Lagrangian problems, constrained point masses

Problem:

A circular hoop of radius r rotates with angular frequency ω about a vertical axis through the center of the hoop in the plane of the hoop.  A bead of mass m slides without friction around the hoop and is subject to gravity.
(a)  Give the Lagrangian in terms of the angle θ shown in the drawing.
(b)  Find the equation of motion in terms of this angle.
(c)  Find the values of θ for which the bead may be stationary with respect to the hoop and determine which of the stationary points are stable.

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Solution:

Similar problems

Problem:

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In the figure above the mass point m rotates about the point s.  At the same time the thread holding the mass point is shortened continuously.  Find the constants of motion.

Solution:

Problem:

An object of mass m slides on a horizontal, friction-free horizontal table.  A light, inextensible string, which passes through a small hole in the table, attaches the mass to a second body of mass M.  The second body hangs below the table where g = -9.8 m/s2 (see the sketch).  At time t = 0, mass m is located at position (r, θ).

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(a)  Determine the differential equations governing the motion of the system.
(b)  For the special case that r = constant, solve the resulting equations and interpret your results.
(c)  What are the constants of motion for this system?

Solution:

Problem:

A rigid wire shaped like an upside-down L is spinning about its vertical segment as shown in the figure.

wire

The angular velocity of the motion is Ω.  A bead of mass m is constrained to slide without friction on the horizontal segment of the wire and is connected by a massless string to an identical bead on the vertical segment.  The string has constant length l and follows the shape of the wire without friction.  The bead on the vertical segment is subjected to gravity (g is downward).
Consider only situations when this geometry is valid.
(a)  In terms of the distance r of the revolving bead from the center of rotation, write down the Lagrangian for this system.
(b)  Find the equation of motion for the bead.
(d)  Solve the equation of motion for the trajectory of the bead.

Solution:

Problem:

Consider the double pendulum consisting of two massless rods of length L = 1 m and two point particles of mass m = 1 kg in free space, with a fixed pivot point.
(a)  Write down the Lagrangian of the system shown in terms of the coordinates θ and α shown and the corresponding velocities.
(b)  Find two conserved quantities.
(c)  Write down the second-order differential equations of motion.

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Solution:

Problem:

A particle of mass equal to 3 kg moves in the xy plane.  The potential energy of the particle as a function of position is given by  U = (36 J/m2)xy - (48 J/m2)x2.  The particle starts at time t = 0 from rest at the point with position vector (x, y) = (10 m, 10 m).
(a)  Set up the differential equations describing the motion and solve them to determine the position of the particle as a function of time.
(b)  Find the velocity as a function of time.

Solution:

Problem:

Obtain Lagrange's equations of motion for a spherical pendulum (a mass point suspended by a rigid, weightless rod).

Solution:

Problem:

Consider a bead of mass m sliding freely on a smooth circular wire of radius b which rotates in a horizontal plane about one of its points O, with constant angular velocity Ω.  Let θ be the counterclockwise angle between the diameter that passes through the mass and the diameter that passes through the point O, with θ = 0 the case where the mass is farthest from O.

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(a)  Find the equation of motion for θ.  Compare this equation with the equation of motion for a simple pendulum (point mass and massless rod).
(b)  For the initial conditions θ = 0, dθ/dt = ω0 at t = 0, describe the θ motion that occurs for |ω0| < 2Ω and for |ω0| > 2Ω.  (Note:  The same equations have the same solutions.)
(c)  Describe the θ motion that occurs for |ω0| << 2Ω.
(d)  Find the force that the wire exerts on the bead as a function of θ and dθ/dt.

Solution:

Problem:

A bead, of mass m, slides without friction on a wire that is in the shape of a cycloid with equations
x = a(2θ + sin2θ),
y = a(1 - cos2θ),
- π/2 ≤ θ ≤ π/2.

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A uniform gravitational field g points in the negative y-direction.
(a)  Find the Lagrangian and the second order differential equation of motion for the coordinate θ.
(b)  The bead moves on a trajectory s with elements of arc length ds.
Integrate ds = (dx2 + dy2)½ = ((dx/dθ)2 + (dy/dθ)2)½dθ with the condition s = 0 at θ = 0 to find s as a function of θ.
(c)  Rewrite the equation of motion, switching from the coordinate θ to the coordinate s and solve it.  Describe the motion.

Solution: