The center of a long frictionless rod is pivoted at the origin, and the rod is forced to rotate in a VERTICAL plane with constant angular velocity \omega. Write down the Lagrangian for a bead of mass m threaded on the rod, using r as your generalized coordinate, where r, \phi are the polar coordinates of the bead. (Notice that \phi is not an independent variable since it is fixed by the rotation of the rod tobe \phi = \omega * t) Solve Lagrange's equation for r(t). What happens if the bead is initially at rest at the origin? If it is released from any point ro > 0, show that r(t) eventually grows exponentially. Explain your results in terms of the centrifugal force m* \omega ^2 *r. This is an altered version of problem 21 in Chapter 7 from Taylor's Classical Mechanics book.Â

from 1609, galileo galilei used the recently invented telescope to observe the sun, moon and planets. he saw the mountains and craters of the moon, and for the first time revealed the planets to be worlds in their own right. galileo also provided strong observational evidence that planets orbited the sun.

his discoveries about the moon, jupiter's moons, venus, and sunspots supported the idea that the sun - not the earth - was the center of the universe, as was commonly believed at the time.galileo turned his gaze toward venus, the brightest celestial object in the sky - other than the sun and the moon.

explanation:

a fuel engine ( known as a gasoline engine in american english )is an internal combustion engine with spark ignition, designed to run on petrol (gasoline) and similar volatile fuels.

.) the observation by semmelweis that more mothers died of childbed fever when their doctors also worked in the morgue

be subject to a force andchange its motion