Question

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 of motion.

(d) Solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time.

(e) Construct the Hamiltonian of the system, H, and write down the Hamilton’s equations (canonical equations) of motion.

Answer #1

Thank you, please rate if it helps :).

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...

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

The block of M=5kg, modeled as a particle, rests on the inclined
surface that makes an angle of θ with the horizontal ground. The
static friction coefficient between the block and the surface is
μs=1/3. A force of P is horizontally applied on the block. Take
g=10m/s/s for convenience.
1. Draw a free body diagram.
2. Write each force expressed in cartesian vector form.
3. Write down scalar compontent form of equilibrium equations in
x and y direction.
4. Find...

The block of M=5kg, modeled as a particle, rests on the inclined
surface that makes an angle of θ with the horizontal ground. The
static friction coefficient between the block and the surface is
μs=1/3. A force of P is horizontally applied on the block. Take
g=10m/s/s for convenience.
Angle of slope is 36.87
1. Draw a free body diagram.
2. Write each force expressed in cartesian vector form.
3. Write down scalar compontent form of equilibrium equations in
x...

Your task will be to derive the equations describing the
velocity and acceleration in a polar coordinate
system and a rotating polar vector basis for an object in general
2D motion starting from a general
position vector. Then use these expressions to simplify to the case
of non-uniform circular motion, and
finally uniform circular motion.
Here's the time-dependent position vector in a Cartesian coordinate
system with a Cartesian vector
basis: ⃗r(t)=x (t)
̂
i+y(t)
̂
j where x(t) and y(t)...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 7 minutes ago

asked 9 minutes ago

asked 9 minutes ago

asked 11 minutes ago

asked 13 minutes ago

asked 13 minutes ago

asked 18 minutes ago

asked 20 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 22 minutes ago