1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Water Pressure in Centrifuge

  1. Aug 26, 2014 #1
    1. The problem statement, all variables and given/known data
    A test tube filled with water is being spun around in an ultra centrifuge with angular velocity ω. The test tube is lying along a radius, and the free surface of the water is at a radius r0.

    (a) Show that the pressure at radius r within the test tube is P = (1/2)rho*ω2(r2-r02). Ignore gravity and atmospheric pressure

    2. Relevant equations
    dP= -ρgdz

    3. The attempt at a solution
    Since the test tube is undergoing circular motion, and the water within the test the test tube lies at a varying radius from center, the acceleration (g) is given by a=(v2)/r = ω2r

    So dP = -ρgdz = -ρω2rdr

    P= -(1/2)ρω2r2 from r to r0

    I think that much is fine, and this gives me the correct answer... But, here is where my understanding is hazy. If I define r to be increasing radially from center, then it is my understanding that my lower bound should be r and my upper to be r0.

    Here's my question: If I define an axis to be increasing in one direction, then should I always integrate from the lower value to higher?

    If I use this same reasoning with a simple pressure vs. water depth example then I run into trouble. If I want to find the pressure at a depth of 30m and I define my z axis to be increasing upwards then it seems to me the integral should look like this:

    dP = ∫-ρgdz from -30 to 0 = -ρg(0 - -30) = -ρg*30

    But the answer should be positive!

    Any help would be much appreciated
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Aug 26, 2014 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    In the formula you have used, the gravitational acceleration is pointing in the negative z direction. If you change the direction of the axis, then you must also change the direction of the field.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted