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