If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. the air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. in long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. if a transmission line has a length of 347 m, a linear density of 4.35 kg/m, and a tension of 65.4 mn, what are (a) the frequency of the fundamental mode and (b) the frequency difference between successive modes

Your starship, the aimless wanderer, lands on the mysterious planet mongo. as chief scientist-engineer, you make the following measurements: a 2.50-kg stone thrown upward from the ground at 14.0 m/s returns to the ground in 5.70 s ; the circumference of mongo at the equator is 2.20×10^5 km ; and there is no appreciable atmosphere on mongo. (a) the starship commander, captain confusion, asks for the following information: what is the mass of mongo? (b) if the aimless wanderer goes into a circular orbit 2.20×10^4 km above the surface of mongo, how many hours will it take the ship to complete one orbit? (c) an unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 44.0 km . to the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 23.0 m/s . if nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?