1. Two blocks each of mass M are connected by an extension less, uniform string of length l. One block is placed on a smooth horizontal surface and the other block hangs over the side of the table with the string passing over a frictionless pulley. Describe the motion of the system:

a) when the mass of the string is negligible
b) when the string has mass m

Use lagrangian/Hamiltonian Dynamics to solve this problem.

2. The motion of two bodies interacting only with each other by central forces can be reduced to an equivalent one body problem. Show by explicit calculation that such a reduction is also possible for bodies moving in an external uniform gravitational field

Problem 2 is trivial. You have to switch to new variables. You go from the positions of the two particles r_{1} and r_{2} to the center of mass

R = [m_{1}r_{1} + m_{2}r_{2}]/[m_{1} + m_{2}]

and the relative position:

r_{1,2} = r_{1} - r_{2}

If you write down the Lagrangian, then the only new term is the potential energy in the external gravitational field, which can be written as:

V = [m_{1}z_{1} + m_{2}z_{2}]g = [m_{1}+m_{2}] Z g

where Z is the z-component of the center of mass.

So, the homogeneous gravitational potential is just that of a point mass with mass equal to the total mass located at the center of mass position. Therefore, if you consider only the relative motion of the particles, this term drops out.
------------------------------------------------------------

A circle of radius R, oriented with the plane of the circle horizontal, is attached to a vertical axis at one point on the circumference of the circle. A bead, of mass m, is attached to the circle and is free to move around the circle, with no frictional losses.
The circle - bead system rotates about the axis at a constant

Find the orbital speed of a satellite in a circular orbit 1870 km above the surface of the Earth.
1870km = 1.87e6 m
I tried using the V^2=GMe/r and came up with 14.6 km/s which is wrong.
I tried finding the velocity of earth 7.91 km/s and subtracting from the 14.6 km/s and still the answer is wrong.
Am I using th

A particle of mass m moves in one dimension under the influence of a force:
F(x, t) = (k/x^2)*e^(-t/T)
Where k and T are positive constants.
a) Compute the Lagrangianand Hamiltonian Functions
b) Compare the Hamiltonian and the total energy.
c) Discuss the conservation of energy for the system.

A lunar landing craft needs to hover just above the surface of the moon, which has a gravitational acceleration of (1/6) g. The exhaust velocity is 2000m/s but fuel amounting to only 20% of the total mass can be used.
How long can the landing craft hover?

Lagrangianand Hamiltonian's Mechanics:
Write down the Lagrangian L for two particles of equal mass m1 = m2 = m, confined to the x-axis and connected by a spring with potential energy U = ½ k x^2. (Here x is the extension of the spring, x = (x1- x2 -l), where l is the spring's outstretched length, and that mass l remains to

A system with one degree of freedom has a Hamiltonian
see attached
where A and B are certain functions of the coordinate q and p is the momentum conjugate to q.
a) Find the velocity q(dot)
b) Find the Lagrangian L(q q(dot)) (note variables)

Problem:
A) Write the Lagrangian for a simple pendulum consisting of a mass m suspended at the end of a massless string of length l. Derive the equation of motion from the Euler-Lagrange equation, and solve for the motion in the small angle approximation.
B) Assume the massless string can stretch with a restoring force F=-k(r

Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity w about its vertical axis, as shown in figure 7.17. Use cylindrical polar coordinates and let the equation of the parabola be z=kp^2. Write down the Lagrangian in terms of p as the

A satellite of mass 205 kg is launched from a site on the Earth's Equator into an orbit at 175 km above the surface of Earth.
(a) If the orbit is circular, what is the orbital period of this satellite?
s
(b) What is the satellite's speed in orbit?
m/s
(c) What is the minimum energy necessary to place this satellite in