Easy fluid mechanics problem?

Question

The planet mercury has been turned into mercury with the
uniform denstiy (p) of liquid mercury. An alien pentagonal 
obelisk (density=p/3) containing a message for earth remains 
at rest near mercury's center until equilibrium is reached at which 
time it moves along the radial coordinate without frictional head. 
In order to reach Earth, the oblisk must reach escape velocity.
Does the obelisk reach Earth?

Data planet
No thermo.
elastic modulus = 3 * 10 ^ 9 N/m^2
p = 13570 kg/m^3

Data obelisk:
h1 = 10m (body)
h2 = 1m 
edge=1 m

Anonymous · Accepted Answer

The PE to escape from the planet's surface is 

PE1 = - mMG/R


Assuming constant density from the surface to the center, the gravitational field varies with 

F = m MG/R^2 *  r/R

or integrating from 0 to R

PE2 = ∫ m MG/R^2 * r/R dr  [From R to 0]

PE2 = m MG/R^2  1/2 r^2/R  [From R to 0]
PE2 = m  -1/2 MG/R

However, Archemides rules and, at least until the obelisk reaches the surface, you will have a net bouyance of - 2/3 m.  (This is just the PE of the mercury moving towards the center and displacing the obelisk up)

So, we just combine everything and see where we are:

Σ PE = - mMG/R + 2/3 m *1/2 MG/R
Σ PE = - 2/3m  MG/R

It will only have 1/3 of the energy it needs to escape or a velocity at the surface of  v = √1/3 v.esc.  Actually, it wouldn't even have enough energy to reach orbit which is v.orbit = √ 1/2 v.esc.

Anonymous · Answer

Easy my @rse.

Easy fluid mechanics problem?

2 Answers