Question

A simple pendulum consists of a mass m attached to a light string of length l. When at rest it lies in a vertical line at x = 0. The pendulum is driven by moving its point of suspension harmonically in the horizontal direction as $xi$ = a cos $omega$t about its rest position (x = 0). There is a damping force $F_d$ = -bv due to friction as the mass moves through the air with velocity v. (a) Show that the horizontal displacement x of the mass, with respect to its equilibrium position (x = 0), is the real part of the complex quantity z where $frac{md^2z}{dt^2} + bfrac{dz}{dt} + momega_0^2z = momega_0^2ae^{iomega t}$ and $omega_0^2 = g/l$. (b) Assuming a solution of the form z = $Ae^{i(omega t-delta)}$, find the phase angle $delta$ between the driving force and the displacement of the mass.

          A simple pendulum consists of a mass m attached to a light string of length l. When at rest it lies in a vertical line at x = 0. The pendulum is driven by moving its point of suspension harmonically in the horizontal direction as $xi$ = a cos $omega$t about its rest position (x = 0). There is a damping force $F_d$ = -bv due to friction as the mass moves through the air with velocity v.
(a) Show that the horizontal displacement x of the mass, with respect to its equilibrium position (x = 0), is the real part of the complex quantity z where
$frac{md^2z}{dt^2} + bfrac{dz}{dt} + momega_0^2z = momega_0^2ae^{iomega t}$
and $omega_0^2 = g/l$. (b) Assuming a solution of the form z = $Ae^{i(omega t-delta)}$, find the phase angle $delta$ between the driving force and the displacement of the mass.

$A simple pendulum consists of a mass m attached to a light string of length l. When at rest it lies in a vertical line at x = 0. The pendulum is driven by moving its point of suspension harmonically in the horizontal direction as xi = a cos omegat about its rest position (x = 0). There is a damping force Fd = -bv due to friction as the mass moves through the air with velocity v. (a) Show that the horizontal displacement x of the mass, with respect to its equilibrium position (x = 0), is the real part of the complex quantity z where fracmd^2zdt^2 + bfracdzdt + momega0^2z = momega0^2ae^iomega t and omega0^2 = g/l. (b) Assuming a solution of the form z = Ae^i(omega t-delta), find the phase angle delta between the driving force and the displacement of the mass.$

Added by Carol P.

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