Thread: Triangle Center Of Gravity

Would anyone know the formula to find the center of gravity of an isosceles triangle. Looking down upon the geometric isosceles triangle from a top view, I would like to anchor something to its balance point.

I am sure there is a formula for that. I just do not know what it is.

I do not know if it is more then just splitting up the square inches. Because of the lever action involved.

A perfect triangle would be easy. Just find the center.


Sincerely,


William McCormick

What you have to do is bisect each side of the triangle, then draw a line from the midpoint of each side to the opposite vertex. The three lines will all meet in the center of gravity.
http://www.jimloy.com/geometry/centers.htm

Centroid: The three medians (the lines drawn from the vertices to the bisectors of the opposite sides) meet in the centroid or center of mass (center of gravity). The centroid divides each median in a ratio of 2:1.

Originally Posted by WilliamMcCormick

center of gravity

I'm just curios. Is there a difference between the center of gravity and the center of mass? Center of mass seems like a better phrase to me.

Originally Posted by William McCormick

A perfect triangle would be easy. Just find the center.

If this is a real physical object we're talking about then that only works if it's of uniform density. Just thought I'd point that out.

http://en.wikipedia.org/wiki/Center_of_mass

In the context of an entirely uniform gravitational field, the center of mass is often called the center of gravity — the point where gravity can be said to act.

Originally Posted by Harold14370

What you have to do is bisect each side of the triangle, then draw a line from the midpoint of each side to the opposite vertex. The three lines will all meet in the center of gravity.
http://www.jimloy.com/geometry/centers.htm

Centroid: The three medians (the lines drawn from the vertices to the bisectors of the opposite sides) meet in the centroid or center of mass (center of gravity). The centroid divides each median in a ratio of 2:1.

It looked good to me on smaller isosceles triangles. But when I drew longer ones, I think it falls apart.




This looks better, and you did say something about the midpoint of the side. You would only need the two lines drawn to find the intersection.

I have not seen what it looks like on smaller triangles though. It still looks wrong though. To much now on the big side.



Sincerely,


William McCormick

The second picture is definitely closer to right. Are you finding the midpoint of the height of the triangle, or its sides?

An extra tip for more complicated objects in case you wanted to have more center of mass fun:


where is the x cordinate of the collective center of mass, is the mass of each object, and is the x cordinate of the center of masses of each object. This process can be repeated for the other three cordinates:




to get the three cordinates of a center of mass. This is also a method that works for your triangle, but since your object is more simple, the above method may be easier.

Originally Posted by MagiMaster

The second picture is definitely closer to right. Are you finding the midpoint of the height of the triangle, or its sides?

That was the midpoint of the sides of the triangle. You cannot see the slight angle created by the two intersecting white lines, from the midpoint of the sides, and perpendicular to the sides.

The second one is correct though. By actual test.

In real life, you just take a pipe, if the triangle is symmetrical and roll the triangle on the pipe until it is balanced. If the triangle was odd shaped you could use a ball bearing, or a marble, if you are just using a wooden template of equal size or scale to your actual triangle. To get a very close placement of the center of gravity.

In most cases when you balance anything, you have to go back and either drill out material or drill and add in heavier material to actually balance it. Or sometimes we just grind material off.

I went to the machinists handbook for formulas, and I made three aluminum triangles and balanced them and then marked them.

I found that actually the center of gravity is surprisingly just a little more towards the tip of the triangle. Then the marks indicate. By actually roughly balancing them. Just 1/16" but noticeable. I even favored the marks a bit to compensate for it. But still the actual balance point is a bit further up.








Sincerely,


William McCormick

Originally Posted by Demen Tolden

Originally Posted by WilliamMcCormick

center of gravity

I'm just curios. Is there a difference between the center of gravity and the center of mass? Center of mass seems like a better phrase to me.

To me center of gravity sounds more scientific. Because center of mass would imply physical measurements would or could play into it.

I know I was surprised how much the lever plays into moving the center of gravity on a long isosceles triangle. Compared to its actual center of the object or mass.





Sincerely,


William McCormick

Originally Posted by Harold14370

What you have to do is bisect each side of the triangle, then draw a line from the midpoint of each side to the opposite vertex. The three lines will all meet in the center of gravity.
http://www.jimloy.com/geometry/centers.htm

Centroid: The three medians (the lines drawn from the vertices to the bisectors of the opposite sides) meet in the centroid or center of mass (center of gravity). The centroid divides each median in a ratio of 2:1.

Harold you were absolutely right. Thank You. A line drawn from each midpoint to the opposite vertex, will intersect with any other line drawn from any other midpoint of any side to an opposite vertex. Giving you a center of gravity.

When you said bisect I am thinking in terms of angles. Usually we use bisect to talk about splitting an angle. As in bisector. But bisect does mean to split in half. I always say midpoint when talking about splitting lines.


Sincerely,


William McCormick

The angle bisectors all intersect in the incenter, the center of the largest circle that can be fit into the triangle, as shown in your first picture.

The perpendiculars from the midpoints of each side intersect in the circumcenter, the center of the smallest circle that fits around the triangle.

There are lots of very interesting geometrical fact about triangles, many of which have been studied as far back as the ancient greeks.

Originally Posted by MagiMaster

The perpendiculars from the midpoints of each side intersect in the circumcenter, the center of the smallest circle that fits around the triangle.

If you have a long thin isosceles like triangle with both sides of different length. The intersection will take place outside the triangle. But does still create the center of the smallest circle that will fit around the triangle.


Sincerely,


William McCormick

Right. (I didn't say that the circumcenter had to be inside the triangle.)

Originally Posted by MagiMaster

Right. (I didn't say that the circumcenter had to be inside the triangle.)

I know you did not.

But someone just trying it for fun, might not think outside the triangle, is a viable possiblity. Ha-ha.



Sincerely,


William McCormick

Here are some quick old drawing tricks that come in handy for anything you do.


http://www.Rockwelder.com/geometry/G...letriangle.pdf


Sincerely,


William McCormick