Question

a) A small block of mass m is confined to move on the inside surface of a cone defined by z = aρ, where (ρ, φ, z) give its position in cylindrical coordinates. The block slides without friction along the surface and feels the gravitational force, which is directed in the negative z direction. a. Write down the Lagrangian for the system in terms of the coordinates ρ and φ

b) Obtain the Lagrange equations of motion for the coordinate

c) Obtain the first integral of the equation from part (b) and give the physical interpretation of this result

d) Obtain the Lagrange equations of motion for the coordinate ρ.

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...

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 small block with mass 0.0400 kg slides in a vertical circle of
radius 0.600 m on the inside of a circular track. During one of the
revolutions of the block, when the block is at the bottom of its
path, point A, the magnitude of the normal force exerted
on the block by the track has magnitude 4.05 N . In this same
revolution, when the block reaches the top of its path, point
B, the magnitude of the...

A small block with mass 0.0475 kg slides in a vertical circle of
radius 0.450 m on the inside of a circular track. During one of the
revolutions of the block, when the block is at the bottom of its
path, point A , the magnitude of the normal force exerted
on the block by the track has magnitude 3.75 N . In this same
revolution, when the block reaches the top of its path, point
B , the magnitude...

A small block with mass 0.0350 kg slides in a vertical circle of
radius 0.450 m on the inside of a circular track. During one of the
revolutions of the block, when the block is at the bottom of its
path, point A, the magnitude of the normal force exerted on the
block by the track has magnitude 3.90 N . In this same revolution,
when the block reaches the top of its path, point B, the magnitude
of the...

A small block with mass 0.0475 kg slides in a vertical circle of
radius 0.550 m on the inside of a circular track. During one of the
revolutions of the block, when the block is at the bottom of its
path, point A, the magnitude of the normal force exerted
on the block by the track has magnitude 3.85 N . In this same
revolution, when the block reaches the top of its path, point
B, the magnitude of the...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 9 minutes ago

asked 14 minutes ago

asked 28 minutes ago

asked 28 minutes ago

asked 29 minutes ago

asked 46 minutes ago

asked 55 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago