Question

3) Under forced oscillation, the phase of harmonic motion of the particle differs from the phase of the driving force . Explain.

4) Explain how Galileo, described the motion of moon with respect to its planet ?

Answer 1

Dear Student

1. A differential equation for the motion of a damped harmonic oscillator can be written
$$\begin{gathered} m\frac{d^{2}x}{d{t}^2}+2m\zeta {\omega }_0\frac{dx}{dt}+m{{\omega }_0}^{2}x=F_{0}Sin\left(\omega t\right) \\ mistheinertialmassofthesystem \\ {\omega }_{0}isthecharactersticsofthefrequency \\ \zeta isthedampingfactor \\ {F}_{0}istheamplitudeofthedrivingforce \\ \omega isthefrequency \\ thestationaryt\to \infty solutiontakestheshapex\left(t\right)=A_0\sin (\omega t-{\phi }_0) \\ thepahsedifference \\ {\phi }_0=arc\tan \left(\zeta \frac{2\omega {\omega }_0}{{\omega }^2-{{\omega }_0}^2}\right) \\ Itisthephaselag,sowiththeimplicitychosenphaseconvention \\ ithastobepositive \\ foranon-zerodamping,intheresonantcase\omega ={\omega }_0 \\ \omega ={\omega }_0,theargumentofthearc\tan functiondiverges, \\ sothephasedifferenceturnsoutinthiscasetobe\pi /2. \end{gathered}$$


2.

Galileo observed with his telescope what he described at the time as "three fixed stars, totally invisible by their smallness", all close to Jupiter, and lying on a straight line through it. Observations on subsequent nights showed that the positions of these "stars" relative to Jupiter were changing in a way that would have been inexplicable if they had really been fixed stars. An observation which he attributed to its being hidden behind Jupiter. Within a few days, he had discovered three of Jupiter four moons. He discovered the fourth on 13 January. Galileo named the group of four the Medicean stars, in honour of his future patron. Later astronomers, however, renamed them Galilean satellites in honour of their discoverer. These satellites are now called lo, Europa, Ganymede and Callisto.


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