# What is inertia

Definition/Summary

Inertia is the phenomenon that a force is required to cause change of velocity. The amount of inertial mass of an object is measured by measuring how much force it takes to accelerate it. The symbol for inertial mass is m.

Equations

Extended explanation

The original meaning of 'inertia' amounts to 'tendency to lag behind', or 'slowness to respond'. A similar expression is 'inert gas' in chemistry.

Mechanical accelerometer

A mechanical accelerometer measures degree of lagging behind. Inside a casing a test object is suspended with springs. When the casing accelerates then initially the test mass inside will not co-move with the casing. As the casing accelerates the springs compress and extend, until a point is reached where the forces that the springs exert upon the test mass add up to the amount that is required to make the test mass co-accelerate with the casing.

In terms of human perception: if you're in a car, and it starts to accelerate then initially you will not co-accelerate with the car; initially the car seat is not exerting enough force upon you. As the car accelerates it moves relative to you, and the springs in the seat compress. When sufficiently compressed the springs exert the required force to make you co-accelerate with the car.

Physical sensation

In the car the physical sensation is one of being pushed into the seat; it feels as if a force is exerted upon you.
The reason for that sensation is the close connection between inertia and gravitation. Gravitation acts in equal measure upon all parts of an object, and gravitational mass is always proportional to inertial mass. (This is exemplified by the following: when an accelerometer is released to free fall it will register zero acceleration. This is called the equivalence of inertial and gravitational mass.)
When you are in a car that is accelerating the perception of being pushed into the seat is identical in nature to the perception of gravity pulling your body into a very soft mattres.

Why inertia is not a force

During acceleration of a car a force is doing work, increasing the car's kinetic energy. Now consider a car equipped with regenerative braking. When the driver switches to regenerative braking the motors start operating as generators, and the batteries are recharged.

The process of converting one form of energy to another is referred to as 'doing work'. When the car accelerates electric potential energy stored in the batteries is converted to kinetic energy of the car. Regenerative braking is the reverse of that process; kinetic energy is converted to electric potential energy.

If you're in a car and it suddenly brakes you feel yourself lunged forward, which is due to inertia. So can we say that in regenerative braking inertia is doing work upon the batteries, recharging them? No, we cannot say it in that form, because inertia cannot be categorized as a force. In newtonian dynamics the Third Law defines what counts as a force and what doesn't. It is counted as a force if it consists of an interaction between a pair of objects, accelerating each other. Example: electric force between charged particles.
Inertia is not part of such a pair as defined by the Third Law, so inertia cannot be categorized as a force. Therefore in the case of regenerative braking we cannot say that inertia is exerting a force, recharging the batteries: that would create a self-contradiction.
The self-consistent description is to say that in regenerative braking the car battery is doing negative work. (In accelerating the car the battery is doing work, in deceleration you get the reverse: doing negative work.)

The key point is that inertia cannot be categorized as a force.

Inertial frame of refererence

By definition an object is in inertial motion if a co-moving accelerometer registers zero acceleration.
However, the equivalence of inertial and gravitational mass presents a practical problem: an accelerometer onboard a spacestation will register zero acceleration. Are we forced to draw the conclusion that the spacestation is not accelerating?
The fact that we have knowledge of the physics that is involved allows a more comprehensive view. When considering a mechanical phenomenon inside the spacestation then for all practical purposes the spacestation frame can be considered an inertial frame.
If the perspective is widened, encompassing the entire Earth and the spacestation in orbit, then the Earth, being far the heaviest, takes prime position, and the satellite is considered to be in acceleration with respect to the Earth.
In a yet wider perspective, encompassing the entire Solar system, the designation of 'inertial frame' goes to the Solar system's center of mass. The common center of mass of The Sun and Jupiter lies just outside the Sun. Roughly said: "Size matters".

Laws of motion

The Solar system's common center of mass is not an observable point of course, but the laws of motion and the law of gravitation enable us to establish its position with the same accuracy as the positions of the celestial bodies themselves.

The uniformity of inertia

The reason that laws of motion can be formulated at all is the fact that inertia is extraordinarly uniform. If an object is in inertial motion then in any direction the same force will result in the same acceleration. Also the acceleration is always exactly proportional to the exerted force, and in the same proportion everywhere.
Try to imagine a universe in which inertia is erratic, randomly changing from place to place and over time. Then no laws of motion would exist.
Laws of motion as we have exist if and only if inertia is perfectly homogenous (evenly distributed everywhere), and perfectly isotropic (the same in all directions).

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