*Notes to a video lecture on http://www.unizor.com*

__Force, Acceleration,__

Inertial Mass

Inertial Mass

While Kinematics studies all aspects of an object's motion, Dynamics studies the cause of the motion,

*forces*, and object's responsiveness or resistance to these

*forces*, measured by its

*inertial mass*.

The

**Law of Inertia**, that should be considered as an axiom confirmed by experiments, states that

**an object**(that is, no change in

*at rest*__stays__*at rest*and an object in*uniform motion*__stays__in this*uniform motion**velocity vector*)

**, unless acted upon by**.

*unbalanced forces*When we first mentioned this principle, we did not explain a concept of

*force*, living it to intuitive understanding. However, from this principle, taken as an axiom, we can conclude that any change in

*velocity*of an object must be caused by something that we can call the

*force*.

This logic of defining an object not by directly pointing to it, but by

specifying its properties, is quite familiar to us. In Mathematics we

never define certain basic objects, like

*point*or

*straight line*,

but describe their properties and use these properties to construct

more complicated objects and prove their properties based on properties

of their components. In Physics we did not define

*space*or

*time*, yet we know how to specify the position in

*space*using coordinates or how to measure

*time*using clocks.

So, not specifically pointing to

*force*, we can specify its main property - it can change the

*velocity*of an object. Or, going from the object side, if it changed its

*velocity*, there must be some

*force*that caused it.

Any change in

*velocity*(magnitude or direction) results in non-zero

*acceleration*, since

*acceleration*is the derivative of

*velocity*by time. If during any time interval the object's

*velocity*has changed, the

*average acceleration*during this interval is non-zero and, therefore,

*instantaneous acceleration*(that is, a derivative of

*instantaneous velocity*) cannot be always zero during this time.

We can conclude, therefore, that the concepts of

*force*and

*acceleration*are strongly related to each other.

Moreover, since we know how to measure

*acceleration*of any object, we can use this quantitative characteristic to measure

*force*that caused this acceleration.

Going back to experiment, we can change the

*velocity*by applying to an object certain effort as an act of

*force*.

For example, we can push a chair that stands on a floor and it will move, which means that its

*velocity*relative to a floor changed from zero to some other value. This change in

*velocity*and, therefore, non-zero

*acceleration*are caused by our push, which we can consider as an act of

*force*.

Let's push now a sofa with the same effort as we used pushing a chair. Most likely, it will move only a little, its

*velocity*will change from zero to some very small value.

So, we made the same efforts in both cases, pushing a chair and a sofa, but got different results.

It prompts us to think that, besides the value of

*force*, which

was related to our efforts and must be the same in both cases, there is

some characteristic of an object that affects the resulting change of

*velocity*. This characteristic, a property of any object, is a measure of how sensitive this object is to the

*force*applied to it, how much it accelerates, when some specific

*force*acts on it.

In other words, since absence of

*force*is associated with

*inertia*, this new characteristic of an object is a measure of how well it maintains its state of

*inertia*, thus resisting the act of

*force*. This characteristics of an object to maintain the state of

*inertia*is called

**.**

*inertial mass*In many cases we will omit the word "inertial" and use only "mass" until we reach the section dedicated to

*gravity*, where the new term

*gravitational mass*will be introduced.

We definitely want to measure both

*force*and

*inertial mass*. We know how to measure

*acceleration*, it's the second derivative of the

*position function*by time. We can also agree that the

*force*is a vector directed exactly as the

*vector of acceleration*. But it's not enough. Without quantitative measuring of the magnitude of the vector of

*force*acting on an object, even knowing its

*acceleration*, we cannot quantitatively evaluate its

*mass*, and without

*mass*we cannot evaluate the

*force*. Seems like a dead end.

Of course, not.

Here is how we can solve this problem.

Firstly, we can learn how to compare two forces to determine whether

they are equal or not. We simply apply them to the same object in the

same state and check if

*vectors of acceleration*achieved by this object are the same in both cases. If it is, the

*vectors of forces are equal*.

Secondly, we can learn how to compare masses of two different objects to

determine whether they are equal or not. We simply apply the same

*vector of force*to them and check if

*vectors of acceleration*of these objects are the same in both cases. If it is, the masses are equal.

Knowing how to compare masses and forces, people have decided to choose some object as a unit of

*mass*- a small cylinder made of platinum-iridium alloy that weighed about the same as a cubic decimeter of water. The

*mass*of this cylinder they called "one kilogram" (

**), and every other object of the same mass would have a**

*1 kg**mass*of

**. From this such derived units as "one gram"**

*1 kg***)**

*1 g = 0.001 kg*Now it was possible to define a unit of magnitude of the

*vector of force*- the one that gives an object of mass of

**an acceleration of**

*1 kg***.**

*1 m/sec²**vector of force*is called "one newton" (

**).**

*1 N*Continuing this logical process of defining how to measure forces and masses, we can make the following steps.

If the same force, applied to two objects, A and B, gives to A an

acceleration N times bigger than to B, we say that the mass of B (a

measure of its resistance to force, a measure of its inertia) is N times

as big as the mass of A. If the mass of A is

*1 kg***.**

*N kilograms*Similarly, if force F gives an object acceleration N times greater than

force G, applied to the same object, we say that force F is N times

greater (stronger) than force G. If the force G is

**.**

*1 newtons***.**

*N newtons*
## No comments:

Post a Comment