What is center of gravity espcially for triangles

Click For Summary

Discussion Overview

The discussion centers on the concept of the center of gravity, particularly in relation to triangles, and how to calculate it. Participants explore both mathematical and physical interpretations, including the distinction between center of mass and centroid, as well as the methods for determining these points in geometric shapes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants define the center of gravity as synonymous with the center of mass, emphasizing its importance in physics for problems involving motion and equilibrium.
  • Others suggest that the center of mass for a triangle can be found without calculus, asserting that it lies on the median and that the medians intersect at this point.
  • A participant proposes calculating the center of mass for a right triangle using integration, indicating that this approach would generalize to other shapes, though they acknowledge the lack of a formal proof.
  • Another participant emphasizes that the term "centroid" is more appropriate for the geometric concept, distinguishing it from the physics-related terms "center of mass" and "center of gravity," which require a density function.
  • There is a discussion about calculating centroids for polygons and polyhedra, noting that for more complex shapes, integration may be necessary.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of calculus for determining the center of mass of a triangle, with some arguing it is not needed while others advocate for its use. The distinction between centroid and center of mass is also debated, indicating a lack of consensus on terminology and methods.

Contextual Notes

Some participants mention the need for a density function when discussing center of mass and center of gravity, highlighting that assumptions about uniform density can affect the interpretation of these concepts. The discussion also reflects varying levels of mathematical rigor in the proposed methods.

Nouny
Messages
1
Reaction score
0
What is ceter of gravity concept and how to calculate it for a triangle
 
Physics news on Phys.org
Nouny said:
What is ceter of gravity concept and how to calculate it for a triangle
This is a pure math question - it should be moved.

To answer the question you need to know elementary calculus.
 
The centre of gravity these days is referred to as the Centre of Mass of an object.
To solve physics problems regarding motion, equilibrium etc it is necessary to consider the mass of objects.
However complicated the shape of a real object may be its mass can be considered as concentrated at a single point... the centre of mass. The force of gravity on the object acts through the point known as the centre of mass.
One consequence of this is that an object can be balanced by a support placed at, or passing through, the centre of mass.
There are mathematical techniques for calculating the position of the centre of mass (applied physics) and there are experimental techniques for locating the centre of mass.
 
Work it out for a simple 45 degree right triangle by integrating. Then relate it to the vertices. The result will generalize though you won't have a proof, just a result.
 
There is no need to calculate integrals. It is easy to see that center of mass of a triangle lies on its median. As a conclusion - the medians must intersect in one point, and this point is the center of mass.
 
AlexLAV said:
There is no need to calculate integrals. It is easy to see that center of mass of a triangle lies on its median. As a conclusion - the medians must intersect in one point, and this point is the center of mass.

Yeah, I don't see why folks go on about needing calculus when this works just fine.
 
Let me point out that what people are discussing here is more properly called the "centroid", a purely geometric concept. the "center or mass" and "center of gravity", which are physics concepts, require a density function. If the density is a constant, then the "center of mass" and "center of gravity" are the same as the "centoid". Yes, the centroid of a triangle is just the point whose coordinates are the average of the corresponding coordinates of the vertices. For a general polygon, it is a little more complicated- you can divide the polygon into triangles, find the centroid of each triangle, the find the weighted average of those, the "weighting" being by the area of the triangles. For more general figures, with curved sides, you will need to integrate.

Same thing in three dimensions. The coordinates of the centroid of a tetrahedron are the average of the corresponding coordinates of the four vertices. Any polyhedron can be divided into tetrahedrons and the centroid is the weighted average of the centroids of the tetrahedron (weighted by their volume) while centroids of figures with curved surfaces require integration.
 

Similar threads

High School Circular wire center of gravity
  • · Replies 9 ·
Replies
9
Views
2K
Undergrad Questions about center of gravity
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad Rotational stability and Fosbury Flop questions
  • · Replies 3 ·
Replies
3
Views
1K
Graduate Yaw stability -- center of gravity must be ahead of center of pressure
  • · Replies 29 ·
Replies
29
Views
3K
High School Does every object rotate around its center of gravity?
  • · Replies 127 ·
5
Replies
127
Views
11K
Undergrad Finding the center of area (centroid) of a right triangle
  • · Replies 7 ·
Replies
7
Views
2K
High School Does the inverse square law equalize gravity at different spots?
  • · Replies 8 ·
Replies
8
Views
2K
High School Why Is an Object's Weight Equal to the Force of Gravity It Experiences?
  • · Replies 51 ·
2
Replies
51
Views
5K
Undergrad Center of gravity of a portion of cylinder
  • · Replies 5 ·
Replies
5
Views
11K
High School How to find the center of gravity of an object
  • · Replies 9 ·
Replies
9
Views
2K