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C.O.M and Inertia of a sphere; top half twice as dense as lower half

  1. Apr 22, 2012 #1
    1. The problem statement, all variables and given/known data
    A sphere of radius L is made up of an upper hemisphere of uniform mass density σ = 2σ0 and a lower hemisphere of uniform mass density σ = σ0. Origin of co-ordinate system lies at centre of sphere with the denser hemisphere above the xy plane. Using spherical polar co-ordinates:


    2. Relevant equations
    (i) Find the mass of the sphere

    (ii) Show the the centre of mass lies at zCOM = L/8

    (iii) The moment of inertia for rotation of the sphere about the z-axis is defined as Iz = ∫∫∫lz2dM, where lz is the perpendicular distance of a mass element dM = σdV from the z-axis. Show that Iz = (4/5)(pi)σ0L5



    3. The attempt at a solution

    (i) M = 4/3(pi)L3(1.5σ0) = 2(pi)L3σ0

    (ii) zCOM = (∫zdM)/∫dM = (1/(2(pi)L3σ0))*∫zdM

    ∫zdM = ∫zσdV
    z = rcosθ
    dV = r2sinθdrdθd[itex]\varphi[/itex]
    ∫zdM = σ∫r3sinθcosθdpdθd[itex]\varphi[/itex]

    I could be doing okay up to this point, but not convinced.

    Now what do I do considering I have the 2 different density hemispheres?

    (iii) No attempt worth writing down



    Cheers :)

    ps. Sorry if my input style is horrible (particularly equations might be messy). It's my first post.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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