An infinitely long, uniform rod of mass μ, per unit length is situated on the z axis. (a) calculate the gravitational force f on a point mass m at a distance rho from the z axis. (b) rewrite f in terms of the rectangular coordinates (x, y, z) of the point and verify that ∇ x f = 0. (c) show that ∇ x f = 0 using the expression for ∇ x f in cylindrical polar coordinates given inside the back cover. (d) find the corresponding potential energy u.

Comedians like to joke that the reason we haven’t been visited by intelligent life from elsewhere in the universe is that aliens have been monitoring earth’s broadcasts of intellectually embarrassing tv programs, like gilligan’s island, fear factor, the jersey shore, and the jerry springer showand so consider us far too primitive to merit a visit. let’s check the assertion that aliens could have been receiving them. tv programs are broadcast at a frequency of about 100mhz with about 100kw of total power in 30frames per second, which emanatesroughly uniformly in all directions. assume that interstellar space transmits these broadcasts without attenuation. no matter how smart they are, aliens would require at least one photon per frame to interpret our signals. findthe number of photons per unit time per unit area reaching a receiver on a possible planet in the nearest star system, which is about 4 light-years away (a light-year is the distance light travels in a year). if aliens aimed a receiver or detector directly at earth, how big (in diameter) would it have to be to receive a photon per frame

Avehicle has an oil leak that is causing the entire oil pan to be wet, but inspection reveals no exact source after cleaning off the oil residue. technician a says to install a fluorescent dye in the crankcase and operate the engine, then re-inspect for leaks with a special light (black light). technician b says the oil leak may be coming from a source at the top of the engine, such as a valve cover gasket. who is correct?

Three charge are arranged as shown in the diagram. the magnitude of the net electrical force acting on the +6 uc charge, rounded to the tenths place, is .