Question

A particle of mass, m, in an isolated environment moves along a line with speed v whilst experiencing a force proportional to its distance from the origin.

a) Determine the Langrangian of the system

b) Determine the Hamiltonian of the system

c) Write down Hamilton’s equations of motion for the particle d) Show that the particle executes simple harmonic motion

Answer #1

A particle of mass m is projected with an initial velocity v0 in
a direction making an angle α with the horizontal level ground as
shown in the figure. The motion of the particle occurs under a
uniform gravitational field g pointing downward.
(a) Write down the Lagrangian of the system by using the
Cartesian coordinates (x, y).
(b) Is there any cyclic coordinate(s). If so, interpret it
(them) physically.
(c) Find the Euler-Lagrange equations. Find at least one
constant...

A
particle moves in a potential field V(r,z)=az/r, a is constant. Use
the cylindrical coordinates as the general coordinates.
1)Determine the Lagrangian of this particle.
2)Calculate the generalized impulse.
3)Determine the Hamiltonian of this particle and the
Hamiltonian’s equations of motion.
4)Determine the conserved quatities of this system.

A particle of mass m moves in a circle of radius R at a constant
speed v as shown in the figure. The motion begins at point Q at
time t = 0. Determine the angular momentum of the particle about
the axis perpendicular to the page through point P as a function of
time.

A particle of mass m moves about a circle of radius R from the
origin center, under the action of an attractive force from the
coordinate point P (–R, 0) and inversely proportional to the square
of the distance.
Determine the work carried out by said force when the point is
transferred from A (R, 0) to B (0, R).

5-7 A particle of mass m moves under the action of gravity on
the surface of a horizontal cylinder.
a) Obtain the Lagrange motion equations for the particle.
b) If the particle slides in a vertical plane having left the
top of the cylinder at a very small speed, find the reaction force
as a function of the position.
c) At what point will the cylinder particle separate?

a particle of mass m moves in three dimension under the action
of central conservative force with potential energy v(r).find the
Hamiltonian function in term of spherical polar cordinates ,and
show φ,but not θ ,is ignorable .Express the quantity
J2=((dθ/dt)2 +sin2 θ(dφ
/dt)2) in terms of generalized momenta ,and show that it
is a second constant of of the motion

Mass m = 0.1 kg moves to the right with speed v = 0.48 m/s and
collides with an equal mass initially at rest. After this inelastic
collision the system retains a fraction = 0.88 of its original
kinetic energy. If the masses remain in contact for 0.01 secs while
colliding, what is the magnitude of the average force in N between
the masses during the collision? Hints: All motion is in 1D. Ignore
friction between the masses and the...

A mass (m) moves along the x-axis with velocity 2v. Another
particle of mass (2m) moves with velocity v along the y-axis.
1) If the two objects collide inelastically and merge, how much
kinetic energy is lost to heat?
2) If instead they collide elastically and it turns out that m
still moves purely along the x-axis, what is it's velocity in the
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It may help to find the velocities of these particles in the
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A particle moves along the x axis. It is initially at the
position 0.150 m, moving with velocity 0.080 m/s and acceleration
-0.340 m/s2. Suppose it moves with constant acceleration for 5.60
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its velocity at the end of this time interval. Next, assume it
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charged particle mass m charge q moves along horizontal surface
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applied such that magnetic force acts into the surface. If initial
speed of particle is u show that time for the particle to stop is
(m/(μqB))*ln(1+quB/(mg))

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