Center of Mass by integration

  1. Show that the center of mass of a uniform semicircular disk of radius R is at a point (4/(3*Pi))R from the center of the circle.

    well I know I am suppose to find this by integration. By this equation

    [tex]M\vec{r_{cm}}=\int\vec{rdm}[/tex]

    But, I am not sure how to find dm in this case...

    do I divide mass by area? or by circumference?

    and since it's a disk, and it was 2 variables, would I have to integrate for 2 variables? (x and y)?
     
  2. jcsd
  3. NateTG

    NateTG 2,537
    Science Advisor
    Homework Helper

    In general, CM calculations end up being double or triple integrals, but this one is pretty straightforward since the disk has constant density.

    I'd be inclined to cut the half-disk into parralel strips and integrate along the CM of each strip.

    Since the density is uniform, you might as well assume that it's one.
     
  4. so I divide M by [tex]2R\sin{\theta}[/tex]?
     
  5. HallsofIvy

    HallsofIvy 40,769
    Staff Emeritus
    Science Advisor

    Since you are talking about a semicircular disk, I would be inclined to use polar (actually cylindrical coordinates).

    The differential of area in polar coordinates is r dr dθ so the differential of volume in cylindrical coordinates is r dr dθ dz.

    Taking the density to be ρ(r,θ,z), dm= ρ(r,θz) dr dθ dz.

    The mass, for semicircular disk of radius R and thickness h would be M= ∫(z=0 to h)∫(θ=0 to π)∫(r= 0 to R) ρ(r,θ,z)r dr dθ dz.

    If ρ is a constant, this is just ρπr2h.

    Since x= r cosθ, the formula for the xcm would be Mxcm= ∫(z=0 to h)∫(&theta= 0 to π)∫(r= 0 to R)(r cos θ)(ρ(r,θ,z)r dr dθ dz)=
    ∫(z=0 to h)∫(&theta= 0 to π)∫(r= 0 to R)(ρ(r,θ,z)r2cosθ dz dθ dz.

    Since y= r sinθ, the formula for ycm would be Mycm= ∫(z=0 to h)∫(θ= 0 to π)∫(r= 0 to R)(&rho(r,θ,z)r2sinθ dz dθ dz.

    The formula for zcm would be Mzcm= ∫(z=0 to h)∫(θ= 0 to π)∫(r= 0 to R)(&rho(r,θ,z)zr dr dθ dz.

    Of course, if ρ is a constant, from symmetry (cosine is an even function) xcm= 0 and zcm= h/2.
     
  6. Ok, I got that one.

    Moving on, the next problem is confusing me as well.

    A baseball bat of length L has a peculiar linear density (mass per unit length) given by [tex]\lambda=\lambda_0(1+x^2/L^2)[/tex]

    so what I've done is
    [tex]\int_{0}^{L}x\lambda_0(1+x^2/L^2)dx[/tex]

    which gives
    [tex]\frac{3\lambda_0L^2}{4}[/tex]

    so I know that M * x_cm = that but how do I find M so I can find x_cm?
     
  7. krab

    krab 905
    Science Advisor

    C'mon. You know the mass per unit length. How can you not know the mass?

    [tex]M=\int_0^L\lambda dx[/tex]
     
  8. HallsofIvy

    HallsofIvy 40,769
    Staff Emeritus
    Science Advisor

    krab: Watch it now! You're starting to sound like me!
     
  9. HallsofIvy

    HallsofIvy 40,769
    Staff Emeritus
    Science Advisor

    By the way, why were these posted under the "classical physics" forum? The look like fairly standard calculus homework problems.
     
  10. well I got it from a "physics for scientists and engineers" from tipler. So I figured it would fit better under physics.
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?