Standard Lagrangian problems (other problems)

One generalized coordinate

Problem:

Consider the system shown below, in which two bodies of the same mass M are connected by a massless string across a massless frictionless pulley.  The coefficient of friction for the mass on the incline is µ and the angle of the incline is 30o with respect to the horizontal.  Define the z-coordinate as shown.  Assume downward motion and µ < 1.
(a)  Determine the Lagrangian L(z, dz/dt) for the system.
(b)  If at t = 0, z = z0, and v = v0, find z(t) until the block is stopped by the pulley.

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Solution:

Problem:

A flexible uniform string of mass M and length L slides smoothly over a circular, frictionless peg of radius R, with the right-hand end moving downward in a uniform gravitational field with g = 9.8 m/s2.  The string is released from rest in a situation where the right-hand end hangs below the left-hand end by an amount d.

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(a)  Find an expression for Z, the vertical position of the right-hand end as a function of time during the interval when the string is still in contact with the peg over the full upper half of the peg.  Assume that the position Z of the right-hand end is measured downward from the equilibrium position Z = 0.
(b)  In the limit R --> 0, find Z also for times subsequent to the loss of contact with the peg.

Solution:

Problem:

For the Atwood Machine shown, all pulleys are smooth and non-rotating and all cords are inextensible.  Work out the accelerations of 3m and 5m.
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Solution:

Problem:

A thin uniform rod of length L and rotates with a constant angular velocity ω around a fixed point A.  (See figure!)   Find the angle of inclination θ.

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Solution:

Problem:

A particle of mass m is attached (so it cannot slide) to the midpoint of a weightless rod of length l.  The ends of the rod are constrained to move along the x and y axes.  A uniform gravitational field acts
in the negative y-direction.  Use θ as a generalized coordinate.  Neglect friction.
(a)  Write the Lagrangian and obtain the equation of motion.
(b)  Solve the equation of motion for small θ, i.e. |θ| << 1, assuming that, at t = 0, θ = θ0 and dθ/dt = 0.

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Solution:

Problem:

A particle of mass m moves in one dimension acted upon by the force
F(x) = -kx + a/x3
where k and a are both positive constants. 
(a) Sketch the potential energy. 
(b) Sketch the velocity phase space portrait for this system.
(c) Calculate the locations of the equilibrium points of the motion.
(d) Indicate whether each of these points is stable or unstable.
(e) Find the Hamiltonian.

Solution:


Multiple generalized coordinates

Problem:

(a)  Give the definition of a cyclic variable.
(b)  How is it related to conservation laws and physical symmetry?
(c)  Give two examples of systems described by Lagrangians with cyclic variables and relate them to part b.

Solution:

Problem:

Consider the 2D problem of a free particle of mass m moving in the xy plane.
(a)  Use the Lagrangian formalism to find the equations of motion of the particle in polar coordinates (r, θ).
(b)  Find the general solution for the orbit r(t).
Hint:  Use energy conservation!
(c)  At t = 0, let r = b,  θ = 0,  dr/dt = 0,  dθ /dt = ω0.  Find r(t) and θ(t).

Solution: