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.