Exercise:
A Wilberforce pulum consists of a mass on a helical spring. The longitudinal motion and the torsional motion are coupled through the spring i.e. if the pulum is started with a purely torsional motion it will start to move up and down. center includegraphicswidthcm#image_path:wilberforcpulum-# center The differential s for the motion of a Wilberforce pulum are given by mddot z & -k z -epsilontheta Jddot theta -kappatheta - epsilon z where m and J are the mass and moment of inertia of the pulum bob k and kappa the linear and rotational spring constant and epsilon the coupling constant for the coupling between the longitudinal and torsional motion. abcliste abc Derive the general coefficient matrix. abc For appropriate parameters the longitudinal and torsional frequencies are equal i.e. k/mkappa/J. Calculate the eigenvalues and verify that the eigenvectors are hat x pmatrix pmfracrsqrt pmatrix abc For a spring with kkO and kappakaO calculate the radius for which the longitudinal and torsional frequency are the same. Calculate the frequencies of the two normal modes for a coupling constant epsilonepsO and a cylinder with mass mO. abcliste

Solution:
abcliste abc The coefficient matrix is bf A pmatrix -frackm & -fracepsilonm -fracepsilonJ & -frackappaJ pmatrix abc Since the two angular frequencies are equal we can write them as omega^ frackm frackappaJ The trace and the determinant of the matrix are then tau -omega^ Delta omega^-fracepsilon^mJ It follows for the eigenvalues lambda -omega^ pm sqrtomega^-omega^+fracepsilon^mJ -omega^pmfracepsilonsqrtmJ The moment of inertia for a cylinder with mass m and radius r is J fracmr^ The eigenvalues are thus lambda -omega^ pm fracsqrtepsilonsqrtm^ r^ -omega^ pm fracsqrtepsilonmr abc From the equality of the angular frequencies omega^ frackm frackappaJ frackappamr^ it follows for the radius r rF sqrtfractimeskak resultrP- The angular frequencies for the first normal modes is omega_ sqrtomega^-fracsqrtepsilonmr sqrtfrackm-fracsqrtepsilonmr sqrtfrackm-fracepsilonmsqrtfrackkappa omaF sqrtfrackmleft-fracepssqrtka times kright resultomaP For the second normal mode we find omega_ sqrtomega^+fracsqrtepsilonmr ombF sqrtfrackmleft+fracepssqrtka times kright resultombP abcliste