = 39 N/m) and
released. At the bottom of the inclined plane...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
A particle of mass m moves in the vertical plane (xOz) under the influence of two forces: gravity: G-m g (g oriented in the -z direction), and the frictional force: F--k V, where V is the instantaneous velocity. The initial conditions are: a) Find x(t) and zit) b) Find the equation z(x) of the trajectory. Under what conditions is this trajectory a parabola? c) Assuming that Xo > 0, and ї0, find the coordinates (xA,-A) of the highest point on...
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,...
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,...
Consider a particle of mass m in an infinite spherical potential well of radius a For write down the energies and corresponding eigen functions ψ--(r,0.9). (3 pt) a) ne that at t-o the wave function is given by o)-A. Find the normalization constant A function in this basis. Solve for the coeffici You may find useful the integrals in the front of the (6 pt) d) Now consider the finite potential spherical well with V(r)- ing only the radial part...
A block of mass m = 3.5 kg is
on an inclined plane with a coefficient of friction
μ1 = 0.31, at an
initial height h = 0.53 m above
the ground. The plane is inclined at an angle θ =
54°. The block is then compressed against
a spring a distance Δx = 0.11 m
from its equilibrium point (the spring has a spring constant of
k1 = 39 N/m) and
released. At the bottom of the inclined plane...
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