Section 19–3 Finding the moment of inertia

(General formula / Parallel axis theorem / Perpendicular axis theorem)

In this section, Feynman discusses the general formula of moment of inertia, and the use of parallel axis theorem and perpendicular axis theorem to find the moment of inertia of an object.

1. General formula:

“Now we must sum all the masses times the x-distances squared (the y’s being all zero in this case) (Feynman et al., 1963, section 19–3 Finding the moment of inertia).”

Feynman expresses the general formula of moment of inertia of an object about the z-axis as I = Sm_i(x_i²+y_i²). He explains that the distance in the expression is not a three-dimensional distance, but only a two-dimensional distance squared, even for a three-dimensional object. We may use the symbol I_z instead of I to emphasize the rotation is about the z-axis in the three-dimensional space. In a sense, Feynman only provides a mathematical definition of moment of inertia. However, it can be theoretically defined as an object’s tendency to resist angular acceleration that is a sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation. The moment of inertia of a complex system can be operationally defined by suspending the system from three points to form a trifilar pendulum.

The total moment of inertia of an object is the sum of the moments of inertia of the pieces. If the object is a rod, I = òx² dm and “the sum” means the integral of x² times the little elements of mass. That is, it is possible to divide the rod into small elements of length dx and the corresponding parts of the mass are proportional to dx. Curiously, Feynman did not explicitly relate the additive property of moment of inertia to any empirical evidence. In volume II, Feynman adds that “[t]he moment of inertia, then, is a tensor of the second rank whose terms are a property of the body and relate L to ω by L_i = S_jI_ijω_j (Feynman et al., 1964).” Importantly, the moment of inertia should not be treated as an isolated concept, but its additive property could be related to equations of angular momentum.

2. Parallel axis theorem:

“In applying the parallel-axis theorem, it is of course important to remember that the axis for I_c must be parallel to the axis about which the moment of inertia is wanted (Feynman et al., 1963, section 19–3 Finding the moment of inertia).”

Feynman states the parallel axis theorem as “[t]he moment of inertia about any given axis is equal to the moment of inertia about a parallel axis through the CM plus the total mass times the square of the distance from the axis to the CM.” He mentions the importance of remembering the axis for I_c must be parallel to the axis about which the moment of inertia is wanted. However, Feynman could have emphasized that the parallel axis theorem is valid only with the axis that passes through the center of mass of the object instead of any other point. Furthermore, this theorem holds for arbitrary non-planar objects that have certain thickness. Simply phrased, we may apply the theorem for objects that are two-dimensional, three-dimensional, or one dimensional.

In deriving the parallel axis theorem, Feynman provides an excellent explanation of the second term (2X_CMSm_ix′_i): x′ is measured from the center of mass, and in these axes the average position of all the particles, weighted by the masses, is zero. The theorem is applicable to any object supported on a pivot at the center of mass such that the object may rotate about an origin and spin about the axis through the center of mass. On the other hand, the parallel theorem could be derived from the perspective of an object whose center of mass rotates about the origin and spins about its center of mass. Thus, its angular momentum can be split into two parts: the motion of the center of mass and the motion around the center of mass. For example, a planet rotating about the sun has an orbital angular momentum as well as spin angular momentum.

3. Perpendicular axis theorem:

“… the moment of inertia of this figure about the z-axis is equal to the sum of the moments of inertia about the x- and y-axes (Feynman et al., 1963, section 19–3 Finding the moment of inertia).”

The perpendicular theorem is stated as “[i]f the object is a plane figure, the moment of inertia about an axis perpendicular to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and intersecting at the perpendicular axis.” Feynman mentions earlier that we shall restrict to two-dimensional objects for the most part of this section. In essence, the perpendicular axis theorem is only applicable to idealized objects that lie entirely within a plane. Physics teachers should compare the two theorems by clarifying that the parallel axis theorem is applicable to non-planar objects, whereas the perpendicular theorem is applicable only to planar objects. Better still, one may define planar objects as two-dimensional objects that have no thickness.

In deriving the perpendicular axis theorem, Feynman shows that I_x = Sm_i(y_i²+z_i²) = Sm_iy_i² and I_y = Sm_i(x_i²+z_i²) = Sm_ix_i² because z_i = 0. Thus, I_z = I_x + I_y and we can say that the moment of inertia of a planar object about the z-axis is equal to its moment of inertia about the x-axis plus its moment of inertia about the y-axis. The proof is based on an ideal condition in which the object has no thickness and thus z = 0 for all points within the object. However, we can still apply the theorem in real life for very thin objects that may give approximately useful results. In other words, the theorem is possibly applicable to objects whose surface area (A) is significantly greater than its thickness (A >> t) from the perspective of scaling property of center of mass.

Questions for discussion:

1. How would you explain the general formula of moment of inertia of a composite system?

2. What are the physical conditions of the parallel axis theorem?

3. What are the physical conditions of the perpendicular axis theorem?

The moral of the lesson: the general formula of moment of inertia is I = Sm_i(x_i²+y_i²) and the parallel axis theorem is applicable to non-planar objects (the axis passes through CM), whereas the perpendicular theorem is applicable only to planar objects.

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

Section 19–2 Locating the center of mass

(Pappus’ area theorem / Pappus’ volume theorem / Centroid of a triangle)

 

In this section, Feynman explains Pappus’ centroid theorems—for surface area and volume—and illustrates the idea with the centroid of a triangle. While the triangle example serves as a simple case, its core idea includes the theorems, which are powerful geometric shortcuts in problem solving. For that reason, the title ‘Locating the Center of Mass’ seems too narrow. A more accurate and appropriate title could be ‘Pappus’s Centroid Theorems.’

 

1. Pappus’ area theorem:

“Then it turns out that the area which is swept by a plane curved line, when it moves as before, is the distance that the center of mass moves times the length of the line (Feynman et al., 1963, p. 19–4).”

 

Pappus' area theorem describes a method for calculating the area of a surface or length of a curve. In simple terms, it states: The surface area generated by rotating a planar curve about an external axis is equal to the product of the curve’s length and the distance traveled by its centroid (center of mass). Formally, if a planar curve of length L is rotated about the external axis that does not intersect the curve, the surface area A generated is: A=L´d where d is the path length traced by the centroid. This is a useful theorem because it bypasses integration: once the curve’s length and centroid path are known, the surface area follows immediately. In short: Pappus reduces a potentially messy calculation to the simple rule: Surface area = curve length × centroid path.

Source: Wikipedia

Necessary Conditions:

The Pappus centroid theorems—both for surface area and for volume—are elegantly simple, but they hold only under specific conditions:

1.      Planar curve or area – The figure to be rotated must lie entirely in a plane. If it is non-planar, such as a helix or the surface of a sphere, the resulting rotation can produce a self-overlapping, ill-defined surface or solid.

2.      External axis – The axis of rotation must lie in the same plane as the figure. If the axis is tilted out of the plane, the centroid traces a skewed three-dimensional path (e.g., elliptical path) rather than a simple circle.

3.      Non-crossing axis – The axis must remain outside the region’s interior. If it passes through the interior, the resulting surface or solid intersects itself, and the formula no longer applies.

These conditions are essential: if any are violated, Pappus’ theorems cease to hold. Some mathematicians might suggest that Feynman could have emphasized these limitations more explicitly in his presentation of the theorems.

 

2. Pappus’ volume theorem:

“…if we take any closed area in a plane and generate a solid by moving it through space such that each point is always 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 that the center of mass moved! (Feynman et al., 1963, p. 19–4).”

 

Pappus' volume theorem gives a direct way to calculate the volume of a solid or area of a surface. In simple terms, it states: The volume of a solid generated by rotating a planar region about an external axis is equal to the product of the region’s area and the distance traveled by its centroid. Formally, if a planar region of area A is rotated about an external axis that does not intersect the region, the volume V of the solid generated is: V = A ´ d where d is the path length traced by the centroid during the rotation. This theorem is useful because it bypasses integration: once the area and centroid are known, the volume follows immediately. In short: Pappus reduces a potentially messy calculation to the simple rule: Volume = planar area × centroid path.

 

“There is another theorem of Pappus which is a special case of the above one, and therefore equally true…... (The line can be thought of as a very narrow area, and the previous theorem can be applied to it.) (Feynman et al., 1963, p. 19–4).”

 

Pappus Centroid Theorem

Some mathematicians may have a different view from Feynman as explained below: Pappus’ two centroid theorems—one for surface area and one for volume—are in fact equivalent and can be seen as two aspects of the same principle. The volume theorem can be derived from the area theorem by viewing a region as a collection of infinitesimal line segments, each generating a strip of surface area whose integration yields the total volume. Conversely, the area theorem can be derived from the volume theorem by treating a curve as the limiting case of an infinitesimally thin strip of area, whose rotated volume reduces in the limit to the surface area generated by the curve. In this sense, each theorem implicitly contains the other, and together they express a single unifying principle: the quantity being rotated—whether length or area—multiplied by the path of its centroid yields the resulting surface or volume. Hence, it is natural to group them under the name Pappus’ Centroid Theorem, with the outcome determined by whether the rotation involves a curve or a region.

 

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, p. 19–4).”

 

Pappus’ volume theorem can be easily applied to a right triangle with 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 this cone, with height H and base radius D is πD²H/3, which is exactly one third the volume of the smallest cylinder that can enclose it. 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, rotating the right triangle about the other axis, we can deduce the y-component of the centroid as Y = H/3.

 

Physical Interpretation

Pappus’ centroid theorem can be understood physically using the idea of mass. Imagine a planar figure rotating about an external axis: each tiny mass element moves along a circular path, contributing a small segment proportional to its distance from the axis. The center of mass (centroid), by definition, is the single “average” position weighted by all these elements. This means we can think of the total motion as if one mass—the sum of all the elements—moves along a circle whose radius is the centroid’s distance from the axis. From this perspective, the surface area or volume generated by the rotation is simply the length or area of the figure multiplied by the distance traveled by the centroid. In short, Pappus’ theorem arises naturally from the principle that the motion of a distributed mass can be represented by the motion of its center of mass, transforming a complicated integral into a neat geometric shortcut.

 

Applying Pappus’ theorem via Work 

Interestingly, Pappus’ theorem can be illustrated using the concept of work (Levi, 2009). When a planar shape rotates around an axis to generate a solid of revolution, imagine filling the volume with a fluid and slowly compressing it with a piston. The infinitesimal work done on a thin disk of thickness dx is dW=P dV =PA dx, where A is the area of the disk. Summing over all disks, the total work done corresponds to the area of the shape multiplied by the distance traveled by its centroid—exactly the volume generated by the rotation. This approach essentially treats the rotational motion as a series of infinitesimal “pushing” operations, showing that the total volume equals the product of the shape’s area and the centroid’s path — a physical demonstration of Pappus’ theorem (see figure below).

Source: (Levi, 2009).

Review Questions:

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

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

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

 

Historical note:

Pappus of Alexandria (4th century AD) first formulated the centroid theorems in his Collection. His reasoning was expressed in the language of Greek geometry, relying on the concept of proportions and the center of gravity rather than integral calculus. By showing that the centroid of a figure traces a circular path under rotation, and relating that motion to the resulting area or volume, he provided a geometrically insightful formulation, though not a step-by-step proof in the modern sense.

In the 17th century, Paul Guldin republished these results in his Centrobaryca (1640–41), embedding them within a systematic theory of centers of gravity. Drawing on Archimedean principles—particularly the concepts of balance and moments—he offered a deductive justification of the theorems. Although Guldin lacked the tools of calculus, his treatment was more rigorous than Pappus’, because it showed why the results must hold from first principles.

Historically, Guldin was accused of plagiarism regarding Pappus’ theorems, since he did not credit Pappus in his work—even though he cited many other sources. In terms of rigor and presentation, Guldin’s proof is “better” than Pappus’, because it provides a logical justification grounded in mechanics rather than intuition. In terms of originality, Pappus deserves credit for formulating the theorems first. Historians often describe Guldin as giving the theorems their first rigorous foundation, while Pappus supplied the original geometric insight (e.g., Mancosu, 1996).

 

In short: Pappus had the idea, Guldin gave it rigor*. This is why the theorems are often referred to as the Pappus–Guldin Theorems in historical and mathematical literature.

 

*An example of Guldin's definition: “A rotation is a simple and perfectly circular motion, around a fixed center, or an unmoved axis, which is called the 'axis of rotation', turning around either a point, or a line, or a plane surface, which, almost as leaving a trace behind it, describes or generates a circular quantity, either a line, or a surface, or a body (Mancosu, 1996, p. 58).”

 

Key Takeway (In Feynman’s style): You see, the trick isn't just that a rotating figure makes a volume. Any fool can see that. The magic is in finding the shortcut. The theorem gives you a way to be lazy, in the very best sense a physicist can be! It’s like this: you don't have to add up every single little piece of the figure. That’s brute force, and it’s messy. Instead, you find the one special point—the centroid—where the whole thing balances. It’s the average location of all the stuff.

Analogy:

· Imagine a team of runners on a circular track. Each runner is at a different lane, tracing a different path around the stadium. Keeping track of all their steps is hopeless.

· Now imagine one “average runner” standing at the centroid position. When this single runner makes a lap, the distance they cover—multiplied by the number of runners—equals the total distance of the whole team.

· That’s Pappus’ trick: instead of handling chaos (calculus) from countless paths, you just follow one “lane” (centroid’s path). One path, multiplied, gives you the total instead of wrestling with the chaos of every path, you just track the path of the centroid. Multiply that single path by the size of the figure, and you’ve got the total.

 

The Moral of the Lesson: Paul Guldin, a Jesuit mathematician, might have smiled at the saying, “It is more blessed to ask forgiveness than permission.” (This so-called Jesuit credo is often used by Nobel laureate Frank Wilczek to justify bold but innovative thinking.) In a well-known controversy, Guldin attacked Cavalieri’s method of indivisibles. Guldin argued that when a surface is generated by the rotation of a line, the surface is not just a collection of lines, and therefore considered Cavalieri’s method flawed. Yet historians favored innovation: Cavalieri’s idea is now recognized as a precursor to integral calculus despite its limitations (Andersen, 1985). Sometimes, what seems boldly ‘wrong’ in the moment may later be perceived as brilliantly right.

 

References:

Andersen, K. (1985). Cavalieri's method of indivisibles. Archive for history of exact sciences, 31(4), 291-367.

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.

Guldin, P. (1641). Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae. Libri IV. Marcus Tudella.

Levi, M. (2009). The mathematical mechanic: using physical reasoning to solve problems. Princeton University Press.

Mancosu, P. (1999). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford University Press.

Pappus of Alexandria. (1986). Book 7 of the Collection (A. Jones, Ed. & Trans.). Springer-Verlag.

Section 19–1 Properties of the center of mass

(Symmetrical properties / Scaling properties / Accelerated reference frame)

In this section, Feynman discusses symmetrical properties and scaling properties of the center of mass, as well as a theorem of motion of the center of mass in an accelerated reference frame.

1. Symmetrical properties:

“… if it is just any symmetrical object, then the center of gravity lies somewhere on the axis of symmetry because in those circumstances there are as many positive as negative x’s (Feynman et al., 1963, section 19–1 Properties of the center of mass).”

Center of mass (CM) is a point “inside” an object where the net external force may produce an acceleration of an imaginary particle at this point as if the whole mass of the object were to be concentrated there. Using circular symmetry, Feynman clarifies that CM does not have to be in the material of a body, for example, the CM of a hoop is in the center of the hoop that is not in the hoop itself. In the case of a rectangle that is symmetrical in two planes, we can easily determine its CM that lies on their line of intersection. Similarly, if a body is symmetrical about an axis, its CM also lies on the same axis. More importantly, we can use possible symmetrical properties of CM to simplify physics problems.

Idealization (or simplification): we idealize a rigid body as a system of discrete particles (or a continuous distribution of matter) and gravitational forces are uniform in a small region of space. Feynman also proves a theorem that simplifies the location of the center of mass by assuming a body is composed of two or more parts whose centers of mass are known. However, one may elaborate that the location of the center of mass of a rigid body is uniquely defined, but the center of mass vector is dependent on the selected coordinate system. Furthermore, the center of mass is defined without reference to the gravity of an object. These properties of the center of mass could have been summarized at the beginning or at the end of the section.

2. Scaling properties:

“…Newton’s law has the peculiar property that if it is right on a certain small scale, then it will be right on a larger scale (Feynman et al., 1963, section 19–1 Properties of the center of mass).”

The concept of scaling is not commonly found in current physics textbooks. Feynman explains that Newton’s laws of dynamics hold for the motion of objects at a higher scale and it becomes more accurate as the scale gets larger. On the contrary, quantum mechanics for the small-scale atoms are quite different from Newtonian mechanics that are applicable to large-scale objects. Historically, Galileo (1638) writes that “if the size of a body be diminished, the strength of that body is not diminished in the same proportion; indeed the smaller the body the greater its relative strength (p. 131).” He investigated scaling properties, for example, the relationship between speed and distance of a moving body as well as the irregular shapes of bones. Feynman (1969) has also contributed to the concept of scaling in his interpretation of experimental results on deep inelastic scattering (electron-proton collisions).

Approximation: according to Feynman, we can approximate the motion of bodies at a larger scale by a certain expression in which it keeps reproducing itself on a larger and larger scale. In addition, Newton’s laws of dynamics are similar to the “tail end” of the atomic laws and they can be extrapolated to a very large scale. Interestingly, the laws of motion of particles on a small scale are very peculiar, but a large number of particles also approximately obey Newton’s laws. Currently, one may prefer Feynman to suggest the need of laws of motion for galaxies, for example, Modified Newtonian dynamics (MOND) is developed for motion of bodies at an even larger scale. It provides an alternative explanation for the motion of galaxies that do not appear to obey Newton’s laws.

3. Accelerating reference frame:

“… the theorem that torque equals the rate of change of angular momentum is true in two general cases: (1) a fixed axis in inertial space, (2) an axis through the center of mass, even though the object may be accelerating (Feynman et al., 1963, section 19–1 Properties of the center of mass).”

Feynman states two validity conditions of the theorem concerning the center of mass in which the torque is equal to the rate of change of angular momentum: (1) a fixed axis in an inertial frame, (2) an axis through the center of mass in an accelerating frame. He elaborates that an observer in an accelerating box would expect the same situation (or experience same magnitude of forces) if an object were in a uniform gravitational field whose g value is equal to the acceleration a. One may add that an inertial force acting on the object is equivalent to an apparent gravitational force based on Einstein’s principle of equivalence. In other words, the theorem involving an external torque acting on an accelerating object is equivalent to the same object that is at rest, but it is now under the influence of apparent gravitational field.

Exception (or limitation): when a small object is supported at its center of mass, there is no torque on it because of a parallel gravitational field. This is not strictly true for a large object because gravitational forces are non-uniform, and thus, the center of gravity of the large object departs slightly from its center of mass. However, Feynman could have included Chasles’ theorem that describes the motion of a body as a sum of two independent motions: a translation of the body plus a rotation about an axis. A special case of this theorem is to choose the axis at the center of mass of the body that allows the angular momentum to be split into two components: the motion of the center of mass and the motion around the center of mass. It helps to connect the discussions of translational kinetic energy and rotational kinetic energy in section 19.4.

Questions for discussion:

1. What are the symmetrical properties of the center of mass?

2. What are the scaling properties of the center of mass?

3. What are the validity conditions of the theorem concerning the center of mass in which an external torque is equal to the rate of change of angular momentum?

The moral of the lesson: the center of mass of a rigid body has symmetrical properties and scaling properties, and there are two validity conditions of the theorem concerning the center of mass in which the torque is equal to the rate of change of angular momentum: a fixed axis in an inertial reference frame and an axis through the center of mass in an accelerating reference frame.

References:

1. Feynman, R. P. (1969). Very high-energy collisions of hadrons. Physical Review Letters, 23(24), 1415-1417.

2. 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.

3. Galilei, G. (1638/1914). Dialogues Concerning Two New Sciences (Trans. by Crew, H. and de Salvio, A.). New York: Dover.