Residents of Flatworld (a two-dimensional world far, far away) have it easy. Although quantum mechanics applies in their world, the equations they must solve to understand atomic energy levels involve only two dimensions. In particular, the Schrodinger equation for the one-electron "flatrogen" atom is, −~22m1r∂∂r(r∂∂r)ψ(r, θ) −~22m1r2∂2 ∂θ2 ψ(r, θ) + U(r)ψ(r, θ) = Eψ(r, θ). (1)
(a) Separate variables by trying a solution of the form ψ(r, θ) = R(r)Θ(θ), then dividing by R(r)Θ(θ). Show that the "θ-equation" can be written as, d2 dθ2 Θ(θ) = CΘ(θ), (2) where C is a separation constant.
(b) To be physically acceptable, Θ(θ) must be continuous, which, since it involves rotation about an axis, means that it must be periodic. What must be the sign of C.
(c) Show that a complex exponential is an acceptable solution for Θ(θ).
(d) Imposing the periodicity condition, find the allowed values of C.
(e) What property is quantized according to the value of C?
(f) Obtain the "r-equation".
(g) Given that U(r) = −b/r, show that a function of the form R(r) = e−r/a is a solution, but only if C ia a certain one of its allowed values.
(h) Determine the value of a, and thus find the ground state energy and wave function of the flatrogen atom.

American eels (anguilla rostrata) are freshwater fish with long, slender bodies that we can treat as uniform cylinders 1.0 m long and 10 cm in diameter. an eel compensates for its small jaw and teeth by holding onto prey with its mouth and then rapidly spinning its body around its long axis to tear off a piece of flesh. eels have been recorded to spin at up to 14 revolutions per second when feeding in this way. although this feeding method is costly in terms of energy, it allows the eel to feed on larger prey than it otherwise could. 1.a field researcher uses the slow-motion feature on her phones camera to shoot a video of an eel spinning at its maximum rate. the camera records at 120 frames per second. through what angle does the eel rotate from one frame to the next? 2. the eel is observed to spin at 14 spins per second clockwise, and 10 seconds later it is observed to spin at 8 spins per second counterclockwise. what is the magnitude of the eels average angular acceleration during this time? 3. the eel has a certain amount of rotational kinetic energy when spinning at 14 spins per second. if it swam in a straight line instead, about how fast would the eel have to swim to have the same amount of kinetic energy as when it is spinning? 4.a new species of eel is found to have the same mass but one-quarter the length and twice the diameter of the american eel. how does its moment of inertia for spinning around its long axis compare to that of the american eel?

Ahot-air balloonist, rising vertically with a constant speed of 5.00 m/s releases a sandbag at the instant the balloon is 40.0 m above the ground. after it is released, the sandbag encounters no appreciable air drag. compute the velocity of the sandbag at 0.250 s after its release.

The coulomb's law explains which of the following? select all the force between two charged objects is inversely proportional to the charge on each object. the force between two charged objects is directly proportional to the charge on each object. the force between two charged objects is directly proportional to the distance separating them. the force between two charged objects is inversely proportional to the square of the distance separat coulomb's law is similar to newton's 2nd law. if the force between two objects is f and the charge on both objects is doubled, the new force between the would now be 2f. if the force between two objects is 2f and the distance separating them is doubled, the new force between the objects would now o the be zf.