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 r0. 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 =12.2 cm 0 v0=0 cm/s String Confinement C=5.4*10-5 J-m m=5.7 g
l=1.00 cm y =23 cm 0 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*10-4 J-m
l=1.00 cm m magnet=5.8 g mbeam=10.0 g Beam length=31.9 cm y =0.88 cm
0 v0=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 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
1.01.0 y0 T y0 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 m magnet=5.8 g Period (T)
mbeam=10.0 g Beam length=31.9 cm y =0.88 cm 0
v0=0 cm/s 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 Thanks for listening!
Small Amplitude Approximation S.H.O., Damping force proportional to velocity: y ( t ) y0 e , where
b t 2 cosd t 2 2
b d o Finite Amplitude Constant friction
Both term Damping force proportional to velocity