Monday, May 7, 2018

Unizor - Physics - Mechanics - Dynamics - Force, Acceleration, Mass

Notes to a video lecture on

Force, Acceleration,

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 at rest stays at rest and an object in uniform motion stays in this uniform motion (that is, no change in 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" (1 kg), and every other object of the same mass would have a mass of 1 kg. From this such derived units as "one gram" (1 g = 0.001 kg) were defined.

Now it was possible to define a unit of magnitude of the vector of force - the one that gives an object of mass of 1 kg an acceleration of 1 m/sec². This unit of magnitude of the 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, the mass of B is 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., the force F is N newtons..

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