# Which law of motion is called inertia

## Sluggish and heavy mass

About the different roles of the crowd, and about the weak principle of equivalence

#### An article by Markus Pössel

The mass plays a peculiar multiple role in classical physics. This creates a difference between gravity and all other forces - a difference that stood at the beginning of Einstein's development of his general theory of relativity.

### Mass and inertia

Mass is a measure of the inertia of a body. Suppose I am in empty space, far away from all major sources of gravity. I'm standing on the outer surface of my space station, held securely by foot clamps, and a box and a small ball are floating next to me. Now I give both the box and the ball a nudge of equal strength. This significantly accelerates the ball and flies away at high speed. The box, on the other hand, could only accelerate my push very little. It drifts away from me at very low speed:

The difference between a ball and a box? The box has a significantly greater mass and therefore flies away much more slowly than the ball with its lower mass under the same impact conditions.

The example gives a rough idea of what the crowd is all about. However, it is only suitable to a limited extent for a definition, because for this we would first have to specify when two impulses have the same strength. Instead, it is useful to consider simple collision processes between different bodies. In space one can simply let the bodies fly towards each other for this purpose. In the earthly laboratory, precautions have to be taken to eliminate the effects of friction, for example by installing the bodies on the sled of an air-cushion track, which is part of the standard equipment of school physics laboratories.

It is easiest if you let the bodies fly towards each other frontally (“central impact”). The movement of the body then takes place on the same straight line before and after the impact:

In general, the speeds of the balls will change on impact. We want this change in velocity for the first body Δv_{1} call the change in speed of the second body Δv_{2} - any change in speed is simply the speed of the body after the impact minus its speed before the impact. The ratio of the changes in speed of the two bodies, as shown by corresponding experiments, is independent of the initial speeds of the bodies. It therefore reflects properties of the body, not a property of the initial situation before the impact. We use it to define the crowd, through the relationship

where the minus sign is needed to get a positive numerical ratio. As soon as we have defined a unit of mass (that is, a body arbitrarily selected as a reference body), we can, in principle, determine the masses of all other bodies by determining changes in speed during impact tests.

This definition corresponds to our intuitive ideas about large and small mass or inertia. For example, if the first body has a much greater mass than the second, the left-hand side of the above equation is a large number. Then the first body will hardly change its speed when colliding with the second, whereas the second body will change its speed even more. This corresponds to the statement of the formula above, because it says that the right-hand side of the equation is also a large number. The animation above shows an example: At the beginning, the initial velocities of the turquoise and violet spheres are opposite and equal. The mass of the violet sphere is, of course, four times that of the turquoise one; the purple ball is therefore only slowed down a little, the speed of the turquoise ball changes drastically: it bounces off and then flies back much faster in the direction from which it came.

### Indolence and forces

The mass defined in this way plays a key role in classical mechanics - the theory of how objects move under the influence of forces. For example, as is customary in school, forces can be measured with the help of suitably selected coil springs: The elongation of a coil spring that is acted on by a force (which you pull with your hand, for example) is a measure of the strength of the force - at least for areas in which this elongation is quite small:

This definition agrees with our subjective perception insofar as stronger pulling also leads to greater elongation of the spring. In addition to the subjective sensation, this arrangement now provides the possibility of relating strengths of forces to one another - we can determine exactly when a force is two, three or four times as strong as another force.

Newton's second axiom of mechanics states how the force acting on a body is related to the acceleration it experiences. In short, the following applies

**Force = mass · acceleration**

or, written the other way around,

**Acceleration = force / mass**.

From this it follows once again that the ball and box, on which I act with the same force, accelerate differently: The mass of the ball is much lower, and the expression "force divided by mass", i.e. the acceleration, is therefore much greater than for the box.

### Without looking at the object: gravitational acceleration

A special phenomenon occurs where gravity is involved. We can observe that here on the surface of the earth, for example. If you let the most diverse objects fall to the ground under the influence of gravity, you will find that they all accelerate in the same way. At least that is true if you switch off the effects of air friction by carrying out the drop tests in a vacuum chamber, for example: Whether it's a spring that you drop or a lead weight - one second after you let go is a speed of around 10 meters reached per second, after two seconds it is already around 20 meters per second. If you let go of the pen and lead weight at the same height, they fall side by side in perfect harmony:

In particular, the gravitational acceleration is independent of the masses of the falling objects.

Similar results are obtained if one looks at the movement of objects under the gravitational influence of a central mass on cosmic scales. The most important example is the movement of the planets under the gravitational influence of the sun - at least in the (quite good) approximation in which the gravitational influences of the planets on each other are neglected. In this case, too, the movement can be explained by the fact that every object or planet experiences a gravitational acceleration that is independent of its characteristic properties, and in particular of its mass. This is of course no longer constant, but varies with the distance to the central mass.

### Mass as gravitational charge

One may accept the fact that gravitation expresses itself in accelerations as natural. In one respect, however, it is remarkable, namely when one wants to describe gravity as a force - so obviously with the same concept with which, for example, frictional forces, impact forces, elastic spring forces, electrical or magnetic forces can be described. For the description of planetary orbits and the like, the introduction of a gravitation*force* it does not make much sense - you can also expect accelerations there. For the calculation of force equilibria, for example to answer the question of how strong a certain spring has to be in order to compensate for the effect of gravity on a certain body, or how stable a wall has to be in order to compensate for the gravitational influence of the earth on the roof above on the other hand, it is useful to introduce a gravitational force.

The gravitational force that acts on an object can be measured with the spring arrangement that we introduced above to measure the force - instead of pulling on the spring, we simply hang an object on the spring:

Based on the expansion of the spring, we can read off how strong the gravitational force is that pulls the object towards the ground. This gravitational force will generally be that *Weight* of the body in question.

However, a gravitational force, which leads to the same acceleration due to gravity for all freely falling bodies, must have very specific properties. The acceleration caused by a force results, as already mentioned, at

**Acceleration = force / (inert mass)**.

So that the acceleration is the same for all objects, the gravitational forces must vary depending on the object's mass - in concrete terms: the gravitational force that the earth exerts on each of these objects, for example, must be proportional to their mass, expressed as a formula:

**Gravitational force = mass g**,

where g is a factor that does not depend on the properties of the object in question. Then the following happens when you calculate the acceleration according to the formula given above:

**Acceleration = gravitational force / mass = (mass g) / mass = g**

- the mass of the falling object is reduced, and the same acceleration g remains for all objects. This acceleration is also called acceleration due to gravity or acceleration due to gravity. It has roughly the same value everywhere on (and near) the surface of the earth.

We regularly use the fact that the gravitational force acting on a body - the weight of the body - is proportional to its mass. We use scales - for example spring arrangements like the one shown above - to measure the weight of objects and deduce the mass directly from them, yes, we usually choose the scale of the scale so that we can read the mass straight away! From a physical point of view this is a horror - “this body has a weight of one kilogram” is simply a false statement, because weight is a force and should be given in units of force, whereas kilograms are the unit for mass. If this equation still works in everyday life, it is only because the weight of a body is proportional to its mass, and because the proportionality factor, the acceleration of gravity, has almost the same value everywhere on the earth's surface.

In the physics of forces and fields, it is the rule that the force strength is proportional to a very specific object property. The relevant object property is called the object's charge associated with the force. For example, the electrical force that acts on an object is given by the product of the electrical charge of the object with a variable that is independent of the object's properties, the electrical field (further information on the concept of field can be found in the advanced topic From force to field).

If the gravitational force acting on an object is the product of the object's mass and a factor that does not depend on the object in question, then mass is a species *Gravitational charge*.

### Mass as a source of gravity

Mass also plays another role. So far we have only introduced the gravitational charge as a measure of how strong the force is that a test object experiences from the gravitation of a given mass such as the earth. On the other hand, the charge of an object also determines how strong the force is that emanates from this object - in this case: the mass as gravitational charge determines the gravitational force with which the object in question acts on the surrounding body. If we want to distinguish these two aspects, we can speak of an active and a passive charge. The active gravitational charge of the sun can be determined, for example, by observing the planetary orbits; the active gravitational charge of a metal ball can be determined experimentally with a structure named after the physicist Henry Cavendish, which is sketched here:

The small balls (gold in the picture) are suspended from a rigid rod (turquoise). They are attracted to the larger test masses (purple) - but there is still the torsion thread (red) on which the rod is suspended and which offers resistance to any attempt to twist it.

The active gravitational charge is also proportional to the mass. The proportionality factor - it can be determined from the Cavendish experiment, in which the mass of the attractive spheres can be measured directly - is the gravitational constant.

### Three things

Mass thus plays three conceptually quite different roles - on the one hand that of an active and a passive gravitational charge, on the other hand that of a measure of the inertia of a body. Or are we not dealing with three different roles, but with three different physical quantities - the *inert mass*that determines the acceleration with which a body reacts to a given force, and the *(passive) heavy mass* or the *(active) heavy mass* as a gravitational charge?

This distinction is not obvious, but it can be made - if only to see whether such differences can be proven experimentally or whether the experiments confirm the equality of the different types of mass within the scope of their measurement accuracy. The result of at least all previous experiments is: Passive heavy mass and inert mass are actually the same within the scope of the measurement accuracy. Objects that are placed next to each other in space and then released fall in the same way and experience the same acceleration, regardless of their composition or their properties. Comparatively few experiments are devoted to the difference between active and passive heavy mass, but they also speak in favor of the equality of the two quantities.

### Theory of Relativity and the Weak Equivalence Principle

In classical physics it is unclear how this equality comes about. It is different in Einstein's theory of gravity, the general theory of relativity. There the reaction of bodies to the action of gravity is explained purely geometrically - masses distort the geometry of space and time, and all bodies follow the straightest possible paths in this distorted space-time. The artificial dichotomy that follows from the concept of force - the force dependent on the heavy mass on the one hand, the reaction to it dependent on the inertial mass on the other hand - is replaced by a law in which the equal treatment of all bodies is built into the lowest level: that all bodies Experiencing the same gravitational acceleration in a given situation is because their motion is directly determined by the properties of the space-time surrounding them.

Conversely, the fact that all bodies experience the same acceleration due to gravity is a basic requirement of Einstein's description. As a so-called “weak equivalence principle” (as part of the more general Einsteinian equivalence principle) it is at the very beginning of every systematic description of general relativity.

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