Math Is Fun Forum

dgiroux48

Member

Registered: 2013-04-18

Posts: 4

Center of Mass Problems

I'm in a multivariable calculus course and we are currently going over center of mass using integrals. I am having trouble with these two problems.

1)Given a, b > 0, determine the center of mass of a homogeneous triangle with vertices (0, 0), (1, 0), and (a, b). Show that it lies at the intersection of the medians.

2)Determine the center of mass of the homogeneous sector 0 ≤ θ ≤ π/6, 0 ≤ r ≤ 1. Determine the moment of inertia of the sector for rotations about the axis passing though the center of mass and perpendicular to the plane of the sector.

I am lost on how to prove it lies on the intersection in 1.

For 2) I have the center of mass but then I kinda get lost in all the calculations in the moment of inertia calculation.

Any help is appreciated! Thanks!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,646

Re: Center of Mass Problems

hi dgiroux48

Welcome to the forum.

For 1, I'd divide the triangle into horizontal parallel strips.  (see diagram below)

I assume you can use the property 'centre of mass of a thin strip' lies at its centre.  If not you'd have to prove that first.

So for each strip the C of M lies on the median at a vertical distance from the x axis, y, that is proportional to its length, L.

(That comes straight from a Euclidean theorem)

So you can integrate in the y direction.

Bob

ps.  I'll come back to the Moment of Inertia later.

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob