Deriving Mass Distribution & Rotation Curve Function

[b_o3]

Hello everyone... can someone help me with this problem please:

The rotation curve V(r) for a mass distribution characterizes the rotational velocity of a test particle in orbit in its gravitational field as a function of radius from its center. Suppose you have a spherically symmetric mass distribution with the mass density p(r)=p0(r0/r)^3/2, where r0 and p0 are constants, derive the expression for M(r), the total mass interior to r. From this derive the rotation curve function. Suppose the mass distribution is trunctuated at some radius R0, what do you expect the rotation curve to look like (i.e as a function of r) at r>R0

any help would be appreciated, thanks :D

 

[Mentz114]

We need to see some attempt at a solution. What is the mass of a sphere with radius r and density p ?

 

[b_o3]

M= density*( (4/3)pi * r^3

 

[Mentz114]

If you put the density expression p(r) into your equation, you've nearly solved the first part.

 

[b_o3]

oh thanks, so i got M(r)= p0(r0/r)^3/2*4/3pi*r^3... now I am supposed to derive the rotation curve function... isn't that just the circumference?

 

[b_o3]

can u help me out with deriving the rotation curve function... from that formula

 

[Mentz114]

To get the required expression for M(r) you must now work out the mass of a thin shell of thickness dr, then integrate that expresion wrt to r from 0 to r.
You need calculus now.

 

[b_o3]

so just replace the r with dr?

 

[Mentz114]

Try

mass of shell = M(r+dr)-M(r)

 

[b_o3]

thanks... ...

 

[b_o3]

should i integrate this formula or not

 

[Mentz114]

Yes, integrate it between 0 and r. This will give the final expression for M(r).

I have to go offline now, so it's over to you.

 

[b_o3]

alryty.. thanks a lot :D

 

[b_o3]

wait a minute...

 

[b_o3]

can u PLZZZZ show me how to do the integration.. its just not working :S ... myt have something to do with my being up all nyt Oo

 

[Mentz114]

Go to bed.

 

[b_o3]

no... :(?...

 

[b_o3]

lol yea... i need this done today tho
seriously can't u just give me the integrated formula i can't get it :S (i dnt take calculus_)

 

[b_o3]

COME On SAVE me ! I've just got this one problem left