Can Inertia Make Us Move?

Abstract

Gravity disturbs certain experiments and reduces the field of investigation of some scientific domains. Gravity is omnipresent on Earth. However, gravity can only exist if the gravitational attraction is thwarted by the presence of a solid surface that prevent the gravitational force of continuing its action of attraction. Although gravitation is omnipresent in the universe, gravity is the exception and its absence, weightlessness, the general rule.

Inertia forces exist in non-inertial reference frames in accelerated motion with respect to an inertial reference frame, in which there are no inertia forces. Gravity (or weightiness) is defined as the phenomenon resulting of gravitation and inertia. In a non-inertial reference frame, the force associated to gravity (or weightiness), weight, is the sum of the gravity force and of all inertia forces. This chapter recalls some notions in physics on inertia, inertial and non-inertial reference frames, gravitation and gravity, the Equivalence principle, mass and weight, that are needed to understand the phenomenon of weightlessness.

Appendices

Appendix 1: A Matter of Words

As seen in Sect. 3, the use of the single word ‘gravity’ can be confusing. So let us summarize what we mean exactly by gravity and gravity.

Gravity (without italics) is the force associated with the phenomenon of gravitation as explained in Sect. 3. Gravity (in italics) is the phenomenon resulting from the phenomenons of gravitation and inertia, and weight is the force associated with the phenomenon of gravity.

The common use of the single word ‘gravity’ to designate both concepts is the source of many confusions since centuries and its back translation in French and other languages has created additional confusion.

Another word exists in the English language, and although rarely used, it can exactly describe what we mean by gravity here. This word is weightiness. If we look at its definition, we find² “The state or quality of being physically heavy or weighty, or the property of being comparatively great in weight; weight; force; importance; impressiveness”.

So, we can extend the definition of gravity (in italics) given in the middle of Sect. 3 to the word weightiness, as the phenomenon resulting from the phenomenons of gravitation and inertia. Weight is the force associated with the phenomenon of weightiness.

However, the use of the word ‘gravity’ has become so widespread in the English language to describe the two concepts of gravitational force and the weightiness phenomenon that we will conform to this usage. Nevertheless, we will italicize ‘gravity’ to signify that we mean weightiness.

Let us note also that other languages have two different words for these two concepts, for example in French, ‘pesanteur’ and ‘gravité’ respectively for weightiness and gravity.

Appendix 2: Mass and Quantity of Matter

Newton defined mass as the measure of quantity of matter contained in a body and expressed as the product of the volume by the specific mass of the body. This definition should not be confused with another notion, called amount of substance that is defined as the number of elementary particles constituting a body (atoms, molecules, fractions of molecules, …) and whose unit is the mole. By convention, the amount of carbon ¹²C whose mass is 0.012 kg represents an amount of substance of 1 mole, which is approximately equal to 6.02214199 × 10²³.

Furthermore, historically, what is called nowadays mass used to be designated formerly by ‘quantitas materiae’ in Latin, which also contributed to the confusion.

Appendix 3: Inertial Reference Frames

To determine a perfect inertial reference frame is impossible. However, the principle of inertia is verified daily at our spatial and temporal scale. The concepts of inertial reference frame and of the inertia principle are tools of mathematical mechanics that allow to apprehend and to describe most physical phenomenons observed at macroscopic scale. This notion of model is important as it gives a description (most of time, a simple one) of an observed physical phenomenon with certain approximations. Practically, one tries to determine at best a reference frame supposedly inertial and adapted to the study of a specific problem with the best possible approximation, such that the error committed because of this approximation would be negligible with respect to the studied phenomenon, i.e. below a threshold arbitrarily fixed to a small fraction of the result obtained while studying this phenomenon.

The committed approximation is generally due to two inertial accelerations, expressed in a reference frame R in a non-uniform rectilinear motion with respect to another reference frame R_o, supposedly fixed (i.e. whose “inertial” character is better that the one of R). To evaluate an order of magnitude, let us suppose that the motion of R with respect to R_o is a pure rotation with a constant angular velocity ω. The vectors of centrifugal acceleration $\overrightarrow {{a_{e} }}$ and Coriolis acceleration $\overrightarrow {{a_{c} }}$ of a material point in R have norms

$$ a_{e} = \omega^{ 2} r \quad \quad \quad \quad a_{c} = { 2}\omega v_{r} $$

(A3.1)

where r and v_r are the distance and relative speed of the point with respect to the origin of R (and assuming further that the displacement acceleration of the origin of R with respect to R_o is negligible and that the relative motion of the point is perpendicular to the rotation axis of R with respect to R_o; a course of Mechanics will provide more details, see books suggested in references).

Four reference frames are generally considered as shown in Fig. 2 and one proceeds by successive approximations.

Fig. 2

(Credit: P. E. Paulis)

Artist’s impression of some inertial reference frames: the Galactic reference frame R_Gal (in galaxy’s centre), Copernicus reference frame R_C (between the Sun and Jupiter), Kepler reference frame R_K (in the Sun’s centre), the celestial geocentric reference frame R_G (in Earth’s centre) and a reference frame attached to Earth’s surface R_E.

3.1 1 Copernicus Reference Frame (R_C)

This reference frame has for origin the centre of inertia (or centre of mass) of the Solar System and for axes three orthogonal axes whose direction are fixed with respect to fixed stars. The reference frame can be considered in excellent approximation as an inertial reference frame. However, it is not exactly inertial as its origin is attached to the centre of inertia of the Solar System which is not fixed in the universe. Furthermore, the fixed stars toward which the three axes are directed are only fixed for a short instant at the astronomical scale, although sufficiently long to study the motion of planets since man observes them.

Instead of the Copernicus reference frame, one could consider a galactic reference frame (R_Gal) attached to the centre of the galaxy and whose axis’ direction would be aligned on extra-galactic stars. Our galaxy describes a slow rotating motion such as the Sun, at 32 000 light-years (≈ 3.03 × 10²⁰ m) from the galactic centre, describes a galactic revolution in 240 million years on average.

This galactic reference frame would be better (i.e. more “inertial”) than Copernicus’ one, but with the inconvenience that it would be extremely difficult to describe the motion of the Sun and planets, and, a fortiori, a phenomenon at the surface of the Earth. Furthermore, this galactic reference frame would not be perfectly inertial as well as our galaxy takes part in the general motion of the local cluster of galaxies and of universe expansion.

Calculating by (A3.1) the committed approximation by considering Copernicus reference frame instead of the galactic reference frame, the orders of magnitude shown in Table 1 are perfectly negligible in front of the value of g_w = 9.81 ms⁻² (one considered a unit velocity of 1 m/s for the relative speed v_r in Coriolis’ acceleration).

Table 1 Approximations of inertial reference frames

3.2 2 Kepler Reference Frame (R_K)

This reference frame has for origin the centre of inertia of the Sun and axes parallel to those of the Copernicus reference frame. This heliocentric reference frame can be considered with a very good approximation as an inertial reference frame, as the Sun centre of inertia is always close to the Solar System centre of inertia.

As the ratio of all planet masses to the Sun’s mass is about 1.3 × 10⁻³ and that the ratio of Jupiter to Sun’s masses is about 9.55 × 10⁻⁴, one can approximate the distance between the origins of Kepler and Copernicus reference frames by the distance between the centre of inertia of the reduced Sun-Jupiter system (C.i.S-J) and the Sun’s centre. The C.i.S-J is at about 7.43 × 10⁸ m from the Sun’s centre on the axis Sun-Jupiter, just outside the Sun whose radius is 6.96 × 10⁸ m. Considering that the C.i.S-J moves with the orbital velocity of Jupiter (whose orbital eccentricity is neglected), one can evaluate the committed approximation in the Kepler reference frame with respect to Copernicus reference frame. The orders of magnitude shown in Table 1 are also negligible. The largest distance from the Solar System centre of inertia in the most unfavourable theoretical case in which all the planets are in conjunction on the same side of the Sun and aligned, i.e. passing through their orbit node. In this configuration, the Solar System instantaneous centre of inertia is at 1.51 × 10⁹ m from the centre of the Sun and the value of a_e in Table 1 is only doubled. Any other planetary configuration would yield a smaller value.

3.3 3 Earth’s Reference Frames

In Kepler’s reference frame, Earth’s trajectory around the Sun is modelled by an elliptical orbit having the Sun in one of its foci. In reality, Earth’s motion is more complicated and can be decomposed as follows:

• (1) The centre of inertia of the Earth-Moon system describes a quasi-elliptical trajectory around the centre of inertia of the Solar System;

• (2) The Earth’s centre of inertia describes a quasi-elliptical trajectory around the centre of inertia of the Earth-Moon system;

• (3) Earth describes a complex rotation motion around its centre of inertia, that can be represented by a rotation movement around an axis, affected itself by several very slow movements, the mains being precession and nutation.

Therefore, Earth’s reference frames (the celestial geocentric reference frame or a reference frame attached to Earth) are far from being inertial!

3.3.1 3.1 Celestial Geocentric Reference Frame (R_G)

Nevertheless, the celestial geocentric reference frame (with an origin at Earth’s centre of inertia and axes parallel to those of Kepler’s frame) can be considered as inertial with a good approximation, due to the motions given in (1) and (2) above.

This approximation is generally sufficient to study Earth’s artificial satellites’ motion, Eastward deflection of falling bodies, gyroscopes, Foucault’s pendulums, winds and ocean currents, but not tides.

Considering that motion (1) can be approximated by Earth’s sidereal revolution with angular velocity ω_E.rev. on a circular orbit of radius r_E.orb. (eccentricity e = 0.017 is sufficiently small to be neglected), one finds in Table 1 the orders of magnitude of inertial centrifugal and Coriolis accelerations due to motion (1) above.

Motion (2) can be analysed as in the case of the reduced Sun-Jupiter system here above. Knowing the Earth-Moon distance (3.844 × 10⁸ m) and the ratio of their masses (≈ 81.3), the centre of inertia of the Earth-Moon system (C.i.E-M) is at 4.67 × 10⁶ m from the Earth’s centre, or at approximately 3/4 of Earth’s radius in direction of the Moon. Assuming a circular Moon’s orbit (its eccentricity e = 0.0549 can be neglected), one can say that Earth’s centre of inertia describes a rotation around the C.i.E-M at the Moon sidereal orbital angular velocity (ω_M.rev). Can we neglect the effect of this offset rotation of the Earth around the inertia centre of the Earth-Mon system? The answer, surprisingly in view of the distance (3/4 of Earth’s radius) is yes: the inertial accelerations given in Table 1 are negligible.

A similar reference frame, called Earth Centred Inertial (ECI) reference frame, is used to describe orbits and trajectories of artificial satellites and spacecraft in astronautics. Its origin is at the Earth’s centre of inertia. The fundamental plane, formed by the X and Y axes, includes Earth’s equator and the X axis points toward the vernal equinox (see Fig. 3).

Fig. 3

(from Ref. 7)

The Earth-Centred Inertial (ECI) reference frame

3.3.2 3.2 Reference Frames Attached to Earth (R_E)

The reference frames attached to Earth (with an origin at the centre or at the surface of the Earth and axes fixed with respect to Earth, assumed solid and not deformable) can also be considered as inertial with an approximation not as good as above, but still sufficient to study local phenomenons on Earth. In this approximation, one neglects the inertial centrifugal and Coriolis accelerations due to the rotation motion (3) above. Considering only Earth’s sidereal rotation around its axis at the angular speed ω_E.rot, the inertial centrifugal acceleration at the equator is approximately 300 times less than the acceleration g_gr of gravity. One can thus generally neglect it in a first approach; however, it should be taken into account for precise measurements (see Appendix 5).

One of the reference frames attached to Earth that is used commonly is the Earth Centred, Earth Fixed (ECEF) reference frame, with the X axis always aligned with the meridian of Greenwich which is defined as longitude 0° (see Fig. 4).

Fig. 4

(from Ref. 29)

The Earth Centred, Earth Fixed (ECEF) reference frame

Appendix 4: Fundamental Forces and Inertial Forces

Except for inertia forces, all forces in the universe are based on four fundamental interactions, namely gravitation, weak, electromagnetic and strong interactions.

The gravitational force acts between masses and is an attractive force. The electromagnetic force acts between electric charges, and can be attractive between charges of opposite signs or repulsive between charges of same signs. The strong and weak forces are nuclear forces responsible for the interactions between subatomic particles and acting only at very short distances. The strong nuclear force is the force responsible for the structural integrity of atomic nuclei, allowing the binding of the protons and neutrons of the nuclei, while the weak nuclear force is responsible for the decay of certain nucleons into other types of particles. The most known effect is the beta decay of neutrons inside the nucleus of atoms and the radioactivity produced by this decay. The standard model of particle physics predicted the unification of the weak and electromagnetic forces in the electroweak theory which observation has confirmed. The unification of the other forces is a current active field of research in physics.

All other forces in nature derive from these four fundamental interactions. For example, friction forces and the spring recall force are manifestations of the electromagnetic force.

On the other hand, inertia forces appear only in non-inertial reference frames. These include centrifugal forces and Coriolis forces and more generally, any forces that appear in a non-inertial reference frame in motion with a non-uniform velocity with respect to an inertial reference frame. These inertia forces are not genuine, like the four fundamental forces above, as they depend on the reference frames.

Inertial forces are sometime called “pseudo-” forces or “fictitious” forces by some authors. According the Cambridge online dictionary³, “pseudo-” means “pretended and not real” and “fictitious” means “invented and not true or existing; false”. This choice of words, “pseudo-” or “fictitious”, is very unfortunate, as inertial forces are “real”, although it is clear that they are not similar to the four fundamental forces deriving from fundamental interactions.

Let us take a simple example. If the passenger walking in the train of Fig. 1 falls due to the acceleration or deceleration of the train (cases 2 and 3), the lump of his forehead will be real and painful, and not “fictitious” or “pseudo-” …

Therefore, to differentiate fundamental forces from forces appearing in non-inertial reference frames, one could call these changing-reference-frame forces, or accelerated-reference-frame forces, or simply… inertial forces (Fig. 5).⁴

Fig. 5

(Credit: xkcd)

Centrifugal force vs centripetal force: a matter of point of view

Appendix 5: How to Measure and Express Weight?

Defined in a non-inertial reference frame, the weight W of a body is a vectorial entity, defined as the product of the body mass by the vectorial acceleration of gravity g_w

$$ \vec{W} = m\vec{g}_{w} $$

(A5.1)

The acceleration of gravity g_w is further defined in Appendix 6.

One measures usually the weight of a body in two ways, with a scale or with a spring apparatus. At the risk of tiring the reader, let us recall how to use a scale. The body whose weight we want to measure is placed on one of the scale platforms. One places on the other platform one or several mass standard(s), which are calibrated and marked masses (and that the common language designates unfortunately and incorrectly by the word “weight”). When the balance beam comes to horizontal by adding or removing a certain number of mass standards, the sum of weights of these mass standards indicates the result of the weighting of the body.

When this method is applied in a uniform gravitational field (i.e. with a scale having sufficiently small dimensions compared to the radius of curvature of the Earth gravitational field), this method comes down to compare two masses:

$$ W_{body} = W_{stand.} \Rightarrow m_{body} g_{w} = m_{stand.} g_{w} \Rightarrow m_{body} = m_{stand.} $$

(A5.2)

Conducted on the surface of another planet, in a different gravitational field, this method always gives the same result, as the local acceleration of gravity $g_{w}$ applied two both platforms of the scale is equal (the simplification by $g_{w}$ on both side of the second equality of (A5.2) is independent from the value of $g_{w}$). Let us note that this method, used since antiquity, has contributed to the confusion between notions of weight and mass, the equality of ones yielded from the equality of the others, but not implying the equality between them. The use of the term ‘weight’ to designate mass is a misnomer and is unfortunately still very common.

The other method is a dynamic method, using apparatus like extension or compression dynamometers, the most common being the bathroom scale. This method does not compare two masses but two forces, the force associated to gravity (the weight) and the force due to the extension of compression of a graduated spring:

$$ \vec{F}_{spring} = - k\vec{x} $$

(A5.3)

where k is a spring characteristic, called the spring constant or spring stiffness, $\vec{x}$ is the vectorial change of the spring length and the negative sign expresses that the spring force is applied in the opposite sense of the change of the spring length.

Conducted on another planet, in a different gravitational field, this method never gives the same result, as the weight depends on the local gravity while the spring force is independent from it. For example, using the same extension dynamometer on Earth and on the Moon, one would have at equilibrium:

$$ W = F_{spring} \Rightarrow \left\{ {\begin{array}{*{20}c} {mg_{E.w} = kx_{1} } & {\left( {\text{on Earth}} \right)} \\ {mg_{M.w} = kx_{2} } & {\left( {{\text{on}}\,{\text{the}}\,{\text{Moon}}} \right)} \\ \end{array} } \right. $$

(A5.4)

with $x_{1} > x_{2}$, as $g_{E.w} > g_{M.w}$, Moon’s gravity being approximately six times less than Earth’s gravity.

Let us conduct now the body weighting experiment with a dynamometer or with a bathroom scale on the surface of the Earth. Which reference frames are we going to consider? In the celestial geocentric reference frame (assumed inertial), the reference frame of the laboratory is non-inertial as it is dragged along by Earth’s rotation (see Fig. 6). This rotation induces in the laboratory reference frame an inertial centrifugal force whose norm is

$$ F_{in} = F_{centrif.} = ma_{centrif.} = m\omega_{E.rot.}^{2} \rho = m\omega_{E.rot.}^{2} r_{E} \cos \phi $$

(A5.5)

where $\rho$ is the distance from the point where the weighting is done to the Earth’s rotation axis. Assuming a spherical Earth of radius $r_{E}$ and the laboratory located at a place of latitude $\phi$, one obtains the last equality in (A5.5).

Fig. 6

One reasons in the non-inertial reference frame attached to the Earth R_GE or in the laboratory reference frame (not represented), in rotation with respect to the celestial geocentric reference frame R_GC, considered as inertial. The body to be weighted is on Earth’s surface at a point of latitude $\phi$ and at a distance $\rho$ ($= r_{E} \cos \phi$) from the rotation axis. The body weight W is the vectorial sum of $F_{gr}$ and $F_{in}$ (whose magnitude has been exaggerated on the drawing for visual clarity). The deviation angle $\varphi$ between the directions of W (direction of the local vertical) and of $F_{gr}$ (direction of Earth radius) is also exaggerated on the drawing

This force is directed perpendicularly to the Earth’s rotation axis and is maximal on the equator ($\phi = 0^\circ$) and nil at the poles ($\phi = 90^\circ$). In the laboratory’s non-inertial reference frame (or any reference frame attached to Earth, whether geocentric or not), the weight is the vectorial sum of the force of gravity and of the inertial centrifugal force. The weight vector is not directed toward the centre of the Earth (assumed spherical) as the gravitational force, but along the vector resulting from the vectorial addition of the force of gravity and the inertial centrifugal force (see Fig. 6). This direction is the local vertical, given by the plumb line, and deviating by an angle $\varphi$ from the direction of the local earth radius. The weight vector and its norm read

$$ \vec{W} = \vec{F}_{gr} + \vec{F}_{in} \Rightarrow W = F_{gr} \cos \varphi - F_{in} \cos \left( {\phi + \varphi } \right) \approx F_{gr} - F_{in} \cos \phi $$

(A5.6)

where the gravity force and the inertial centrifugal force are projected on the direction of $\vec{W}$. The deviation angle $\varphi$ is nil (i.e. the local vertical is along the earth radius) at the equator (where the gravity and inertial centrifugal forces are aligned but in opposite senses) and at the poles (where the inertial centrifugal force is nil). For other latitudes, one shows that this angle $\varphi$ is small, the maximum value is less than $0.2^\circ$ arising at latitudes $\phi = \pm 45^\circ$ (for a spherical model of the Earth) and decreases when coming closer to the equator or to the poles. One can then neglect $\varphi$, like in (A5.6). Note as well that the negative sign in front of $F_{in}$ shows that this inertial force must be subtracted from the gravity force. Replacing the norms of W, $F_{gr}$ and $F_{in}$ by (A5.1) and (A5.5) yield

$$ W = mg_{w} = m\left( {\frac{{GM_{E} }}{{r_{E}^{2} }} - \omega_{E.rot.}^{2} r_{E} \cos^{2} \phi } \right) $$

(A5.7)

This expression of the weight is correct in its principle, but it can still be corrected for the values of $r_{E}$ and $\omega_{E.rot.}$ (see Appendix 6). The inertial centrifugal force due to Earth’s rotation is implicitly measured in the experiment of measuring the weight conducted in a non-inertial reference frame attached to the Earth, geocentric or not.

One will recall that the mass of a body is invariable and stay the same anywhere in the universe (if there are no nuclear reactions in the body and if one can neglect relativistic effects), while the weight depends on the location in the universe.

Appendix 6: The Acceleration g

The acceleration g_gr defined in relation (5) is indeed the acceleration of gravity. However, by an unfortunate but current language abuse, the average value of 9.81 ms⁻² given at the end of Sect. 3 is in reality the average value of the acceleration of gravity, expressed in a reference frame attached to the Earth and considered as non-inertial. As shown in Appendix 5 in (A5.7), the average acceleration of gravity in a point on Earth’s surface of latitude $\phi$ can be written in a first approximation as

$$ g_{w} = g_{gr} - a_{centrif.} \cos \phi = \frac{{GM_{E} }}{{r_{E}^{2} }} - \omega_{E.rot.}^{2} r_{E} \cos^{2} \phi $$

(A6.1)

The accelerations $g_{w}$ and $g_{gr}$ given here above depend on Earth’s radius, i.e. on the distance of Earth’s attractive centre to a point located on Earth surface. This distance varies in function of latitude $\phi$ as Earth is not exactly spherical but can be represented in a second approximation by an ellipsoid of revolution flattened at the poles. The values of Earth’s equatorial and polar radii recommended by the International Astronomical Union are respectively 6 378 136.6 and 6 356 752.3 m. Relation (A6.1) yields then the values in Table 2.

Table 2 Approximate values of gravity accelerations

These values are approximate but generally sufficient for calculations in the first order.