Problem 14 Magnetic Spring Reporter Hsieh TsungLin Question

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

Force Field 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. Fig. Potential diagram for the upper magnet

Horizontal Dimension Experimental Setup Experimental Result Vertical Dimension Analysis Summary

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

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

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)

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 p. The curve at the bottom turning point is sharper p. Amplitude decays p. Period reduces

Beam Confinement C=6. 4*10 -4 J-m l=1. 00 cm mmagnet=5. 8 g mbeam=10. 0 g Beam length=31. 9 cm y 0=0. 88 cm v 0=0 cm/s

Experimental Results p. Almost frictionless p. 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 r l

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

Equation of Motion : Moment of Inertia

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

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

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

Comprehensive Solution of 1. 0 y 0 ↑，T↑ y 0 → 0， T →Ts l →large，T X l 1. 4 2. 2

Usage of the Solution Diagram C=6. 39*10 -4 J-m l=1. 00 cm mmagnet=5. 8 g mbeam=10. 0 g Beam length=31. 9 cm y 0=0. 88 cm v 0=0 cm/s Period (T)

Finite Damping

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

Summary Confinements Tube String Beam Analytical Modelling Numerical Modelling 1. 0 1. 4

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Small Amplitude Approximation S. H. O. , Damping force proportional to velocity: , where

Finite Amplitude Constant friction Both term Damping force proportional to velocity