Problem 14 Magnetic Spring Reporter: Hsieh, Tsung-Lin

Question  Two magnets are arranged on top of each other such that one of them is fixed and the other one can move vertically.  Investigate oscillations of the magnet.

Outline  Horizontal Dimension (Force field)‏  Experimental Setup  Experimental Result  Vertical Dimension  Analysis  Summary

Horizontal Dimension (Force field)‏  Experimental Setup  Experimental Result  Vertical Dimension  Analysis  Summary

Forces  Magnetic force  Gravitational force  Dissipative force

 Cylindrical magnet can be interpreted by a magnetic dipole.  When the upper magnet is at the unstable equilibrium position, the separation is said to be r 0. Force Field Fig. Potential diagram for the upper magnet

 Horizontal Dimension Experimental Setup  Experimental Result  Vertical Dimension  Analysis  Summary

Tube Confinement  Large friction  Start with large amplitude Side view Top view Tube

String Confinement  Large friction  Start with large amplitude Side view Top view String

Beam Confinement  Almost frictionless  Start with small amplitude

Experimental Procedures  Perturb the upper magnet  Record by camera  Change initial amplitude  Change length (l)‏  Change mass (m)‏

When Rod is Long and Light… y y

 Horizontal Dimension  Experimental Setup Experimental Result  Vertical Dimension  Analysis  Summary

Tube Confinement  C=6.4*10 -4 J-m  m=5.8 g  l=1.00 cm  y 0 =12.2 cm  v 0 =0 cm/s

String Confinement  C=5.4*10 -5 J-m  m=5.7 g  l=1.00 cm  y 0 =23 cm  v 0 =0 cm/s

Experimental Results  with Period  The curve at the bottom turning point is sharper  Amplitude decays  Period reduces

Beam Confinement  C=6.4*10 -4 J-m  l=1.00 cm  m magnet =5.8 g  m beam =10.0 g  Beam length=31.9 cm  y 0 =0.88 cm  v 0 =0 cm/s

Experimental Results  Almost frictionless  Periodic motion  T=0.17 ±0.00 s

 Horizontal Dimension  Experimental Setup  Experimental Result Vertical Dimension  Analysis  Summary

Magnetic Force vs. Separation

Verifying the Equation l l r

 Horizontal Dimension  Experimental Setup  Experimental Result  Vertical Dimension Analysis  Analytical  Numerical  Summary

Equation of Motion : Moment of Inertia

Small Amplitude Approximation Small oscillation period T s = The force can be linearized.

Finite Amplitude, Thus, there are only three parameters,,.

Numerical Solution Finite oscillation period T=f (T s,, )‏

Comprehensive Solution of  y 0 ↑ ， T↑  y 0 →0 ， T →Ts  l →large ， T X l

Usage of the Solution Diagram Period (T)‏  C=6.39*10 -4 J-m  l=1.00 cm  m magnet =5.8 g  m beam =10.0 g  Beam length=31.9 cm  y 0 =0.88 cm  v 0 =0 cm/s

Finite Damping

 Horizontal Dimension  Experimental Setup  Experimental Result  Vertical Dimension  Analytical Modelling  Numerical Modelling Summary

 Confinements  Tube  String  Beam  Analytical Modelling  Numerical Modelling Summary

Thanks for listening!

 S.H.O.,  Damping force proportional to velocity: Small Amplitude Approximation, where Fig. Analytical result Fig. Tube confinement result Thus,

 S.H.O.,  Constant friction:  Damping force proportional to velocity: Small Amplitude Approximation, where Fig. Analytical result Fig. Tube confinement result, Thus,

Finite Amplitude  Constant friction Damping force proportional to velocity Both term