Find the center of mass of a pentagon with five equal sides of length a, but with one triangle m

Find the center of mass of a pentagon with five equal sides...

Key Concepts

Center of Mass

The center of mass is the point where the total mass of a body or system may be considered to be concentrated for many practical purposes. It is calculated as the weighted average of the positions of all the mass elements in an object. This concept is crucial when dealing with irregular or composite bodies since it provides a simplified way to analyze the object's behavior under various forces.

Composite Bodies

A composite body is an object constructed from two or more simpler shapes. When the body is not a simple geometric figure, it often becomes necessary to break it down into parts where standard formulas can be applied. This approach allows one to compute overall properties, such as the center of mass, by dealing with each simpler shape separately and then combining the results.

Decomposition Method

The decomposition method involves dividing a complex shape into simpler components, such as triangles, rectangles, or circles, for which the center of mass can be readily determined. Once the centers of mass for the individual components are established, a weighted average calculation provides the center of mass for the overall object. This method is particularly useful in engineering and physics when working with irregular shapes.

Subtractive Mass Approach

The subtractive mass approach is used when dealing with bodies that have parts removed from them. Instead of calculating the center of mass for the entire shape, one can treat the removed portion as having negative mass. This method simplifies the problem by allowing the computation of the center of mass for the complete shape and then subtracting the contribution of the missing part, effectively using a weighted average that includes negative values for the absent mass.

Transcript

00:01 Okay, so we're trying to find the center of mass of a pentagon with one triangle missing.

00:07 And then they suggest to do that.

00:11 We could look at this example.

00:14 So let me just draw this out.

00:17 Think a bit about it.

00:31 So maybe a good starting question is what is the center of mass of just a regular pentagon without part of it missing? so you can argue that it has to be right in the center.

00:47 I guess there's a few different ways.

00:50 So the center of mass of a regular one is just going to be right in the center.

00:53 You can think about it as breaking it up into five different triangles.

01:03 And then the center of mass of each of these triangles is going to be a bit towards the bottom.

01:07 Right here, right here, right here, right here, right here.

01:12 And then if you want to find sort of the average position of all of these by symmetry, it has to be right in the middle.

01:21 So just because you could rotate, you have discrete symmetry and you can rotate the shape by, i guess, however many degrees.

01:31 And then it would look the same.

01:33 So therefore, the center of mashit and change and the only way point that has that property is right in the middle.

01:42 For example, if it were up here and you rotated it, it would be over here, which doesn't make sense because then the fundamental shape hadn't changed during that rotation, yet the center of mass moved.

01:53 So we know that would be the position of center of mass if you didn't have this triangle.

01:57 Now we have the triangle.

01:59 But keep in mind what we can do, if in general defined center of mass, you can say the center of mass is equal to the sum of all the masses multiplied by the center of mass.

02:12 Of each of those.

02:15 And usually we only, or often we use this just, oh, divided by, sorry, divided by the total mass.

02:21 And often we only use this if each mass is like a ball or a particle.

02:26 This is applicable if you know this for composing an object of any amount of objects, given that you are using the center of mass and then the mass of that object.

02:35 So what we can do is we can treat this center, this pentagon with a triangle missing as a full, pentagon, or sorry, we can treat the pentagon.

02:52 So we can say x center of mass of the pentagon with the triangle, with no triangle missing, let's say, no triangle missing is equal to, let's treat it as the triangle.

03:18 So the mass of one triangle.

03:20 Times the center of mass of one triangle plus the mass of the pentagon with one missing.

03:30 Let's just say, sorry for all the notation.

03:36 So this is the mass of the pentagon.

03:40 I'll use mp for massive pentagon with triangle missing.

03:50 And we want to multiply by the center of mass of that pentagon.

03:55 And this thing is what we're after.

03:57 Oops, question mark here.

04:00 This is what we're after.

04:01 And then we want to divide it by the mass of the pentagon plus the mass of the triangle.

04:07 Now, like i said, we know that this center of mass of this pentagon, let's put our origin right here.

04:15 So if we make our coordinate system like this, this is, let's let this equal zero.

04:23 We can set this equal to zero.

04:25 And then we can say that it's equal to the mass of the triangle, position of the triangle plus mass of the pentagon with the triangle missing times the center of mass of that with it missing.

04:39 And we just want to solve for this.

04:44 So we can think that we can figure out the mass of the triangle.

04:47 That's just going to be one fifth of the mass of the pentagon, given that you can divide it into these five triangles that i drew.

04:57 So let's go ahead and write that out.

05:01 Mp over 5.

05:03 And then you can cancel these mps.

05:07 Just divide both sides by mp.

05:09 Oops.

05:12 Really wanted to cancel it, i guess.

05:15 And then what you get is that the mass of the, or the center of mass of the pentagon is just one -fifth of the negative one -fifth.

05:35 Of the center of mass of the triangle.

05:42 So let's find the center of mass of the triangle.