A Bead on a Rotating Hoop

Transcription

1 Ryan Seng and Michael Meeks May 15, 2008

2 Outline Table of Contents

3 Figure: Hoop Diagram Introduction Introduction & Background The behavior of the bead will vary as it travels along the hoop, the dependent factor being the hoop s angular velocity.

4 Introduction Forces Diagram

5 Frictionless Introduction Frictionless Introduction Acceleration Forces Fictitious Force - Centrifugal Force - Gravitational Force -

6 Frictionless Calculations Frictionless Calculations Begin with Newton s Equation F = ma Substitute in Forces and Acceleration mg sin φ + mrω 2 sin φ cos φ = mr d 2 φ dt 2

7 Frictionless Calculations Making Equation Dimensionless Introduce new variable τ, τ = t/t. dτ/dt = 1/T. Taking the derivative of φ with respect to time we get dφ dt = dφ dτ dτ dt dφ 1 dτ T We then take the second derivative of φ with respect to time d 2 φ dt 2 = d ( ) dφ d ( ) 1 dφ dt dt dt T dτ ( d 1 dτ T ) dφ dτ dτ dt 1 T d 2 φ 1 dτ 2 T 1 d 2 φ T 2 dτ 2

8 Frictionless Calculations Frictionless Calculations Cont. Replace d2 φ with 1 d 2 φ dt 2 T 2 dτ 2 mg sin φ + mrω 2 sin φ cos φ = mr 1 T 2 d 2 φ dτ 2 (1) Divide through by mg and let T equal b/mg Introduce γ and ε ( ) rω 2 sin φ + sin φ cos φ = g ( m 2 gr b 2 sin φ + γ sin φ cos φ = ε d 2 φ dτ 2. ) d 2 φ dτ 2.

9 Frictionless Equilibrium Solutions Frictionless Equilibrium Solutions Two equilibrium solutions Figure: Two Equilibrium Solutions.

10 Frictionless Equilibrium Solutions Frictionless Equilibrium Solutions Potential for three equilibrium solutions Figure: Three Equilibrium Solutions.

11 Frictionless Examples Frictionless Examples 4 γ = Velocity φ

12 Frictionless Examples Frictionless Examples Cont. γ=1 2 Velocity φ

13 Frictionless Examples Frictionless Examples Cont. γ = Velocity φ

14 Frictionless Examples Frictionless Examples Cont. x 10 3 γ= Velocity φ

15 Friction Introduction Friction Introduction Acceleration Forces Fictitious Force - Centrifugal Force - Gravitational Force - Friction

16 Friction Calculations Friction Calculations Pick up where we left off on Equation (??) Introduce friction b T mg sin φ + mrω 2 sin φ cos φ = mr 1 T 2 d 2 θ dτ 2 dφ dτ mg sin φ + mrω2 sin φ cos φ = mr 1 T 2 d 2 θ dτ 2

17 Friction Calculations Friction Calculations Cont. Divide through by mg and sub in b/mg for T (1) dφ dτ sin φ + Introduce γ and ε again ( rω 2 g ) sin φ cos φ = ( m 2 gr b 2 ε d 2 φ dτ 2 = dφ sin(φ) + γ sin(φ) cos(φ). dτ ) d 2 θ dτ 2

18 Friction Equilibrium Solutions Friction Equilibrium Solutions To find the Equilibrium Solutions we deal with the first ODE. bdφ/dt = mg sin(φ) + mrω 2 sin(φ) cos(φ) We then set φ equal to 0 and factor mg sin(φ) ( 1 + rω2 g Setting the first part equal to 0 This happens at 0 and π mg sin(φ) = 0 ) cos(φ) = 0

19 Friction Equilibrium Solutions Friction Equilibrium Solutions Cont. Setting the second part equal to 0 Solving for φ we get Subbing γ back in we get 1 + rω2 g cos(φ) = 0 φ = ± cos 1 ( g rω 2 ) φ = ± cos 1 ( 1 γ ) This creates another equilibrium solution, but it is not in a specific position

20 Friction Equilibrium Solutions Friction Equilibrium Solutions Cont. Two obvious Equilibrium Solutions Figure: Two Equilibrium Solutions.

21 Friction Equilibrium Solutions Friction Equilibrium Solutions Cont. Potential for Three Equilibrium Solutions Figure: Three Equilibrium Solutions.

22 Friction Examples Friction Examples 0.4 γ = Velocity φ

23 Friction Examples Friction Examples Cont. 0.1 γ = Velocity φ

24 Friction Examples Friction Examples Cont. 0.1 γ = Velocity φ

25 Supercritical Pitchfork Bifurcation Supercritical Pitchfork Bifurcation γ 1, Equilibrium Solution at 0 γ > 1, Equilibrium Solution 0 < φ < π/2 ( ) φ = ± cos 1 1 γ

26 Supercritical Pitchfork Bifurcation Supercritical Pitchfork Bifurcation Cont. Supercritical Pitchfork Bifurcation 2 φ γ

27 Supercritical Pitchfork Bifurcation References Arnold, David. Department of Mathematics. College of the Redwoods. Spring Brizard, Alain J. Lagrangian Mechanics. Department of Physics. Saint Michael s College. March 15, 2008 Bundschuh, R. Therotical Mechanics. Department of Physics. Ohio State University. Spring March 15, 2008 Frederic Moisy. Supercritical bifurcation of a spinning hoop. American Journal of Physics (2003): Research Library. ProQuest. College of the Redwoods Library, Eureka, CA. 25 Mar

28 Supercritical Pitchfork Bifurcation References Cont. Rosales, Rodolfo R. Bead moving along a thin, rigid, wire. Department of mathematics. Massachusetts Inst. of Technonlogy, Cambridge, Massachusetts, MA. October 17, March 15, Strogatz, Steven H. Nonlinear Dynamics and Chaos. 3.5 Overdamped Bead on a Rotating Hoop Perseus Books Publishing.