A microbe is ejected from its home planet by a violent event such as a volcanic eruption or a meteorite impact. The subsequent fate of the microbe depends on the balance of the two main forces that act on it, namely, the gravitational attraction of the star around which the planet orbits and radiation pressure due to the luminosity of the star. Required:
Model the microbe as a dark sphere of radius a at a distance r from the star. Assume that the total power radiated by the star P = 1.0 x 1031 J/s (greater than that of the sun by a factor of 30,000). Derive an expression for the radiation force on the microbe. Give your answer in terms of p, r, c and a.

Aballoon is rising vertically upwards at a velocity of 10m/s. when it is at a height of 45m from the ground, a parachute bails out from it. after 3s he opens his parachute and decelerates ata a constant rate 5m/s.when. (a) what was the height of the parachutist above the ground when he opened his parachute? (b)how far is the parachutist from the balloon at t=3s? (c)with what velocity does the parachutist hit the ground? (d)after how long does the parachutist hit the ground after his exist from the balloon?

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.

Low areas cut into a gun barrel are called