Erratum: “Free falling inside flattened spheroids: Gravity tunnels with no exit” [Am. J. Phys. 86, 924–933 (2018)]

In a recent article published in this journal,¹ the free-fall trajectory of a body inside a rotating spheroidal Earth was computed. The main result was provided in Eq. (11), which gives the general equation for the fall in a terrestrial reference frame, with $z′$ axis along the rotation axis and $(x′,y′)$ in the equatorial plane. The trajectory was then expressed in the (rotating) terrestrial frame, with the origin at the drop point D, and a vertical defined as the direction of the weight at the dropping point (called “vertical stick” in the paper). In the limit of short falling times, the trajectory has an eastward deviation and a meridional deviation (southward when D is in the northern hemisphere).

The meridional deviation S was first expressed with reference to the vertical stick, and then with reference to the position of a plumbline hanging from the dropping point. There was a mistake in this very last step: the article incorrectly states that “When a plumbline is at rest in the terrestrial frame, its shape follows the effective force field lines (gravitational plus centrifugal).” The actual shape of a plumbline depends on the linear mass of the wire and on the mass of the object attached to its end. In the limit of a massless wire, the wire has a rectilinear shape, in the direction of the weight (gravitational force plus centrifugal force) at the position P of the object attached to its lower end. In this erratum, we compute the correct meridional deviation of the plumbline (Fig. 1).

The coordinates of P in the (rotating) terrestrial frame can be computed as follows (we use the same notations as in the paper). The position D of the higher end of the plumbline has coordinates $(xD′=r0 cos λ,yD′=r0 sin λ)$⁠, where λ is the geographic latitude. The line PD is colinear with the effective force field at P (see last equation of page 925)

under the condition

where $Δx′=x′−xD′$ and $Δz′=z′−zD′$ were introduced, to follow the same steps as in Sec. VI C of the paper. This equation gives the locus of the plumbline positions, for all possible lengths, in the reference frame with axes pointing towards the equator and towards the pole. In the more natural frame defined by the vertical stick (with the use of the first equations of p. 931), it is found that

where Z_p is the depth measured along the vertical stick, S_p is the southward deviation of the plumbline, and θ is the geodesic latitude. When the height of fall Z_p is small, this yields

The deviation of the plumbline relative to the vertical stick is twice as large as that given in the article in the equation that precedes Eq. (29), and computed under the assumption that a plumbline follows the line of force of the weight. This expression depends on e through the e² term and through θ and r₀. If we consider that e ≪ 1, as is the case for our Earth, we find that the last equations of Sec. VI should be replaced by

for the deviation of the plumbline relative to the vertical stick, and

for the deviation of the falling body with respect to the plumbline.

There also was a typo in Eq. (24) that did not affect any of the subsequent analysis. It should read