Gravitational Dynamics of Titanium-Devouring Aliens on a Spherical Asteroid

[PhysicsIsNotMe]

Titanium-Devouring Aliens-Gravitational Field,Potential Energy

(Titanium-Devouring Aliens) A spaceship of titanium-
devouring little green people has found a perfectly spherical asteroid that
they confirm is made of homogeneous titanium. A narrow shaft was bored
from some point on the surface to the center of the asteroid. But then, an
accident occurs when one of them falls into the hole at the surface, down the
shaft, and dies upon impact. However, these aliens being devoid of human
sympathy, continue working until out of the narrow shaft they have excavated
an entire sphere with a diameter equal to the asteroid's radius. Then again,
at this point, another accident occurs. From the same hole at the surface,
another green person falls, and again dies upon impact.
Now the Supreme Commander of the little green people is alarmed by
these events and sends an expert to investigate. In particular, the expert is
asked which of the two accidents had a larger impact speed. The expert does
more though; she sends the exact ratio between these speeds. What is this ratio that she sends? (Hint: It may help to consider a sphere of "negative
mass density".)

Ok so this is a question that was created out of thin air obviously by my teacher but whatever. The only thing I can think of is that the gravitational field in the asteroid must be equal everywhere, despite the shape of the shaft in which the alien falls. If this is the case wouldn't the ration be zero? Please help!

 

[Justin Lazear]

If your teacher made this up on the spot, your teacher is quite imaginative.

The question is basically asking you how much potential energy each alien has at the surface relative to the center of the larger sphere. How would you go about calculating this?

--J

 

[PhysicsIsNotMe]

I don't understand how you got that the question is asking you for the potential energy each alien has at the surface relative to center of larger sphere. Anyways I barely understand the question and the concepts behind it. *sigh* I hate these questions my teacher just makes up.

 

[Justin Lazear]

First things first. Energy is conserved. So what's the alien's energy at the top? We know that energy can be written as the sum of the potential and kinetic energy,

E_top = KE + PE

but initially, the alien isn't moving. So KE = 0. This leaves, at the top,

E_top = PE

Then, at the bottom, we repeat the process.

Finally, we use energy conservation and say

E_top = E_bottom

and from this equation we can solve for velocity, since KE depends on velocity.

--J

 

[PhysicsIsNotMe]

If at the bottom KE=0, then PE(top)=PE(bottom) but isn't the mass and everything else constant throughout. Doesn't this equation just give a ratio of one? If PE=-W, and W=1/2mv^2, wouldn't 1/2mv^2(top)=1/2mv^2(bottom) give a ratio of one. I guess I just really don't understand these concepts

 

[Justin Lazear]

Well, clearly at the bottom, KE is not zero, since the aliens were moving fast enough to die.

You do seem to be having a little trouble with the whole energy thing. Are you comfortable with energy arguments on, say, the Earth? For instance, can you solve the Earth's escape velocity problem?

--J

 

[PhysicsIsNotMe]

oh! well by at the bottom i was assuming you were saying the aliens had already landed and died and were just lying there motionless. I know how to work with equations and numbers and all, just have trouble when it comes to concepts and obscure questions about aliens :-/

ok so the equation I'm working with now is PE(top)=PE(bottom)+KE(bottom) which is mgy=mgy+1/2mv^2?

 

[Justin Lazear]

Okay, then turn them into balls of playdough and planets made of whatever material you're happy with. It doesn't change the concept.

Your equation is right, except you need to distinguish between the heights (since they're clearly different). Now that you have that, solve for v.

--J

 

[PhysicsIsNotMe]

well would the height at the top be 0 and the height at the crash be h, since we're not given numbers. If so then the equation would become 0=mgh+1/2mv^2, so then v=sqrt(-2gh) lol that can't be right

 

[Justin Lazear]

It's not as big a problem as you think, since h in your coordinate system would be negative, so that first negative will cancel away. A more natural coordinate system might be to set y = 0 at the center and y = h at the top, and then you wouldn't have that confusing negative.

Also, that expression for potential energy is valid only for the column hole, but not the spherical hole. You must use a different one for that.

--J

 

[PhysicsIsNotMe]

ok so for the column in which the alien falls the velocity is given by v=sqrt(2gh), now for the spherical one our teacher told us to consider the sphere as having negative charge density, how does this differ from doing it like the column.

I believe you use the formula v=sqrt((2GM)/R), where M=p[(4(pi)r^3)/3], where p is the density of sphere and thus the density of titanium?

 

[Justin Lazear]

It would seem that the teacher is asking you to calculate the potential energy supposing there were a complete sphere and then subtract off (i.e. add the negative, hence the negative density) twice the energy from a sphere made of the same stuff but that's only half the radius.

I'm not so sure if that'd a valid method to use, but I'm not willing to check at the moment, so you're on your own. Sorry.

--J

 

[PhysicsIsNotMe]

ummm ok? anyone else that could help with the sphere part please do :)