In this problem we will perform a linear stability analysis of the fluid equations of motion for gravitational stability, with the effect of rotation included. Consider a rotating, self-gravitating cylinder, rotating with uniform angular velocity Nez, where e, is the unit vector in the z direction. Consider an adiabatic equation of state, so that 8P/8p=0", where a is the sound speed. The equation of continuity is, of course, ap/at + V. (pu) = 0
The equation for linear momentum, including the centrifugal and Coriolis forces, is now
au/at + 1/2Vu^2 + (V x u) x u = -VP/p + V + Rπ^2er - 2ex x u
and the gravitational potential satisfies Poisson's equation
V^2 = 4πGp
The first item is to characterise the equilibrium state. Assume po = constant. Po = constant, uo = 0 Show that the equilibrium is satisfied if
0 = 1/2R^2^2 + constant
and = 2πGp0.
Let us now consider linear perturbations to these equations with a specific frequency and wavenumber. That means we replace a/at with w and each a/ax with kx and similarly for y, x.
First consider the case for perturbations along the rotation axis, z, such that u1 = (0,0,u1). In this case, show that the perturbations must satisfy a dispersion relation w2 = k2a2 - 4πGp0. Derive the condition for instability and compare it to the Jeans mass we discussed in class.
Secondly, consider perturbation perpendicular to the rotation axis. In this case we can consider a Cartesian xy system. Show that the dispersion relation is now w2 = k2a2 - 4πGpo + 4π2. Does the addition of rotation make the system more stable or less stable?

First to answer all gets a donation question 3(multiple choice worth 1 points) (04.01 lc) which of the following levels is lower than cells in the hierarchy of organization of living things? molecules organs organ systems tissues question 4(multiple choice worth 1 points) (04.01 lc) which of these represents an individual form of life such as an animal, plant, or single-celled life form? organ organism organ system tissue question 5(multiple choice worth 1 points) (04.03 lc) what are the green structures inside plant cells where photosynthesis takes place called? chloroplast mitochondrion nucleus vacuole question 6(multiple choice worth 1 points) (04.03 lc) which of the following best matches a part of a living cell with the function it performs? cytoplasm: fluid which carries the nucleus and other organelles vacuole: fluid which carries the nucleus and other organelles cytoplasm: directs all of the cell's activities vacuole: directs all of the cell's activities question 7(multiple choice worth 1 points) (04.03 mc) read the table below. organelles of plant cells present in both plants and animals present in plants but not animals small in animals but large in plants organelle a organelle b organelle c which statement is most likely correct? organelle a is chloroplast, organelle b is cytoplasm, and organelle c is nucleus. organelle a is mitochondrion, organelle b is cytoplasm, and organelle c is nucleus. organelle a is cell wall, organelle b is chloroplast, and organelle c is mitochondrion. organelle a is mitochondrion, organelle b is chloroplast, and organelle c is vacuole. question 8(multiple choice worth 1 points) (04.03 mc) what will most likely happen in the absence of a nucleus? genetic information will not be transmitted by the cell. energy will not be released during cellular respiration. the cell will not store food, water, nutrients, and waste. the cell will lose its shape and structure.

For a function f(x), a) if f(7) = 22 then f^-1(22) = b) f^-1(f(c)) =

Is experimental probibilty the same as the observed frequency in math? i need the answer asap!