The answer is 42, actually 42 minutes, but what was the question? There are, in fact, a number of physics questions, not at first glance closely related to one another, to which 42 minutes (or perhaps 84 minutes) is the answer. This paper was prompted by one such question, a Figuring Physics item in a recent issue of this journal,¹ which called for a description of the motion of a block released on a flat frictionless plate tangent to the Earth's surface. The correct answer given in Figuring Physics was that "the block will oscillate to and fro,"² but I was disappointed to see that the time of oscillation was not called for. So let us refine the question: "For small-amplitude oscillations, how long does it take for the block to travel from one extreme to the other, i.e., what is the half-period of the motion?" This is the question (actually just one of the questions) to which the answer is 42 minutes, as we can readily calculate. Those who are familiar with The Hitchhiker's Guide to the Galaxy [Douglas Adams (Harmony Books, New York, 1979)] will understand why I chose to ask initially for the half period (42 min) rather than asking the more obvious question, "What is the period?"³

Throughout this paper I will assume a complete absence of friction, air resistance, and other nuisances from the real world that tend to spoil physics homework problems. I will also take the Earth to be stationary, nonrotating, and perfectly spherical, i.e., without hills or other imperfections. (Any other assumption, e.g., reality, would spoil the fun.) The mass distribution of the Earth will also be taken to be spherically symmetric. Thus, the gravitational force on any object at or above the surface is equal to that exerted by a point object of mass M located at the center of the Earth, with M the total mass of the Earth. For the most part, it will not be necessary to assume a homogeneous Earth. When we come to consider the motion of objects in tunnels inside the Earth, then at that point we will take the Earth's mass distribution to be not only spherically symmetric but also homogeneous — contrary to fact. Further, in any such "tunnel problem," the tunnel itself will be assumed to be of negligible diameter; removal of dirt to dig the tunnel will have a negligible effect on the gravitational force on an object in the tunnel. (Just as the gravitational force on an object on the surface is primarily due to the distant parts of the Earth — the amount of distant stuff is so large that it overwhelms the nearby stuff in spite of the inverse square law — likewise in a tunnel, the nearby material is of no importance relative to the distant parts of the Earth. The only exception occurs when the motion is restricted to the immediate neighborhood of the exact center of the Earth.)

Any one-dimensional oscillatory motion (well, almost any) is simple harmonic for small amplitudes, i.e., the physics is characterized by a restoring force proportional to the displacement from the equilibrium position. The canonical example is the motion of a mass on a spring, F = –kx, with a resulting period:

To calculate the period, all we need is the coefficient of x in the force law, together with the mass. It may be simpler in some cases to find the period by using the equivalent language of energy. For small displacements, the potential energy will be proportional to x², and if the energy is written in the form

then the period is again found from the ratio of the coefficients in the kinetic and potential energy terms.

Now look at Fig. 1. With M and R denoting the mass and radius of the Earth and m the mass of the block, the gravitational force is (very nearly) GmM/R², and to find its component toward x = 0, we need only multiply by sin x/R. Thus:

and so the analog of the spring constant, k, is (GmM/R³), and the period of oscillation is

a half period of 42 minutes.

Figure 1.

Since g = GM/R², one can also write = 2, the period of a hypothetical simple pendulum in a uniform gravitational field of strength g, whose string is of length R. (See section "A Very Long Pendulum" for a treatment of a "real" pendulum whose string is of length R and whose bob grazes the Earth's surface.)