Question

The mass m_{1} on a smooth table is connected to the
mass m_{2} which moves only in the vertical direction by an
inextensible L-length rope passing through a hole in the table.
Answer the following by ignoring the friction in the system.

a) What are the generalized coordinates of the system. Write the
Lagrangian of the system using these coordinates.

b) Find the conservative values of the system.

c) Obtain the equations of motion of the system and discuss what
these equations mean.

d) Solve the Lagrange equations. Assume that the masses do not pass
through the hole.

Answer #1

Objects with masses m1 = 12.0 kg and
m2 = 8.0 kg are connected by a light string
that passes over a frictionless pulley as in the figure below. If,
when the system starts from rest, m2 falls 1.00
m in 1.48 s, determine the coefficient of kinetic friction between
m1 and the table.
Express the friction force in terms of the coefficient of kinetic
friction. Obtain an expression for the acceleration in terms of the
masses and the...

Two masses m1 = 5 kg and m2 = 3 kg are connected by a
light rope dropped over a pulley as shown in the figure
a) Neglecting friction, what is the acceleration of these
masses?
b) What is the tension of the rope?
c) What will be the acceleration if the coefficient of friction for
the mass m1 is μk = 0.4?

Two masses M1 and M2 are connected by an inextensible
flexible massless string. The mass rests on a table with frictional
co-efficient 0.2. The string passes over a smooth fixed pulley as
shown. Initially the string is slack The mass M2 is allowed to fall
freely from rest. The string becomes taught when the mass M2 falls
through 50 cm. The two masses subsequently move with common
velocity. Determine how far mass M1 will travel on the table before
coming...

Two blocks with masses M1 and M2 are connected by a massless
string that passes over a massless pulley as shown.
M1 has a mass of 2.25 kg and is on an incline of θ1=46.5∘ with
coefficient of kinetic friction μ1=0.205.
M2 has a mass of 6.05 kg and is on an incline of θ2=33.5∘ with
coefficient of kinetic friction μ2=0.105.
The two‑block system is in motion with the block of mass M2
sliding down the ramp.
Find the magnitude...

Masses of m1=1.25kg and m2=0.32kg are connected by strings
through pulleys, respectively. m1 When stationary, it is 2.39
meters away from the floor in a vertical direction. Find the time
it takes for m1 to reach the floor when placed in a stationary
state, but the pulley is very small in mass and free of friction
and g = 9.80 m/s2.

Objects with masses m1 = 11.0 kg and
m2 = 8.0 kg are connected by a light string
that passes over a frictionless pulley as in the figure below. If,
when the system starts from rest, m2 falls 1.00
m in 1.54 s, determine the coefficient of kinetic friction between
m1 and the table.

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 mass (M1=5.0kg) is connected by a light cord to a mass
(M2=4.0kg) which slides on a smooth frctionless surface. The
pulley( a solid disk r=0.20m) rotates about a frictionless axle.
The accleration of M2 is 3.5 m/s^2. What is the mass of the
pulley?

A mass m1 = 1 kg on a horizontal table is pulled by a string
looped over a massless frictionless pulley and attached to another
hanging mass m2. The coefficient of kinetic friction between the
table and mass m1 is 0.5 and the coefficient of static friction is
0.6.
a) What is the minimum mass m2 that will start both masses
moving?
b) What is the acceleration of the system with this minimum
mass?

Two masses are connected by a massless string, passing over a
massless, frictionless pulley as shown in the diagram. Mass
m1 = 5 kg, and is released from rest at a
height h = 4 m above the table. Mass
m2 = 3 kg, and starts at rest on the table.
Ignore friction and air resistance. Take the system to be the two
masses and the earth. What is the potential energy of this system,
in Joules? Next, you let...

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