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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
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Problem 14Magnetic 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 r0. Fig. Potential diagram for the upper magnet
Horizontal Dimension Experimental Setup • Experimental Result • Vertical Dimension • Analysis • Summary
Tube Tube Confinement • Large friction • Start with large amplitude Top view Side view
String Confinement String • Large friction • Start with large amplitude Top view 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 • y0=12.2 cm • v0=0 cm/s
String Confinement • C=5.4*10-5J-m • m=5.7 g • l=1.00 cm • y0=23 cm • v0=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*10-4J-m • l=1.00 cm • mmagnet=5.8 g • mbeam=10.0 g • Beam length=31.9 cm • y0=0.88 cm • v0=0 cm/s
Experimental Results • T=0.17 ±0.00 s • Almost frictionless • Periodic motion
Horizontal Dimension • Experimental Setup • Experimental Result Vertical Dimension • Analysis • Summary
Verifying the Equation l r l
Horizontal Dimension • Experimental Setup • Experimental Result • Vertical Dimension Analysis • Analytical • Numerical • Summary
: Moment of Inertia Equation of Motion
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 • y0↑,T↑ • y0→0, T→Ts • l →large,TXl 1.0 1.0 1.4 2.2
Period (T) 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 • y0=0.88 cm • v0=0 cm/s
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
, where Thus, Small Amplitude Approximation • S.H.O., • Damping force proportional to velocity: Fig. Analytical result Fig. Tube confinement result
Finite Amplitude Damping force proportional to velocity Constant friction Both term