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| ISBN-13: | 9780486137858 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 10/02/2012 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 416 |
| File size: | 21 MB |
| Note: | This product may take a few minutes to download. |
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CHAPTER 1
THE CENTER OF MASS
1.1 INTRODUCTION
Many geometric facts may be made quite vivid and intuitive with the help of the physical notion of the center of mass. It is the purpose of this chapter to illustrate some of the techniques involved. But, lest the reader jump to hasty conclusions, we offer a note of caution. It should not be thought that the methods of this chapter merely give "physical proofs." Rather, the center of mass may be used as motivation for certain parts of geometry. Furthermore, as we shall see, this notion can be made mathematically precise. To consider a rough analogy, geometry may be defined and taught without a single picture. Doing so may be considered by some to be a great advance, but it is wrong to consider geometry with pictures as merely giving "artistic proofs." In any event, pictures (and physical models) are strongly suggestive and helpful.
1.2 SOME PHYSICAL ASSUMPTIONS AND CONVENTIONS
A mass-point is visualized as a particle having a certain mass m located at a point P of space. It is, of course, an idealized physical notion. We shall be concerned with bodies consisting of finitely many mass-points. These notions are used by physicists every day. Thus a ball is often idealized as a mass-point, and in astronomical contexts it is perfectly reasonable to take the earth and stars as mass-points. Furthermore, a large solid mass is approximated by cutting it up into (finitely many) small pieces and considering each part as a mass-point.
For many purposes, a solid body of total mass m may be replaced by a single mass-point of mass m, located at the center of mass of the body. We are going to make various assumptions about the center of mass. These will be made quite explicit in Section 1.5, but for the present we prefer to work "intuitively."
An especially simple mass consists of two mass-points located at A and B with masses m1 and m2, respectively, as in Fig. 1.1. If we wish, we may imagine them connected by a bar, but we should think of the bar as massless in order not to introduce any new masses. We shall assume that the center of mass of this system is located on the bar at the point "where the see-saw balances." This is the point P on the segment AB at which m1AP = m2PB (see Fig. 1.1). Thus,
AP/PB = m2/m1 (1)
This has the effect of pulling P closer to B when the mass of B gets larger. (If m2 increases, so does AP.) Therefore, by the device of "mass-points," we shall be able to deal with points P on a segment AB which divide AB in a given ratio.
We may easily remember the rule (1) which locates P by noting that AP (distance to A) is in the numerator and so is m2 (mass of B). Diagrammatically, we merely place the mass of B on the segment opposite B, and similarly for the mass of A. This is indicated in Fig. 1.2 where we read AP/PB = 5/3 and we say that P divides AB in the ratio 5/3. The numbers 5 and 3 do not represent the distances AP and PB but give the ratio AP/PB, and any proportional numbers will do as well.
It is convenient to use a similar device for broken lines. In this case, the numbers along a line refer only to ratios along that line. Thus, in Fig. 1.3, we have AB = BC and CD = 2DE; but, for example, we do not necessarily have DE = BC. We also make a point of writing, for example, CD = 2DE rather than CD = 2ED. Thus we make sure that the directions CD and DE are the same.
If a mass-point is located at A and has mass m, we shall refer to the mass-point mA. Thus we may refer to a system of mass-points 3A, 4B, 9C instead of the more cumbersome "system of mass-points located at A, B, C with masses 3, 4, and 9, respectively." When we wish to replace them with a single mass-point located at their center of mass, we replace them with 16X, where X is the center of mass of this system, and 16 = 3 + 4 + 9 is the total mass. In the next few sections, we shall learn how to find such a center of mass.
1.3 PHYSICAL MOTIVATIONS IN GEOMETRY
In this section are various examples which illustrate how the notion of a center of mass may be used in geometry.
Example 1 (The Centroid of a Triangle). Consider a triangle ABC and place a mass 1 at each vertex (see Fig. 1.4).
Let M be the center of mass of this system. To find it, replace 1A and 1B with their "equivalent," 2M1, located at their mid-point. Then replace 2M1 and 1C with their equivalent. The answer is 3M, the point on the median from C and two-thirds of the way from C (see Fig. 1.4). But clearly this same M might have been obtained in two equally effective ways, as in Fig. 1.5. The result is the pretty theorem about the medians of a triangle: The medians of a triangle intersect in a point two-thirds of the way along each median from each vertex. This common point of intersection has the physical significance of being the center of mass of three mass-points of equal mass located at the vertices of the triangle. It is called the centroid of the triangle. It is the planar analogue of the mid-point of a segment, which is, analogously, the center of mass of two mass-points of equal masses located at the end-points of the segment.
Example 2. Let masses 1,2, and 3 be placed at the vertices A, B, and C of the triangle in Fig. 1.6. As in Example 1, we find the center of mass by first replacing two of the mass-points with their equivalent mass-point and then using the other mass-point. In Fig. 1.7, 2B and 3C are replaced by 5A1 where A1 divides BC in the ratio 3/2. Then M, the center of mass, is on AA1 and divides it in the ratio 5/1. The same figure gives two other ways of proceeding. For example, M is also the midpoint of CC1 and divides BB1 in the ratio 4/2 = 2/1.
We are using a very important physical assumption here. Briefly, it is that a system of mass-points has only one center of mass, and it can be found by individually working on parts of the system. This will be made quite formal in Section 1.5.
Example 3. In Fig. 1.8, P divides AB in the ratio 5/1 and Q divides PC in the ratio 2/3. BQ is extended until it meets AC at R. Find AR/RC and BQ/QR.
The method is to take Q as the center of mass of a suitable physical system. Give A mass 1 and B mass 5. Then P is the location of the center of mass of 1A and 5B (see Fig. 1.9). By giving C mass 4 we obtain Q as the center of mass of 6P and 4C. The previous examples point the way. By observation, AR/RC = 4/1 and BQ/QR = 5/5 = 1.
Example 4 (Ceva's Theorem). Let P be inside ABC. Suppose AP is extended until it meets BC at A1, and that B1 and C1 are found similarly. Then (see Fig. 1.10),
AB1 x CA1 x BC1 = B1C x A1B x C1A
The method of "proof" is clear from Fig. 1.11. We treat P as the center of mass of mA, nB, pC for suitable masses m, n, p. We have three ways of finding P and, according to Fig. 1.11, we have
AB1/B1C = p/m, CA1/A1B = n/p, BC1/C1A = m/n
By multiplying we obtain
AB1/B1C x CA1/A1B x BC1/C1A = p/m x n/p x m/n = 1
Clearing of fractions, we have the result.
Example 5. In Fig. 1.12, find the ratios AG/GE and AF/FC.
We should like A and B to have mass 2 and mass 3, respectively. Clearly, any proportional numbers can be used. Since B and C need mass 5 and mass 2, we see that B "wants" to have mass 5 as well as 3. Choose 15, the least common multiple for the mass of B (in order to avoid fractions), and we quickly obtain Fig. 1.13, where we read
AG/GE = 21/10, AF/FC = 6/10
In the exercises which follow, try to work as "physically" as possible, where this can be done.
Exercises
1. In Fig. 1.14, AP = 2PB and QC = 2PQ. Compare AR and RC.
2. In Fig. 1.15, find the ratios BP/PQ and AP/PR.
3. In Fig. 1.16, the letters a, b, c, d, and e represent actual distances. Express QO in terms of them. (Hint: First ignore e, and use only the ratios a/b and c/d. Then find the ratio QO/OT and substitute OT = e.)
4. Menelaus's theorem deals with the ratios into which an arbitrary transversal divides the three sides of a triangle. In Fig. 1.17, A1, B1, C1 are the points of intersection of a transversal with BC, CA, AB, respectively. "Prove" Menelaus's theorem:
AC1/C1B x BA1/A1c x CB1/AB1 = 1
5. The proof of Ceva's theorem assumed that any point P inside a triangle ABC is the center of mass of A, B, and C with suitable masses. Show that this is so, and show further that the ratios of the masses are unique, and the weights themselves are unique, if the total mass is 1.
6. State and prove the converse of Ceva's theorem.
7. A heavy wire is bent in the form of a triangle. Where is its center of mass? (Note: It is not at the centroid of the triangle.)
8. Problems such as Exercises 1 and 2 above often distress students because the given information does not serve to determine the figure. Comment on this.
1.4 FURTHER PHYSICAL MOTIVATIONS
The methods of the previous section may also be applied to solid geometry. Furthermore, even in the plane, some algebra (rather than the arithmetic of the last section) is often convenient. As in the previous section, we illustrate the techniques by examples.
Example 1 (The Centroid of a Tetrahedron). Let ABCD be a tetrahedron (not necessarily a regular tetrahedron). Assign the mass 1 to each vertex and let M be the center of mass of the resulting configuration. There are quite a few ways to find M. As in Fig. 1.18, we may replace three mass-points 1A, 1B, 1C with 3M1, where M1 is the centroid of ABC. Then M is found along M1D by dividing M1D in the ratio 1/3. In Fig. 1.19, 1A and 1B are replaced by 2X1 (where X1 is the mid-point of AB). Similarly, X2 is found along CD, and M is then the mid-point of X1X2. (Note again that it is strongly assumed that there is only one center of mass M.) Hence we have "physically proved" the following theorem:
The lines connecting a vertex of a tetrahedron to the centroid of the opposite face all meet in a common point M, called the centroid of a tetrahedron. It is located three fourths of the way from each vertex to the centroid of the opposite face. Furthermore, M bisects each segment which joins the mid-points of opposite edges of the tetrahedron.
Example 2. In Fig. 1.20, EF/FC = 4/3 and AF/FD = 7. Show how F may be realized as a center of mass involving the vertices A, B, and C, and in this manner compute BD/DC.
We proceed as in the last section. The mass at D (which is the replacement of the masses at B and C) is 7 times that of A. Let x = mass at A, 7x = mass at D. Similarly, the masses at E and C are in the ratio 3/4, so we let 3y = mass at E, 4y = mass at C. (Here x and y are proportionality factors.) Now, letting z = mass at B, we read from Fig. 1.21 :
3y = z + x
7x = z + 4y (1)
Eliminating z by subtracting, and then simplifying, we obtain
3y - 7x = x - 4y (2)
7y = 8x (3)
Now the values of x, y, z are determined only up to a proportionality factor. This is physically clear by considering what would happen if all masses were, for example, doubled. Equation (3) gives the ratio of x to y. For simplicity, we choose x = 1 and y = 8. Then by using (1), we obtain z = 3y - x = 24 - 7 = 17. Therefore, the mass system giving Fig. 1.20 is given in Fig. 1.22. Thus, BD/DC = 32/17, which is the answer to the problem.
Example 3. Transversals may be obtained by breaking up a mass into parts. Thus, in Fig. 1.23, the mass-point 3A is split into two parts, 1A and 2A. 1A and 1C may be replaced by 2D, while 2A and 3B may be replaced by 5E, and in this way the center of mass M may be obtained. By considering Fig. 1.24 we see, for example, that DE and the median CF meet at M and divide each other in the indicated ratios. (Recall that ratios may be read only along a line. Thus we do not say, for example, that MF/ME = 1 /2.) The next example exploits this method of treating transversals.
Example 4. In Fig. 1.25, AD/BD = 1/2, AE = EC and DF/FE = 2. Find BG/GC and AF/FG.
We proceed algebraically, keeping in mind the previous example. The transversal DE is considered to come from a split of the mass at A, combining some with B (yielding D) and some with C (yielding E). Hence, let x = mass at C = part of mass at A combining to give E. Similarly, a mass y at B and 2y at A may be replaced by the mass 3y at D (see Fig. 1.26). Comparing Figs. 1.25 and 1.26 with respect to DF/FE, we have
DF/FE = 2/1 = 2x/3y
Hence, 6y = 2x. We choose x = 3, y = 1. (As usual, any proportional numbers can be used.) The mass at B is 1 and at C it is 3. Therefore, BG/GC - 3. Furthermore, the mass at A is x + 2y = 3 + 2 = 5. We leave to the reader the problem of finding AF/FG from these data.
Example 5. In Fig. 1.27, ABCD is a tetrahedron and M is the mid-point of EF. Construct a line through M which meets both AB and CD.
As usual, M is conceived of as a center of mass. Let 2x and x be the masses of A and C (so that 2xA and xC may be replaced by 3xF), and let 3y and 2y be the masses of B and D (see Fig. 1.28). M is the center of mass of the system if 3x = 5y. Choose x = 5, y = 3 to obtain Fig. 1.29. By first replacing 10A and 9B with 19G, and then 5C and 6D with 11H, we see that M is on the line GH where G and H are on AB and CD and divide them, respectively, in the ratios 9/10 and 6/5. This is the solution of the problem.
Exercises
1. In Fig. 1.30, determine masses at A, B, and C in such a way that G is the center of the mass of the resulting system.
2. In Fig. 1.31, find PT/TQ.
3. In Fig. 1.32, find BI/IC.
4. In Fig. 1.33, a, b, c, d, e, are given lengths. Find x.
5. In Fig. 1.34, C' divides AB in the ratio 1/2, and similarly for A' and B'. By assigning suitable masses to A, B, and C, and by suitably breaking them up, show that the centroid of A'B'C' is also the centroid of ABC. Generalize and state a converse.
(Continues…)
Excerpted from "A Vector Space Approach to Geometry"
by .
Copyright © 1993 Melvin Hausner.
Excerpted by permission of Dover Publications, Inc..
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Table of Contents
1. The Center of Mass1.1 Introduction
1.2 Some Physical Assumptions and Conventions
1.3 Physical Motivations in Geometry
1.4 Further Physical Motivations
1.5 An Axiomatic characterization of Center of Mass
1.6 An Algebraic Attack on Geometry
1.7 Painting a Triangle
1.8 Barycentric Coordinates
1.9 Some Algebraic Anticipation
1.10 Affine Geometry
2. Vector Algebra
2.1 Introduction
2.2 The Definition of Vector
2.3 Vector Addition
2.4 Scalar Multiplication
2.5 Physical and Other Applications
2.6 Geometric Applications
2.7 A Vector Approach to the Center of Mass
3. Vector Spaces and Subspaces
3.1 Introduction
3.2 Vector Spaces
3.3 Independence and Dimension
3.4 Some Examples of Vector Spaces: Coordinate Geometry
3.5 Further Examples
3.6 Affine Subspaces
3.7 Some Separation Theorems
3.8 Some Collinearity and Concurrence Theorems
3.9 The Invariance of Dimension
4. Length and Angle
4.1 Introduction
4.2 Geometric Definition of the Inner Product
4.3 Proofs Involving the Inner Product
4.4 The Metrix Axioms
4.5 Some Analytic Geometry
4.6 Orthogonal Subspaces
4.7 Skew Coordinates
5. Miscellaneous Applications
5.1 Introduction
5.2 The Method of Orthogonal Projections
5.3 Linear Equations: Three Views
5.4 A Useful Formula
5.5 Motion
5.6 A Minimum Principle
5.7 Function Spaces
6. Area and Volume
6.1 Introduction
6.2 Area in the Plane: An Axiom System
6.3 Area in the Plane: A Vector Formulation
6.4 Area of Polygons
6.5 Further Examples
6.6 Volumes in 3-Space
6.7 Area Equals Base Times Height
6.8 The Vector Product
6.9 Vector Areas
7. Further Generalizations
7.1 Introduction
7.2 Determinants
7.3 Some Theorems on Determinants
7.4 Even and Odd Permutations
7.5 Outer Products in n-Space
7.6 Some Topology
7.7 Areas of Curved Figures
8. Matrices and Linear Transformations
8.1 Introduction
8.2 Some Examples
8.3 Affine and Linear Transformations
8.4 The Matrix of a Linear Transformation
8.5 The Matrix of an Affine Transformation
8.6 Translations and Dilatations
8.7 The Reduction of an Affine Transformation to a Linear One
8.8 A Fixed Point Theorem with Probabilistic Implications
9. Area and Metric Considerations
9.1 Introduction
9.2 Determinants
9.3 Applications to Analytic Geometry
9.4 Orthogonal and Euclidean Transformations
9.5 Classification of Motions of the Plane
9.6 Classification of Motions of 3-Space
10. The Algebra of Matrices
10.1 Introduction
10.2 Multiplication of Matrices
10.3 Inverses
10.4 The Algebra of Matrices
10.5 Eigenvalues and Eigenvectors
10.6 Some Applications
10.7 Projections and Reflections
11. Groups
11.1 Introduction
11.2 Definitions and Examples
11.3 The "Erlangen Program"
11.4 Symmetry
11.5 Physical Applications of Symmetry
11.6 Abstract Groups
Index