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Find the centre of mass for this sheet of paper with a cutout

  1. Jul 22, 2017 #1
    1. The problem statement, all variables and given/known data
    W3D2WxY.png
    A uniform, rectangular sheet with sides of lengths a and b has a hole of dimensions a/4 by b/4 punched in it as shown below. Find the centre of mass of the sheet after the hole is made.

    2. Relevant equations
    CM= ∫xdm/∫dm

    3. The attempt at a solution
    I'm confused on what to input into the above equaiton
    I do know that it will like on the axis of symetry so on the line y=b/2

    many thanks :)
     
  2. jcsd
  3. Jul 22, 2017 #2

    ehild

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    You can consider the system as the big rectangle and the small rectangle, but with negative mass.
     
  4. Jul 24, 2017 #3
    thanks for the reply, so would the answer be
    CM= ∫xthe whole sheetdmthe whole sheet/∫dmthe whole sheet - ∫xthe cut out dmthe cut out/∫dm the cut out
     
  5. Jul 24, 2017 #4

    ehild

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    No, it is not simply the difference between the CM-s.
    Supposing you have two small balls at x1 and x2, of masses m1 and m2, how do you get the common center of mass?
    And no need to integrate, the CM of a rectangle is in the centre.
     
    Last edited: Jul 24, 2017
  6. Jul 24, 2017 #5
    hmm to answer your question x1*m1 + x2*m2 / m1 + m2 ?

    in regards to my question so would it be the centre coordinates minus a (xcentre of cut out*ρ*area of cut out)/(ρ*area of cut out)
     
  7. Jul 24, 2017 #6

    ehild

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    Do not forget the parentheses. Xcm=(x1*m1 + x2*m2) / (m1 + m2 )
    No.
     
  8. Jul 25, 2017 #7
    okay then, what would it be then?
     
  9. Jul 25, 2017 #8

    ehild

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    Choose a system of coordinates, for example, that in the picture. Point P is the CM of the big rectangle, Q is the CM of the small one.
    You can consider the mass of a rectangle compressed in the CM.
    If the density and thickness of the plate is homogeneous, the mass is proportional to the area. The CM of a symmetric shape is on alll symmetry axis, in case of the rectangle, it is on the middle.
    What are the positions of the CM-s of both rectangles?
    What are the masses (areas)?
    Then apply the formula XCM =(x1m1+x2m2)/(m1+m2) for the masses at P and Q. ( consider m2 negative)

    upload_2017-7-25_13-21-47.png
     
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