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Bead Sliding on Uniformly Rotating Wire in Free Space

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Bead Sliding on Uniformly Rotating Wire in Free Space. Straight wire, rotating about a fixed axis  wire, with constant angular velocity of rotation ω . Time dependent constraint! Generalized Coords: Plane polar:  x = r cos θ , y = r sin θ , but θ = ω t , θ = ω = const

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bead sliding on uniformly rotating wire in free space
Bead Sliding on Uniformly Rotating Wire in Free Space
  • Straight wire, rotating about a fixed axis  wire, with constant angular velocity of rotation ω.
    • Time dependent constraint!
  • Generalized Coords: Plane polar:

 x = r cosθ, y = r sinθ, but θ = ωt, θ = ω = const

  • Use plane polar results:

T = (½)m[(r)2 + (rθ)2] = (½)m[(r)2 + (rω)2]

  • Free space  V = 0. L = T - V = T

Lagrange’s Eqtn:(d/dt)[(L/r)] - (L/r) = 0

 mr - mrω2 = 0  r = r0 eωt

Bead moves exponentially outward.

example from marion s book
Example (From Marion’s Book)
  • Use (x,y) coordinate system in figure to find T, V, & L for a simple pendulum (length  , bob mass m), moving in xy plane. Write transformation eqtns from (x,y) system to coordinate θ. Find the eqtn of motion.

T = (½)m[(x)2 + (y)2], V = mgy

 L = (½)m[(x)2 + (y)2] - mgy

x =  sinθ, y = -  cosθ

x =  θ cosθ, y =  θ sinθ

L = (½)m(θ)2 + mg  cosθ

(d/dt)[(L/θ)] - (L/θ) = 0

 θ+ (g/) sinθ = 0

example from marion s book4
Example (From Marion’s Book)
  • Particle, mass m, constrained to move on the inside surface of a smooth cone of half angle α(Fig.). Subject to gravity. Determine a set of generalized coordinates & determine the constraints. Find the eqtns of motion.

Worked on blackboard!

example from marion s book6
Example (From Marion’s Book)
  • The point of support of a simple pendulum (length b) moves on massless rim (radius a) rotating with const angular velocity ω. Obtain expressions for the Cartesian components of velocity & acceleration of m. Obtain the angular acceleration for the angle θ shown in the figure.

Worked on

blackboard!

example from marion s book8
Example (From Marion’s Book)
  • Find the eqtn of motion for a simple pendulum placed in a railroad car that has a const x-directed acceleration a.

Worked on blackboard!

example from marion s book10
Example (From Marion’s Book)
  • A bead slides along a smooth wire bent in the shape of a parabola, z = cr2 (Fig.) The bead rotates in a circle, radius R, when the wire is rotating about its vertical symmetry axis with angular velocity ω. Find the constant c.

Worked on

blackboard!

example from marion s book12
Example (From Marion’s Book)
  • Consider the double pulley system shown. Use the coordinates indicated & determine the eqtns of motion.

Worked on

blackboard!

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