Exercise:
Pohl's pulum consists of a ring which rotates about its center and a spiral spring producing a restoring torque on the ring. We can ase the torque tau to be proportional to the deflection angle phi of the disc: tau-kphi. vspacemm As a numerical example we consider a ring with outer radius roO inner radius riO mass mO and a spring with elastic constant kO. abcliste abc Show that the period of Pohl's pulum is given by T pisqrtfracmleftr_i^+r_a^rightk Calculate the period for the numerical values given above. abc Asing linear damping tau_d-beta dotphi e.g. via eddy currents derive the formal expression for the damping coefficient for this system. Calculate the value for beta and for the damping ratio zeta if the half-life is ThO. abcliste

Solution:
The basic for rotational motion can be written as taut -kphit Jddotphit labelrot The moment of inertia for a ring hollow cylinder with respect to the its center is J fracmleftr_i^+r_a^right labelinertia From refrot and refinertia it follows ddotphit -frackJphit This is the differential for a simple harmonic motion with angular frequency omega_ sqrtfrackJ sqrtfrackmleftr_i^+r_o^right The period is thus T fracpiomega_ TF pitimessqrtfracmtimesleftri^+ro^righttimesk T approx resultTP abc The total torque is ssctautott tau+tau_d -kphit-betadotphit Jddotphit Solving for the angular acceleration leads to ddotphit -frackJphit-fracbetaJdotphit -omega_^phit-deltadotphit It follows for the damping coefficient delta fracbetaJ The half-life is given by T_/ fracln delta fracln Jbeta After solving for the coefficient beta we find beta fracln JT_/ beF fracln times mleftri^+ro^rightTh be approx resultbeP- The damping ratio is zeta fracdeltadelta_c fracln /T_/omega_ fracln T_/sqrtfracJk zeF fracln Thtimessqrtfracmtimesleftri^+ro^righttimesk ze approx resultzeP