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    From June 2012, question 7. (a):

    A uniform lamina of mass m is in the shape of a triangle ABC. The perpendicular distance of C from the line AB is h. Prove, using integration, that the moment of inertia of the lamina about AB is (1/6)mh^2. (7 marks)

    I'm okay with isosceles and equilateral triangles, but I'm not sure what to do here.
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    (Original post by ombtom)
    From June 2012, question 7. (a):

    A uniform lamina of mass m is in the shape of a triangle ABC. The perpendicular distance of C from the line AB is h. Prove, using integration, that the moment of inertia of the lamina about AB is (1/6)mh^2. (7 marks)

    I'm okay with isosceles and equilateral triangles, but I'm not sure what to do here.
    Look at my attached triangle.

    Ratios: length of strip/(h - x) = b / h
    Attached Files
  1. File Type: doc m of i of triangle.doc (36.0 KB, 60 views)
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    (Original post by ombtom)
    From June 2012, question 7. (a):

    A uniform lamina of mass m is in the shape of a triangle ABC. The perpendicular distance of C from the line AB is h. Prove, using integration, that the moment of inertia of the lamina about AB is (1/6)mh^2. (7 marks)

    I'm okay with isosceles and equilateral triangles, but I'm not sure what to do here.
    For a discrete set of masses m_i, we have I = \sum_i r^2_i m_i where r_i is the distance of m_i from the axis of rotation. However, for a continuous object, we can't form a sum, but rather we have to compute the limit of a similar sum, which gives us an integral:

    I = \int r^2 dm

    So you need to split up the triangle into elemental pieces of mass dm where dm is some function of distance from the axis, then integrate over appropriate limits to ensure that you take account of the whole mass of the triangle. Here, your "elemental pieces" will probably be thin strips parallel to AB, since that ensures that each part of the strip is the same distance from the axis.

    If you haven't come across this before, this may sound incomprehensible, in which case I would advise that you refer to your textbooks, or to do a little googling for example.
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    (Original post by tiny hobbit)
    Look at my attached triangle.

    Ratios: length of strip/(h - x) = b / h
    (Original post by atsruser)
    ...
    Exactly what I needed; thank you.
 
 
 
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