Section 19–2 Locating the center of mass

(Pappus’ first theorem / Pappus’ second theorem / Centroid of a triangle)

In this section, Feynman discusses Pappus’ centroid theorem for volume, Pappus’ theorem for surface area, and how to locate the centroid of a triangle.

1. Pappus’ first theorem:

“One such trick makes use of what is called the theorem of Pappus (Feynman et al., 1963, section 19–2 Locating the center of mass).”

Feynman mentions a mathematical trick that is known as the theorem of Pappus. This theorem of center of mass (centroid) can be stated as “[i]f a plane area rotates about an axis in its own plane which does not intersect it, the volume generated is equal to the area times the length of the path of its centroid (Symon, 1971, p. 224).” That is, if we take any closed area in a plane and generate a solid by rotation such that each point is moved perpendicular to the plane of the area, the resulting solid has a total volume equal to the area of the cross-section times the distance moved by the centroid. However, Feynman did not prove Pappus’ theorem formally. There could be a simple example using a rectangular area (A) to generate a cylindrical volume (V) in which V = (2πr)A, where 2πr is the distance moved by the centroid.

Pappus’ centroid theorem for volume (or Pappus–Guldin theorem) shows a connection between surface area, volume of revolution, and centroid. Feynman elaborates that the theorem is true for a curved path because the outer parts go around farther, but the inner parts go around lesser, and these effects balance out. Importantly, two necessary conditions of Pappus’ theorem are (1) a plane sheet of uniform density, and (2) the generated solid does not intersect itself during rotation. Feynman says that the theorem is true if we move the closed area in a circle or in some other curve. About seven years after Feynman's lecture, Adolph Winkler Goodman and Gary Goodman (1969) developed a generalization of Pappus’ theorem that allows the closed area to move in a natural manner on any sufficiently smooth simple closed curve.

2. Pappus’ second theorem:

“There is another theorem of Pappus which is a special case of the above one, and therefore equally true (Feynman et al., 1963, section 19–2 Locating the center of mass).”

Pappus’ centroid theorem for surface area is sometimes described as a special case of Pappus’ first theorem. The surface area that is swept by a plane curve after rotation is equal to the distance moved by the centroid times the length of the curve. Pappus’ second theorem can be stated as “[i]f a plane curve rotates about an axis in its own plane which does not intersect it, the area of the surface of revolution which it generates is equal to the length of the curve multiplied by the length of the path of its centroid (Symon, 1971, p. 224).” This theorem can be proved using A = ò_C 2py ds = 2p ò_C yds = 2pYs in which Y is the y-coordinate of the centroid and s is the length of the curve. There could be a simple example using a straight line of length (s) to generate a cylindrical area (A) in which A = (2πr)s, where 2πr is the distance moved by the centroid.

Feynman suggests applying Pappus’ centroid theorem to a semicircular piece of wire with uniform mass density to find its centroid. In this case, there is no mass in the center of the semicircular wire, except only within the wire. However, he did not show how to solve this problem during his lecture. We may idealize the wire (length: πR) as having a very narrow thickness, and apply Pappus’ theorem to generate a spherical surface area (4πR²): (2πY)(πR) = 4πR². Solving this equation, we obtain Y = 2R/π where Y is the y-coordinate of the centroid and R is the radius of the semicircular wire. Alternatively, we can locate the centroid of the semicircular wire using the general formula, Y = òydm/M = ò (Rsin q)(rR dq)/(rπR) = 2R/π, and by taking limit from π radians to zero.

3. Centroid of a triangle:

“… if we wish to find the center of mass of a right triangle of base D and height H (Fig. 19–2), we might solve the problem in the following way (Feynman et al., 1963, section 19–2 Locating the center of mass).”

Pappus’ theorem can be easily applied to a right triangle of base D and height H. By rotating the triangle 360 degrees about an axis through H (Fig. 19–2), it generates a cone. The volume of the cone with height H and radius D is πD²H/3, which is exactly one third the volume of the smallest cylinder that the cone can fit inside. Based on Pappus’ theorem, the cone’s volume (πD²H/3) is equal to the area of the triangle (½HD) times the circumference (2πX) moved by the centroid. Thus, (2πX)(½HD) = πD²H/3, and we can obtain the x-component of the centroid as X = D/3. Similarly, by rotating the right triangle about the other axis, we can deduce the y-component of the centroid as Y = H/3.

Feynman has selected a good example because the centroid of any uniform triangular area can also be located at where the three medians (the lines from the vertices through the centers of the opposite sides) intersect. As an alternative, we can locate the centroid of a triangle by constructing the perpendicular bisectors of any two sides. More important, Feynman gives a clue that involves slicing the triangle into a lot of little pieces, each parallel to a base without providing the workings. The centroid of the right triangle can be calculated using the formula R = òr dm/M (or Y = òy dm/M) where M = rV = r(HBt) and dm = rty dx = rt(xH/B) dx in which H is the height and B is the base of the triangle. Using integration and taking limit from x = B to x = 0, we can obtain the y-coordinate of the centroid that is H/3.

Questions for discussion:

1. How would you state Pappus’ centroid theorem for volume?

2. How would you state Pappus’ centroid theorem for surface area?

3. How would you locate the centroid of a right triangle?

The moral of the lesson: Pappus’ centroid theorem is a useful mathematical trick that helps to locate the center of mass of an object.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Goodman, A. W., & Goodman, G. (1969). Generalizations of the theorems of Pappus. The American Mathematical Monthly, 76(4), 355-366.

3. Symon, K. R. (1971). Mechanics (3rd ed.). MA: Reading: Addison-Wesley.