Grace asked in Science & MathematicsPhysics · 10 months ago

Why would the centroid of a triangle also be its center of mass?

3 Answers

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  • 10 months ago

    ..............A

    ............/.*.\

    ........../...*...\

    ......../.....*.....\

    ....B/......M......\C

    ..../........ *........\.

    ../...........*..........\

    D...........F...........E

    Consider the triangle ADE, F is the mid-point

    of the side DE=>AF is the median. M is a point

    at the intersection of line-segment BC & AF.

    Tri. AMB~ Tri. AFD=>

    AM/AF=MB/FD=>FD=AF*MB/AM

    Similarly, FE=AF*MC/AM.

    FD=FE=>AF*MB/AM=AF*MC/AM=>MB=MC

    => M is also a mid-point of BC. If Tri.ADE is

    made of some material, then the weight of

    Tri. ADF=that of Tri. AEF.

    Similarly, M is a point on the median from AD

    to E. Also M is a point on the median from AE

    to D=> M is the intersection point of the 3 medians; i.e. the centroid =the center of mass.

  • oubaas
    Lv 7
    10 months ago

    Yesssssssssir :

    Centroid coordinates = i(x1+x2+x3)/3 ; j(y1+y2+y3)/3

    C. o. M. coordinates = i(x1*m1+x2*m2+x3*m3)/(3*(m1+m2+m3)) ; j(y1*m1+y2*m2+y3*m3)/(3*(m1+m2+m3))

    since a geometrical figure is homogeneous allover its surface , then m1 = m2 = m3 = m , then :

    C. o. M. = m*i*(x1+x2+x3)/(3*m) ; m*j(y1+y2+y3)/(3m)

    mass m cross

    C. o. M. = i*(x1*m1+x2*m2+x3*m3)/3 ; j*(y1*m1+y2*m2+y3*m3)/3...which is exactly the formula of the centroid

  • 10 months ago

    Because it has uniform density. Remember that the mass would balance on the head of a pin at that point.

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