Basic notes on Physics.
In the series: Note 1.

Subject: A simple note on some highlights of Classical Mechanics.

Date : 12 October, 2016
Version: 0.11
By: Albert van der Sel
Doc. Number: Note 1.
For who: for beginners.
Remark: Please refresh the page to see any updates.
Status: Ready.


This note is especially for beginners.

Maybe you need to pick up "some" basic "Physics" rather quickly.
So really..., my emphasis is on "rather quickly".

I am not sure of it, but I hope that this note can be of use.
Ofcourse, I hope you like my "style" and try the note anyway.

It helps to understand this note, if you have a certain level in math.

It's best if you, from my "basic math series", have read note 1 to and including note 5
and note 12, or comparable material.
That means that you are reasonably comfortable with linear- and quadratic equations, sin(x)
and cos(x) functions, and the basics of vectors.

This note: Note 1: Some highlights from Classical Mechanics.


What is classical mechanics anyway? It's in fact a large part of the "typical physics" that one can learn in Highschool.
Usually, it covers Forces on objects, the equations of velocity and acceleration, the potential and kinetic energies,
what "impuls" is, the harmonic oscillator, waves, and other topics.

It's called classical, because since Einsteins relativity theories, and the advent of Quantum Mechanics, it is known
that Classical Mechanics does not adequately can describe very small physical systems (like atoms, elementary particles),
and physical systems moving with a speed that is a relevant fraction of the speed of light.

However, it is the physics that really works in daily life, and it therefore it is a main topic in physics, engineering,
and many other sciences.

Also, classical mechanics is the starting point before one goes into other disciplines of Physics.

Many physicists and mathematicians have contributed to classical mechanics, but one name really stands out: Sir Isaac Newton.
That's why classical mechanics is also often called "Newtonian mechanics".

Optional reading: A remark on the Units of measurement in Physics:

Most countries have adopted the "metric system of units" (mks), or stated somewhat more generally, the "SI" system (International System of Units).
These are units like the meter (m) for length, the kilogram (kg) for mass, the second (s) for time, the Newton (N) for force,
the Joule (J) for energy etc.. Ofcourse, I will explain the units like the "Newton" and "Joule" (and others) in this note.

A few countries (up to now) stick to other unit systems, like the CGS system of units. Most often, the differences are not
too shocking, since it's not very confusing if you like to measure lenght in centimeters, instead of meters, or mass in grams instead of kilograms.
In these cases, the difference are characterized by powers of "10", so it's quite easily survivable.

However, in some cases there can be some confusion, for example, if someone expresses force in "dyne" instead of "Newton".

But keep in mind that such issues never "hurts" the fundamentals of the theories. It simply can't, since we are only talking about "units".
I will stick to the metric/SI system of units, but even if you are acustomed to CGS, you won't be bothered by that fact.

This note has the following content:

Chapter 1. Motion, velocity, and acceleration.
Chapter 2. Forces in Mechanics.
Chapter 3. The concept of Momentum.
Chapter 4. Kinetic Energy, Potential Energy, and Work.
Chapter 5. Harmonic oscillators in classical Mechanics.
Chapter 6. Gravitation and potential Energy.



Chapter 1. Motion, velocity, and acceleration.

In this Chapter, we are going to investigate several types of motion that an object may have.
We are especially interested in equations that relate the distance covered, the speed, and the time elapsed.

While "velocity" actually is a vector, the "magnitude" of that vector is associated with "speed".

In the first few sections of Chapter 1, that is 1.1 up to 1.3, I will use the term "speed" in the theory and examples.
The reason is that in those sections we talk about the "linear uniform motion" (where the speed is constant),
and the "linear uniform acceleration" (where the acceleration is constant), and where the motion of an object is in one direction only.
However, as of section 1.4, we go for the more "formal" way.
For the "true" physical treatment of "motion", see section 1.5.

1.1. Uniform motion with a constant speed (v), S=vt.

Suppose we have an empty, and vacuum Space (just like outerspace). Suppose that there is no forcefield in that Space, whatsoever.

Now, imagine that we have an object that moves through that Space, while no force is present. If that object has
a speed of, say, 30 m/s (meters per second), then it will keep that speed indefinetely.

Likewise, if that object was in rest, and no force acts upon it, then it will stay in rest.

The upper statements are actually a description of Newton's first law of motion.

Suppose an object has a constant speed (v). Then there exists a very simple, and obvious relation between
the distance covered, and the time that have elapsed.
This relation is:

S = vt       (equation 1)

If something travels with a speed of 10 meters per second (m/s), then after 5 seconds, it has covered 50 meters.
If it later travels with a speed of 100 meters per second (m/s), then after 5 seconds, it has covered 500 meters.
The equation "S = vt", is indeed logical and probably agrees with your experiences in daily life.

It's actually the same sort of linear equation as "y=ax" we encountered in "Math note 1".

Example 1:

Suppose a car travels with a constant speed of 20 meter per second. What is the distance covered in 5 seconds?

That distance is:

S = vt = 20 x 5 = 100 m.

Or, the same calculation, but this time also including the units:

S = vt = 20 m/s x 5 s = 100 m.

Note that the speed is measured in "m/s" and time in "s", so in a multiplication, we are only left with "m",
which is correct, since a distance uses the meter as a unit.

Also note that S is a linear function of t, like in S(t) = vt.
Since v is constant, the function S(t) is exactly similar to the well-known linear notation as "f(x)=ax".

Example 2:

Suppose a car travels with a constant speed "v".
After 5 seconds, it has covered a distance of 400 meters.

So, what is it's speed?

We know that S = vt.

In this case, S and t are known values. We are looking for the speed v.

If S = v t, then v = S/t = 400/5 = 80 m/s (which is about 288 km/h).

Or with units:

If S = v t, then v = S/t = 400 m / 5 s = 80 m/s

Example 3:

Suppose a man runs with a constant speed of 3 meter per second. At t=0, we start measuring the distance covered.
Task: Make a graph of S and t.

We know that S(t) = 3t, since the constant speed is 3 m/s.
This is precisely similar to the function "f(x) = 3x", which should be quite easy to plot:

Figure 1. S(t)=3t



For example, if t=3, we have S = vt = 3 * 3 = 9, which we can also read from the graph in figure 1.

1.2. The derivative of S=vt.

If an object travels with a constant speed "v", the rate that the distance "S(t)" changes over time, is constant.
This should also show up, if we calculate the derivative of S(t) with respect to "t".

d S(t)
----
dt
=   d (vt)
------
dt
=   v

This no different from the usual linear relation f(x)=ax in math, and if we would take the derivative of f(x):

d f(x))
----
dx
=   d ax
----
dx
=   a

You know that the derivate "dS/dt" is the rate of change of "S" relative to the change of "t".
Also, a simple example will effectively illustrate this:

If an object travels with a constant speed of 3 m/s, then:

After 1 second, the distance is 3 meters.
After 2 seconds, the distance is 6 meters.
After 3 seconds, the distance is 9 meters.
After 4 seconds, the distance is 12 meters.
etc...

The differential, is only the infinitesemal variant of:

Δ S(t)
----
Δ t

Suppose you take the difference in distance between t=2 and t=4 (as shown in the listing above), then we have:

Δ S(t)
----
Δ t
=   6
-
2
=   3

So, for a motion with constant speed we have:

d S(t)
----
dt
=   d (vt)
------
dt
=   v       (equation 2)

You can ofcourse also use the "accent" notation for taking the derivative, which is the same thing:

If:

S(t) = vt

then:

S'(t) = v

Note: "dS(t) / dt = speed", is universally true.

Above we had a relatively simple relation S(t) = v t. So in this case, dS(t) / dt = v.

However, even if the relation "S(t)" was very complex, the "format" of the differential is universally true.
Suppose S(t) is a quadratic function of "t", or possibly even of a higher power, or even containing goniometric parts,
then the statement "dS(t) / dt = speed" still holds, as long as S(t) is a well-behaved continuous function of "t".

It's in part just a consequence of the theory of differentials.

1.3. Motion with a constant acceleration.

If, as of a certain moment, a constant force starts to act upon an object, then it starts moving.
Unless ofcourse there exist so much friction (in some way), that this force cannot overcome that friction.
Then the object stays at the same place.
For now, we just think of objects in free empty Space, where there is no friction.

But, once the object starts moving, it goes faster and faster as time progresses. In other words: the speed increases.
That is, the object undergoes an acceleration since the speed keeps increasing.

Let's start our next argument, considering an object with mass "m", which is in rest first.
Then, at a certain moment, we apply a constant force on that object.

Just from experience, we know that if the mass is very low, the object picks up speed quickly, and in a short time,
goes faster and faster.
On the other hand, if that mass is considerable, then it slowly starts moving, and slowing picks up more speed.
The above is true ofcourse, if in both cases the same force was applied.

Newton concieved the following equation, which really is the "heart" of classical mechanics:

F = m a       (equation 3; which is Newton's second law)

Where "F" is the Force applied, "m" is the mass of the object, and "a" is the resulting acceleration.

Or: "Force" = "mass" times "acceleration". Although the equation is not complex, it's quite hard to prove it mathematically.

Experimentally, it works like a charm. That is: if you vary forces and masses over a large number of experiments,
all experiments obeys the same formula.

Experiment:

Suppose we have an object of 2kg. Suppose we apply 16 Newton of Force. The acceleration then is 8 m/s2.
Next, suppose we have an object of 4kg. Suppose we still apply 16 Newton of Force. The acceleration then is 4 m/s2.
Next, suppose we have an object of 8kg. Suppose we still apply 16 Newton of Force. The acceleration then is 2 m/s2.

Indeed, we will always find "F = ma", even if we largely vary "F" and "m".

In chapter 2, I will go deeper into the concept of force and Newton's second law.

The few paragraphs above, was actually about how an object gets into motion.

What we really want to do in this section, is the find the equations that relate distance, speed, acceleration and time,
just like we did with the "uniform motion" in section 1.1.

speed and acceleration:

If an object experiences a constant acceleration, then it's speed increases.
That's rather obvious, let's try to make it more precise.

A constant acceleration simply means a constant rate of increase of the speed.
That is, per time unit (say, per second), the same increment (Δ v) of the speed takes place.
This "unit of increase", is the acceleration.

So, it boils down to: Increase of speed Δ v = a Δ t

In reality, the increase of speed, due to the constant acceleration "a", is continuous with time, so
so we may actually skip the Δ.

Thus:

v(t) = at       (equation 4)

or

v(t) = v0 + at       (equation 5)

Equation 5 applies, if our object already had some initial constant speed "v0", before the acceleration started.

Note that equations 4 and 5 are linear relations too. The acceleration is a certain constant (due to some constant force).
So, they truly resemble the well-know linear functions from math, namely "f(x) = ax" and "f(x) = ax + b".

Example 4

Suppose an object is in rest.

At t=0, a force starts to act on the object, that results in a constant acceleration of 5 m/s2.

What is the speed after 10 seconds?

Answer:

v(10) = at = 5 * 10 = 50 meters per second.

Or in units:

v(10) = at = 5 m/s2 * 10 s = 50 m/s.

Example 5

Suppose an object has a initial constant speed of 30 m/s.

At t=0, a force starts to act on the object, that results in a constant acceleration of 5 m/s2.
In this case, the direction of the force is exactly the direction of how the object was already moving.

What is the speed after 10 seconds?

Answer:

v(10) = at + v0 = 5 * 10 + 30 = 80 meters per second.

Or in units:

v(10) = at v0 = 5 m/s2 * 10 s + 30 m/s = 50 m/s + 30 m/s = 80 m/s.

Example 6

Suppose an object has a initial constant speed of 30 m/s.

At t=0, a force starts to act on the object, that results in a constant acceleration of 5 m/s2.
However, in this case, the direction of the force is exactly OPPOSITE the direction of how the object was already moving.

What is the speed after 2 seconds?

Answer:

This time, it makes sense to talk about a "+x" and a "-x" direction.
You know, in physics you need to be cunning once in a while, and "rearrange" a situation so that it is
most comfortable for you.

The motion of the object will be as follows. Due to the opposite force, it starts to slow down.
Then, for an extremely short moment, it comes to rest (v=0), and then immediately accelerates in the opposite direction.

You may choose whether initially the object moved to the "+x" or a "-x" direction.
It really does not matter, since this situation is completely symmetric.
I will choose the "-x" direction as the initial direction that the object was moving along.
So, the object initially had a speed of -30 m/s (due to our choice of + or - direction).

In general, we have:

v(t) = v0 + at

Thus:

v(2) = -30 + 5 * 2 = -30 + 10 = -20 m/s (in the -x direction)

What is the speed after 5 seconds?

v(5) = -30 + 5 * 5 = -30 + 25 = -5 m/s (in the -x direction)

What is the speed after 10 seconds?

v(7) = -30 + 5 * 10 = -30 + 50 = 20 m/s (in the +x direction)

Example 7 (neglecting air resistence)

The force of gravity, near the Earth's surface, results in a constant acceleration of about 9.8 m/s2.
Suppose you are on a very high cliff, and you have an iron ball. You drop it. Now you want to know it's speed after 8 seconds.

It goes exactly as before:

v(8) = at = 9.8 * 8 = 78.4 meters per second.

Or in units:

v(8) = at = 9.8 m/s2 * 8 s = 78.4 m/s.

This answer is ofcourse not fully correct, since (mainly) the air resistence will have a certain "slowing-down" effect.
So, it really makes a difference whether we are in a free, vacuum Space, or we are in another environment,
like a Planet with an atmosphere of a certain density.

Distance covered of an accelerated motion:

Since "v(t) = at", we know that the speed keeps on increasing, so we expect that the distance covered per unit of time, (like per second), gets larger and larger too.

Since it universally holds that:

S'(t) =   d S(t)
----
dt
=   speed

Then in this type of motion we can say:

S'(t) =   d S(t)
----
dt
= at

Thus it follows that:

S(t) = ½ at2       (equation 6)

This is true, since the derivative of "½ at2", is "at".

You probably reckognize that S(t) = ½ at2, is a "parabola".

Example 8:

Suppose an object is initially at rest. At t=0, a force acts upon the object,
resulting in an acceleration of 8 m/s2.

Question: what is the speed, and distance traveled, after 8 seconds?

Answer:

-After 8 seconds, the speed is:

v(8) = a t = 8 * 8 = 64 m/s.

-The distance traveled is:

S(8) = ½ at2 = ½ 8 * 82 = 256 m.

1.4. Speed and Velocity.

"Velocity" is a vector, and the associated "magnitude" (the value, or length) is called "speed".

In all of the above, I used velocity and speed, as if there is no difference. But there is!
"Velocity" is a vector, thus it has a certain "length", but a direction too (in your coordinate system).

However, all examples above were "one dimensional", with no attention for direction.
Those are all OK really, and we do not need to change anything in the examples above,
since we assumed, at each example, that the motion was in one direction only (e.g. the x-axis).

Whenever we must treat velocity as a vector, It does not add much complexity. However, we must treat the theory a little differently.

Once it gets apparent that you really are dealing with a vector, then use the theory from "math note 12", section 1.3.
If needed, consult that note again, since it's a gentle (I hope) introduction in vectors.

In a orthonormal Cartesian frame of reference, it's not hard to calculate the "magnitude" of a vector.
Suppose the vector is called A, then it's magnitude (lenght/speed) is often denoted by |A|.

Examples:

In R2, the length of vector A is |A|=√(x2 + y2)
In R3, the length of vector A is |A|=√(x2 + y2 + z2)
In RN, the length of vector A is |A|=√(x12 + x22 + .. + xN2)

You can also "project" such a vector onto one of the Coordinate axes, by using the inner product of the vector
and the corresponding unit vector. Please see math note 12.
Here is an illustrative example:

Example 9:

Suppose we have the vector A=(1, 3, 2).

Question:

Calculate the magnitude |A|.

Answer:

|A| = √(12 + 32 + 22) = √ (1 + 9 + 4) = √ 14     (is about 3.7).

Question:

Project A onto the y-axis.

Answer:

┌ 1 ┐
│ 3 │
└ 2 ┘
· ┌ 0 ┐
│ 1 │
└ 0 ┘
= 3

The component of A onto the y-axis, Ay, is (0, 3, 0) which has a length of "3".
So |Ay| = 3. This is the magnitude of that component.

Example 10:

Suppose, a plane flies with a constant horizontal velocity of 30 m/s in the +x direction.
Suppose further that we may neglect the air resistenc.

Now, at t=0 you drop an object out of the plane. Describe the components of the total velocity vector.

Immediately after you drop the object, your object has a constant horizontal velocity component vx of 30 m/s.
Since we may neglect resistence, it will keep that constant horizontal velocity component.

So, vx = 30 m/s   (all the time).

However, also immediately after you drop the object, Earth's force of gravity will act, resulting in
a constant vertical (downwards) acceleration, often called "g", along the z-axis.
This acceleration will increase the vertical velocity component, according to the formula:

vz = gt

At any time, the total velocity vector vT is:

vT = vx + vz

where the addition above, is a vector addition of vx and vz

At any time, you can calculate the magnitude of both components.
The trajectory of the object, is a parabola as depicted in figure 2.

Figure 2. The trajectory, and the horizontal and vertical velocity components, of an object dropped from a plane.



As an example of calculating the total "speed" (or magnitude of vT), let's try this:

Question:

What is the total speed (the magnitude of the vector vT) at t=5?

Answer:

|vx| = 30 m/s at all times.

If we approximate "g" as to be 10 m/s2, then:

|vz| = gt = 10 * 5 = 50 m/s.

Using the "Pythagorean theorem", we can now calculate the total "speed" |vT| (at t=5):

|vT| = √ (302 + 502) = √ (900 + 2500) = √ 3400 (which is about 58).

The more seconds we go from t=0, the higher the |vz| speed component will be, ofcourse.

Maybe you like to try a similar calculation for t=10?

1.5. A better and more general description of position, velocity, acceleration.

If you are a "beginner" in physics, this section might seem intimidating...
But it's really not. Just try it. It's just a generalization of the former theory.

This time, we do not make any assumption on the dimensions of the Space we are working in.
Indeed, sections 1.1, 1.2, and 1.3 dealt with a motion in one direction only.
But now, we are going for general description of the motion of an object.

Please take a look at figure 3 below. The position of an object is represented by a vector r(t),
and in figure 3, you notice 3 example positions denoted by r(t1), r(t2), and r(t3).

By using a vector for the position r(t), all derivatives like the velocity v(t), will "automatically"
be vectors too.

Using a position vector, should not surprise you at all. In "vector calculus", it was just common practice to do so.
Indeed, for the point (1, -3, 4) in R3, you simply drawed an arrow from the Origin to that particular point.

Figure 3. General description velocity and position vectors.




If you look at figure 3, you may wonder how to describe the "trajectory" or path of the object.
We may have all position vectors as time elaspses, but what about the true "path" the object is following?

You notice the "orange" section between the endpoints of r(t1) and r(t2)?

It's actually the vector r(t2) - r(t1), which we may call Δ r(t).

The smaller the difference we take (and also the smaller Δ becomes), the more Δ r(t) becomes
the "tangent" to the path of the object.

But that's remarkable! Remember "math note 5", about the derivative of functions?

We know that a good approximation for the velocity of an object (between t2 and t1) is
comparing the "distance" to "the time elapsed". Indeed, "distance traveled, divided by time", is a very general expression for velocity.
So, we would have:

Δr(t)
----
Δt
=   Δv(t)       (equation 7)

But as it always goes with differentials, if we make the Δ smaller and smaller, we arrive at the true derivative:

dr(t)
----
dt
=   v(t)       (equation 8)

Equation 7, is a general expression for relating the velocity v(t) to dr(t) / dt.

This is also even compatible to the fairly simple equation "S=vt" we used in section 1.1
That's the equation for a linear motion with a constant speed. Note that "v=S/t" is completely
in line with equation 8. Since "v=S/t" says that v is "distance divided by time"

Now what about acceleration? As a general statement, we know that acceleration equates to the change in velocity
over a certain time interval.
Completely similar as above, we may thus say:

Δv(t)
----
Δt
=   Δ a(t)       (equation 9)

Again, as it always goes with differentials, if we make the Δ smaller and smaller, we arrive at the true derivative:

dv(t)
----
dt
=   d2r(t)
----
dt2
=   a(t)       (equation 10)

Where a(t) represents the acceleration.

Equations 8 and 10 are more or less quite "general". They deliver us the "instantaneous" velocity and accelertion on a certain time.
Note that for the velocity vector, it holds that at each moment, it is the tangent vector at the path the object is following.

However, the equations 7 and 9 for Δv(t) and Δa(t) can certainly be useful at times.
since you can view them as the average velocity and average acceleration over a relatively short time interval Δ t.

------------------------------------------------------------------

Optional Reading: Addition to Equation 8.

It probably looks a bit silly to come with a "addition to" the equations 8 and 10.
Then, why don't do it right the first time?

Equations 8 and 10 are certainly OK, but they are only written in a "general" format.
If we take a more closer look on the coordinate system in use, then some difference may get apparent.

In R3 for example, we may use the familiar orthonormal basis of unit vectors, which are assumed fixed.
In such a case, just simply take equations 8 and 10 literally, and take the derivatives of r(t).
In a moment, we will see some examples.
However, what if the unit vector may change too? Then it is a tiny bit different.
Here, I will present a really general equation for d r(t) / dt.

Suppose we write r(t) in the following way:

r(t)   =   |r(t)|   rU(t)   =   r(t)   rU(t)  

Where | r(t) | = r(t), is the lenght (or magnitude of r(t) ) and rU(t) is the unit vector.

Then, based on the mathimatical rule for derivative of a product of functions: (fg)' = f'g + fg', we must write:

v(t) =   d r(t)
----
dt
=   d r(t)
----   rU(t)  
dt
+   d rU(t)
------  r(t)
dt
      (equation 8' - note the accent)

Where:

d rU(t)
------ 
dt

is the differential (the change) of the unit vector rU(t).

I just found it neccessary to enhance the theory a bit.

------------------------------------------------------------------

Let's apply the theory of equations 8 and 10, in a few examples.

=> Examples calculating the "instantaneous (tangent) velocity":

Example 11: Parameterized r(t) function:

Suppose we have some vector function t -> r(t), meaning that this is a function R => Rn.

In math note 17, Chapter 1, I listed a few examples of such functions.
For example:

r(t) =
┌ 2cos(t) ┐
└ sin(t)  ┘

Some folks call it "parameterized", since for each component of the vector (like the x part, y part etc..)
we actually have some function of "t".

dr(t)/ dt would then yield:

v(t) = dr(t)
----
dt
= ┌ -2sin(t) ┐
└ cos(t)   ┘

For each "t" for which the components are valid, we get a "tangent" vector to the curve determined by r(t).

So, indeed, we have r(t) described in such a parameterized vector form, we can calculate the instantaneous velocity for each "t".
If the parametrized vector form of r(t), it's not too hard to determine the instantaneous velocity.

However, it's always important to analyze the domain, that is, for which "t" the vectorcomponents of r(t), are valid.

Example 12: Working directly with cartesian coordinates and components:

Often, it's like shown in example 10. In many cases, you have only a subset of v components, and must determine
the total (sum) vector vT. Maybe you like to review example 10 again?
Many other variants are possible too ofcourse.

Here we consider a complete example in R3:

Suppose:

r(t) = x(t) ex + y(t) ey + z(t) ez

where ex, ey, and ez, are the unit vectors along the
x-, y-, and z-axis respectively.

Then finding the velocity goes using equation 8:

dr(t)
----
dt
=   dx(t)
---- ex
dt
+   dy(t)
---- ey
dt
+   dz(t)
---- ez
dt

And indeed, each of the differentials above, is a component of the velocity vector. Thus:

v(t) = vx(t) ex + vy(t) ey + vz(t) ez

The "speed" at any "t", can be found taking the magnitude |v(t)| of the velocity vector.

Example 13: Polar" coordinates in R2 and R3:

I will discuss this in the chapter on harmonic oscillators.

=> General approach for calculation the the "average velocity":

Suppose that you do not know the (parametrized) function of r(t).
But, suppose you do know the factual positions of an object (e.g. in R3) to be r(t1) at t1,
and as the second point, r(t2) at t2.

Then:

Δt = t2 - t1   (as the time interval)

and

Δr = |r(t2) - r(t1)|   (as the length, or distance between the 2 points).

Then, using equation 7, filling in the numbers, would give you the average velocity.

Example 14: Calculation of the "average" velocity:

Suppose at t=0, we see a bee flying at r1=(1, -2, 6). Two seconds later, at t=2,
the bee is found to be at r2=(3, -1, 4).

Question: what was it's average velocity over those 2 seconds?

We already know Δ t, which is 2 seconds.

What is Δ r? It's |r2 - r1|, or the length of the difference of those 2 vectors.

By the way, the magnitude |r2 - r1| is the same as |r1 - r2|

Thus:

Δ r = √ ( (1-1)2 + (-2 -1)2 + (6 - 4)2 ) = √ 13 = (about) 3.6

And we find as the average velocity Δ r / Δ t = 3.6 / 2 = (about) 1.8 m/s


Recap Chapter 1:


1. Theory from sections 1.1 - 1.3:

Solutions for problems, can be solved as shown in the Examples 1 to 8,
using formula's as shown below.

1.1 Uniform motion (one direction), no force, constant speed (v):

v is constant.

S(t) = vt      

S'(t) =   d S(t)
-----
dt
=   d (vt)
------
dt
=   v      

1.2. Constant force, constant (uniform) acceleration (one direction):

v(t) increases with time, due to the constant acceleration.

v(t) = at      

v(t) = v0 + at     (if already an initial constant speed v0 was present )

S(t) = ½ at2      

We also have:

S'(t) =   d S(t)
-----
dt
=   d
--  
dt
½ at2   =   at =   v

2. Theory from sections 1.4 - 1.5 (vectorial):

Solutions for problems, can be solved as shown in the Examples 9 to 14.

Velocity and acceleration must be represented by vectors if we are not working 1-dimensional.

Speed is actually the magnitude of the vector velocity.

Fundamental relations:

dr(t)
----
dt
=   v(t)

dv(t)
----
dt
=   d2r(t)
----
dt2
=   a(t)      



Chapter 2. Forces in Mechanics.

In physics, a number of forces are believed to be really fundamental, like the "strong (nuclear) force",
the "electromagnetic force", the "weak force", and "gravity".

Classical mechanics however, is not about research, or theoretical treatment, on those fundamental forces.

It's much more involved in the actions due to forces, and thus what effects there are on macroscopic objects.

We will talk on the following subjects:

-Discussion on the unit of Force, the "Newton".
-How to get the "result" Force, if multiple forces are working on an object. This will be surprisingly easy, I am sure.
-The force of gravity on an Object on plane with a certain slope α.
-Some further Discussion on F = ma and "inertia".

2.1. The unit of Force, the "Newton" (N).

One "Newton" (N) is defined as the Force on object of 1kg, which then undergoes an acceleration of 1 m/s2.

Although units are (in some way) "just" chosen, the motives for a certain choice are ofcourse "reasonable" and are based on certain
good considerations.

This is true for the Newton as well. We speak of an object of 1 kg (already an SI unit), and an acceleration
of 1 m/s2 (and the number "1" also fits nicely).

One N of force, is, on a human scale, not a gigantic force. That's why in engineering and such siences, you often
see problems formulated in such way, that people talk in numbers of "kN" (kilo Newton, or 1000 N), or for example 105 N.

-If the Force is constant, and keeps working one direction, then the formula's for a motion of a constant acceleration (section 1.3)
can be used.

-Newton's general equation "F = ma", implies that if the Force varies in time, then so does the acceleration
of the object where the force acts upon.

Note that "F = ma" is a linear equation with respect to the variables "F" and "a", if "m" is constant.

So, if we have a certain object with a certain mass, then:
If "F" increases/decreases with a factor "c", then "a" increases/decreases with the same factor "c".

"Force" is a vector, just as "velocity" and "acceleration" are.

However, if the Force works in one direction, and the motion of the object is in that direction too (say the x-axis),
then the lenght (or magnitude) of those vectors can be determined directly. We do not need to add or determine vector components etc..
The projection of those vectors on the x-axis, are directly the magnitude of the vectors themselves.
Then we can easily speak of "speed" and "force" in magnitudes (scalars) only. The "direction" is not so much important here.

Example 15:

At t=0, an object of 10kg starts to move in the +x direction, with a constant acceleration of 3 m/s2.
Suppose that the force works only in the +x direction too.

Question: what is the force?

Answer:

F = ma = 10 kg * 3 m/s2 = 30 N.

Example 16:

Suppose at t=0, a force of 6N starts to act on an object of 3kg. Initially, the object was in rest.
The direction of the force is exactly in line with the motion of the object.

Question:

what is the speed of the object at t=5?

Answer:

F = ma, thus a = F/m.

So, in this case, the constant acceleration is a = 6/3 =2 m/s2.

The speed after 5 seconds is v = at = 2 * 5 = 10 m/s.

Example 17:

Suppose at t=0, a force of 6N starts to act on an object of 3kg.
Initially, the object had a constant speed of 6 m/s in the +x direction.
The direction of the force is exactly in line with the motion of the object.

Question:

what is the speed of the object at t=5?

Answer:

F = ma, thus a = F/m.

So, in this case, the constant acceleration is a = 6/3 =2 m/s2.

The speed after 5 seconds is v = v0 + at = 6 + 2 * 5 = 16 m/s.

Indeed, the initial speed of v0 was 6 m/s.

2.2. Forces as vectors, and determining the resultant Force.

Whichever dimension you are working in (like R2, or R3, or higher), vectors have a direction
and magnitude.

If multiple forces act upon an object, simply treat them as vectors, and follow the rules for vector addition.
Simply put: using "linear algebra" works in all cases.
However, in the first subsection below, we will try "commen sense" first, and then quickly go to "linear algebra".

Note: the very basics of linear algebra, like vector addition, can be found in "math note 12".

1. The Simplest case: the vectors are all along one dimension (for example the x-axis):

Figure 4.




In figure 4, you see two examples of force addition, when the forces are aligned in one dimension only.
The best policy, as in any vector addition, is to determine the resultant vector.
Once having that, we can determine the length of that vector (or the magnitude of that vector).

- In the upper case in figure 4, the two forces are working in the same direction.

We can see that:

|F1| = 10N and
|F2| = 20N.

then it's obvious to see that the magnitude of the result force |FR| = 30N, since both are working
in the same directon.

- In the lower case in figure 4, the two forces are working in the opposite directions.

In this case, we cannot just add the individual magnitudes. That would result in |F1| + |F2| = 18N.

That's wrong! Magnitudes are always positive numbers. And we can clearly see that the magnitude of the resulting force just has to be 6N.

This time we really add them as vectors first:

F1 is 6N to the -x direction
F2 is 12N to the +x direction

Thus FR = 6N to the +x direction

So, thus we have |FR| = 6N

This sort of problems can be handled by "eye" and "common sense".

-However, The most consequent way to handle problems like this, is as follows:

Let's use linear algebra. The lower case shown in figure 4, suggests we have a Force in the -x direction and one in the +x direction.
It can be seen that we are working in R2.

So we can write:

F1 = ┌ -6 ┐
└ 0  ┘

F2 = ┌ 12 ┐
└ 0  ┘

Thus the resultant vector FR is:

FR = ┌ -6 ┐
└ 0  ┘
  +   ┌ 12 ┐
└ 0  ┘
  =   ┌ -6 + 12 ┐
└ 0  + 0  ┘
  =   ┌ 6 ┐
└ 0 ┘

This is correct. We have a result vector (of magnitude 6) along the +x direction.

The length or magnitude is |FR| = √ (62 + 02) = √36 = 6.

2. The General case: vectors may have any directions:

Here, you may think of any combination of vectors acting on an object, pointing in any direction.
So, this could be in R2, R3, or Rn.

Figure 5 illustrates this for an R2 plane.

Figure 5.




This time, we go for using linear algebra directly. If you glance through highschool books, you may see
several other methods. However, using linear algebra is the most universal way to handle vector "stuff".

Suppose on an object we have two vectors acting on it. We want to find the resultant force and it's magnitude.

The example below is in R2. But it works the same way in R3, or Rn.
As an example, say we may represent those two vectors as:

F1 = ┌ -2 ┐
└ 5  ┘

F2 = ┌ 1  ┐
└ -3 ┘

The sum of F1 and F2 is:

FR = ┌ -2 ┐
└ 5  ┘
  +   ┌ 1  ┐
└ -3 ┘
  =   ┌ -2 + 1 ┐
└ 5 + -3 ┘
  =   ┌ -1 ┐
└ 2  ┘

So, there is our answer. We can immediately "plot" FR in R2.

However, the answer is ofcourse the resultant force in Newton, so we must find the "magnitude". If F1 and F2
were expressed in Newton units too, then |FR| is:

|FR| = √ ((-1)2 + 22) = √5.

2.3. Object on a tilted plane

This example can be found in any textbook on Mechanics.

Suppose we have an object resting on a plane, while that plane is not horizontal,
but instead, has a certain angle α with the horizontal plane.

Then, what are the vector components of the Force of gravity on that object?

First of all, we know that on any object with mass "m", on Earth's surface, it is true that the force of gravity
on that object is:

Fg = mg

where "g" is the "acceleration due to gravity".

So, if you climb a high tower, and you drop an object, our famous equation "F = ma" will then be "F = mg",
since the force of gravity is the only force acting on the object, with it's constant acceleration "g".

Even if an object is resting on a table, then Fg is present, although we don't observe
any acceleration. Ofcourse, the table is (usually) strong enough to resist the force, so that the object
will not simply pass through that table.

Now, the following example is actually no more than an example of the fact that a vector maybe the sum of other vectors.

Indeed, we have often seen in the mathematical notes that in many cases we can write C = A + B
where the vector C is the vectorsum of A and B.

Or, if we have some vector A in R2, which we may write as (a, b), then we can represent it as a linear combination of the
basis vectors, as A = a (1,0) + b (0,1) or, using a different notation, as A = a êx + b êy.

It actually always amounts to the "superposition" principle of vectors.

Now, take a look at figure 6.

Figure 6. Object on plane with a slope α, and the components of Fg



If the table is horizontal, we just have Fg = mg

If the table get's a slope α, then we may always (whatever angle α), create the vector sum:

Fg = Fp + Fn


Where Fp is the vector component parallel to the table, and Fn is
the component perpendicular to the table.

Note that if α get larger, then the Fp component increases, while Fn decreases.

Note also, that if α get's large enough, then Fp get's larger than any possible "friction"
from the table, and the object starts to move.

It's also possible, that the scenario is considered to be "friction-less", but then, in reality the object should
get into motion with any angle α.

Now, what we really are interested in, is the "length" or "magnitude" of the vector Fp,
since that component is along the only possible path where the object might start moving.

The magnitude (or length) of Fp is |Fp|.

Once we have identified a right-angled triangle defined by the segments |Fp|, |Fn|, and |Fg|,
the whole problem collapses into a simple goniometric problem.

From "math note 4" we know that:

sin(α) = |Fp|
----
|Fg|

Thus:

|Fp| = |Fg| sin(α)

|Fn| = |Fg| cos(α)

This way, it has become quite easy to calculate the magnitudes of the components, if |Fg| is known in "Newton" units.

Example 18:

Suppose we have a scenario as sketched in figure 6.

A mass "m" of 4kg rests on a tilted table, with an angle of α = 60 degrees.
Suppose further that "g" is 10 m/s2.

Question: calculate |Fp|

Answer:

|Fg| = mg = 4 * 10 = 40 Newton.

Thus:

|Fp| = 40 sin(60) = (about) 34.6 Newton.

Next, let's go to a short discussion of the equation F = ma.

2.4. Discussion of F = ma.

What exactly "mass" is, is not simple to answer.
Many philosophers and physicists broke their heads on concepts as "mass" and something called "inertia".

It's certainly not "weight". An object weights a certain amount on Earth, but it weights much less if we
would place that same object on Mars.

"Weight" is no more than a measure of how hard the local "force of gravity" is pulling on it.
So, that force of gravity, on a brick for example, is different on the Moon, if you compare it with the force of gravity
on that same brick on Earth.

Even an stronger argument: the force of gravity on Earth varies slightly with the position on Earth.
So, very small variations in weight can be observed too. However, the mass of an object is constant.

From theoretic physics, and particle physics, we know that the scalar Higgs field (which is everywhere), defines an interaction
with elementary particles, that relate "somehow" to the mass of such elementary particle.

However, such Higgs field has nothing to do with the "inertia" we see with macroscopic objects.

This concept of "inertia", is the resistance to change in momentum, that is, "it" does not want to change
it's velocity, unless a force is applied.
We know this concept very well in our macroscopic world. If you have 1kg of some stuff, you need a certain amount
of Force to set it in motion. It get's much harder to set it in motion, of you would have 10 times more of that "stuff".

In the context of classical mechanics, if we then view Newton's second law F = ma, then that "m", the mass,
must be understood as the "inerta" of that object.
So in classical mechanics, "mass" and "inertia" are likely to be the same one concept.
This is also how highschool physics books treat "mass".

On a macroscopic scale, experimentally, the equation F = ma is easy to verify, and it holds all the time.

But we have not really "explained" what is behind "inertia".

Optional reading.

If you have a certain object, then physically, a lot of it's intrinsic "energy" is due to several components,
but a large contribution is from all the "quark-gluon" soup-stuff, inside all the protons and neutrons.
This energy relates to "rest mass" according to Einstein's equation E = mc2.
Yes indeed ! Energy equals mass.

The discussion get's an additional perspective, if we consider "relativistic" machanics.
For a long time now, since Einstein's special relativity (1905), we know that the mass of an object increases
when it's velocity increases. When that velocity get's closer and closer to the speed of light, the mass get's
quickly higher and higher too..., asymptotically to infinity. Experimentally, this has been proven too in,
for example, particle physics using particle accelerators.

You see that? The energy increases, and so does it's mass. In whatever frame of reference you are, mass equals energy.

Einstein already gave you an answer as to what mass is: mass equals energy.

Before we can discuss other important concepts in classical mechanics, namely
Kinetic Energy, Potential Energy, and Work, we need to deal with the concept of "Momentum" first.
So "Momentum" is the subject of the next (relatively short) chapter.

Chapter 3. The concept of Momentum .

If you are really new to physics, then this Chapter may seem intimidating too, at first sight.
But it isn't really. Just try it.


There is a physical property of a system called "momentum", and it's absolutely indispensable for physicists.

Here is just an example: when performing "high-energy particle physics", collisions occur between elementary particles.
One main consideration when studying the results, is that physicists know that "momentum" is conserved between
the particles involved, which have always lead to insights and conclusions.
This is true for the past, the present, and undoubtly for the future too !

Momentum is not only important in Classical Mechanics. In Quantum Mechanics, Relativistic Mechanics etc..,
momentum plays an important role too.

So, what is it?

Suppose over a time interval Δt, we consider a constant force acting on an object with mass "m".

Suppose that before t BEFORE, there was no force whatsoever acting, and the constant velocity of the object was v BEFORE.

During Δt, the force acts on the object, and the object accelerates.

Suppose that after t AFTER, there is again no force whatsoever acting, and the constant velocity of the object now is v AFTER.

The timespan, over which the force acted is Δt = t AFTER - t BEFORE.
The increase in velocity, during Δt, is Δv = v AFTER - v BEFORE.

Now, the constant acceleration over Δt is:

a = (v AFTER - v BEFORE)
-------------
    Δ t

Thus we have also:

m a  =  m (v AFTER - v BEFORE)
-------------
    Δ t

Which is the same as:

F  =  m (v AFTER - v BEFORE)
-------------
    Δ t

Which is the same as:

F Δ t  =  m (v AFTER - v BEFORE)  =  mv AFTER - mv BEFORE       (equation 11)

I did not introduced any new formula here. Actually, it's only stuff we already knew before this section.
But the perspective might have changed a bit.

If we take a closer look at the "meaning" of

F Δt = mv AFTER - mv BEFORE

then we might interpret this as folows. The force F acts only for a duration of Δt,
which has as a direct consequence that the property "mv" changed by mv AFTER - mv BEFORE.

The value of "mv" can be viewed as "the quantity of motion" which an object has. You may also see it as a measure
for "mass in motion".

If a force acts upon an object, for a certain duration Δt, it's amount of motion changes by mv AFTER - mv BEFORE
The quantity "mv" is called momentum "p", so:

p = mv       (equation 12)

Since "velocity" is a vector, we conclude that "momentum" is a vector too.

The magnitude of momentum is expressed in "kg m/s", due to the units in "mv".

If we have a system, of say 2 or more objects, and there is further no additional interaction (e.g. no external force),
except for those objects, then if they interact (like colliding), then total momentum is conserved.

So, in the case of 2 objects of mass "m1" and mass "m2":

p1 (before) + p2 (before) = p1 (after) + p2 (after) = Constant       (equation 13)

which is the same as:

m1v1 (before) + m2 v2 (before) = m1v1 (after) + m2v2 (after) = Constant       (equation 14)

Or. more generally, for "n" objects (like particles):

Σ pi = Σ mivi = constant       (equation 15)


The "change" in momentum, as we can see from upper equations, is due to:

F Δt = mv AFTER - mv BEFORE = m Δ v       (equation 16)

F Δt is sometimes called the "impuls" or the "action" responsible for the "change" in momentum.

Indeed:
- When a force is acting on an object, the momentum changes.
- When no force is acting on an object, the momentum does not change.

A nice illustration:

The animation below shows how the momentum (mv) from the first ball, is fully "transferred" to the last ball,
and the other way around.

Figure 7. Newton's "cradle"


(figure: from wikimedia commons).

Remember, p = mv, so p is a vector.
Since "m" is just a scalar (a number), the vectorial component "sits" in the velocity v.

In solving problems, you should always realize, that the total momentum of a system is the vector sum
of all the individual momenta of the particles, which make up the system.
Only when the particles before and after any collision, stays along 1 dimension, you can work with the magnitudes directly.

Note on "elastic" and "in-elastic" collisions:

In principle I should also explain "elastic" and "in-elastic" collisions.
Although the terms pretty much speak for themselves, there are some subtleties.
I like to wait until we have seen the concept of Kinetic Energy, which make both terms easy to explain.

Let's try some examples.

Example 19:

A ball of 10kg rolls on a frictionless table. It's velocity is 10 m/s in the +x direction.

Question: what is it's momentum?

Answer:

The magnitude of the momentum |p| = 10 kg * 10 m/s = 100 kg m/s.

The momentum as a vector is p is 100 kg m/s in the +x direction.

Or, written differently:

p = 100 ex (if ex is the unit vector along the x-axis). The magnitude of p = |p| = 100 kg m/s.

Example 20:

This example stays in "one" dimension too.

Particle P2: is in rest on the x-axis. It's mass is 2kg.
Particle P1: moves from the left along the +x direction. Speed is 8 m/s. It's mass is 5kg.

At a certain moment, particle P1 collides against particle P2.
Directly after the collision, both are moving in the +x direction. Particle 2 now has a speed of 6 m/s.

Question: What is the speed of Particle P1?

Answer:

p1 (before) + p2 (before) = p1 (after) + p2 (after)

or:

m1v1 (before) + m2 v2 (before) = m1v1 (after) + m2v2 (after)

Thus:

8 * 5 + 0 = 5 * v1 + 2 * 6     =>

40 = 5 v1 + 12     =>

28 = 5 v1     =>

v1 = 28/5 = (about) 5.6 m/s

Example 21:

Suppose we have a collision of 2 particles.

This time we will work in multiple dimensions, so we need to take into account
that we are working with vectors.

Let's work in R2. There is no fundamental difference if you would work in R3 or Rn.

Figure 8. Adding momenta in R2



Before collision:

particle 1: mass m1, velocity v1.
particle 2: mass m2, velocity v2.

Total momentum: p1 + p2 = m1 v1 + m2 v2 = C

The "black" vector (left) in figure 8, represents the "total momentum".

After collision:

particle 1: mass m1, velocity u1.
particle 2: mass m2, velocity u2.

Total momentum: p '1 + p '2 = m1 u1 + m2 u2 = C

The "black" vector in figure 8 (right), represents the "total momentum".

Note that the "black" vector remains the same. That's conservation of momentum.

You see that? Again, linear algebra can solve a problem which at first sight looks pretty intimidating.

We know that before the collision, that the total momentum is p1 + p2.
But, after the collision, the total momentum is the same !

Thus:

m1 v1 + m2 v2 = m1 u1 + m2 u2

This is the general principle. There are several ways to find a certain unknown variable.
As we have seen before, just use vector algebra.

As an important law of physics, we have seen that the total momentum is conserved.


Chapter 4. Kinetic Energy, Potential Energy, and Work.

4.1. Short discussion on the concept of "WorK".


4.1.1 When the Force and the Displacement are "aligned" (one direction):

If a force acts on an object, and this object is in motion due to that force, then that force performs "work".

When you drag a large heavy box along some road, and you have covered 10m, you know that you have invested
quite some energy. If you cover 10m more, then the amount of invested energy has doubled.
It seems as if "Work = Force * Distance". Indeed, in it's most basic form:

W = F * s       (equation 17)

Since the unit of force is the "Newton" (N), and the unit of distance is the "meter" (m), work is expressed in "Newton meter" (Nm).

As an equivalent unit, the "Joule" (J) was defined, which is the work done by a force of 1 N over a distance of 1 m.

Equation 17, must be a "scalar" variant of "something" more general, since as we know that "force" is a vector,
as well as is the position r(t) (as a vector).

However, if the force and direction are aligned along one dimension (one direction), we really can use equation 17
by simply using the "magnitudes" of the terms. So, actually we should re-write equation 17 as:

W = |F| * s       (equation 17, revised)

Example 22:

Suppose a constant force of 15N acts upon an object. This force operates for some time, and in that time,
the object is displaced over 10m.
The direction of the force, and the displacement of the object, are along the x-axis.

Question: what is the work done by that force?

Answer:

Here we may apply "W=Fs", thus:

W = 15 N * 10m = 150 Nm = 150 J.

4.1.2 When there exists an angle α between the Force and Displacement:

The amount of "Work" is always a number (a scalar). In this subsection too, I will only use
the magnitude of a force vector (a scalar), and the displacement (as a scalar).
In fact, as you will see, it is sufficient to take the magnitude of the parallel vector-component of F.

How that will work out, will be shown in this subsection.

First, with any force under consideration, the only component of the Force that will do work
is the component parallel to the displacement.

For example, Consider an object in rest on a table. We know that the force of gravity is present at all times, but yet,
it does not perform work since the object stays in rest. But here, it is obvious that the force of gravity
is perpendicular to the possible path of motion of that object.

However, we can generalize that: if you have a Force pointing in any direction, then only use the component
of the force parallel to the displacement.
Here is an example:

Figure 9. Vector components of F, with a perpendicular and parallel component.



Take a look at figure 9 above. The force F is a vector, and has an angle α with x-axis.
Once we have identified a right-angled triangle defined by the lengths of |F|, |FP| and |FT|.

From "math note 4", we know that:

cos(α) = |FP|
----
|F|

Thus:

W = |FP| s = |F| cos(α) s       (equation 18)

Let's try an example:

Example 23:

Suppose a constant force F of magnitude 15N, acts upon an object. This force operates for some time, and in that time,
the object is displaced over 10m.
However, the angle between F and the x-axis is 30 degrees.

Question: what is the work done by that force?

Answer:

Only the parallel vector component of F, performs work.
We can immediately use equation 18:

W = |FP| s = |F| cos(α) s = 15N * cos(30) * 10m = 150 * cos(30) Nm = (about) 129.90 Nm = (about) 129.90 J.

4.1.3 General approach for a possible variable Force with "s", along some curved path "s":

This subsection is optional reading.

This subsection will also use the "inner product" of two vectors.
If you need to refresh your memory on the "inner product", you might want to take a look again at "Math Note 12, section 2.1".

This time, we spend a few words on the general case, of how to calculate the work a force does, over some distance "s".

However, maybe that path "s", is some "strange" curve in space, insteas of a nice straight line.
Also, the force may depend on the position in space, so it's direction and magnitude might vary per location.

Instead of some displacement "s" as a scalar (like the displacemnt along the x-axis), we should this time use
a position vector r, just like we do when we draw an arrow to some point in, for example, R3.

Note how much equation 18, already starts to looks like the inner product of F and r,
if we instead would use r instead of just "s".

In "Math Note 12, section 2.1", it was made "plausible" that if we have two vectors A and B,
then the inner product may be written as:

A · B = |A| |B| cos(α)

This scalar might be interpreted as the length of the "projection" of A on B.

This time, we have some curve defined by r, and a variable Force F(r), depending on the position r.

What we can do is to divide r into very small segments, and per segment, determine the parallel component of F(r)
at the center of that small segment. Then, determine the product of F(r) at that location and the small segment,
and then, add all products up over the trajectory where you need to calculate the "work".

This "smells" a lot like having an "Integral", if we really would make the length segments infinitesemal in size.
I'am not sure how much you are aquinted with Integrals, but usually an Integral is no more than a very large summation
of very small segments.

If the segments Δr really get's very small, then Δr will go to dr.

Figure 10. Very small segments "Δr" and the parallel component of F at Δr.



I hope you might find the following argument (reasonably) plausible.

Suppose we want an approximation of the work that a force has done, over a curved trajectory,
defined by the "starting point" P1 and "end point" P2 on that trajectory.

This trajectory will be diveded into a large number of small Δr's, so that at such Δri, the
trajectory is nearly straight. Then, at such Δri, we can calculate the product of the distance Δri,
with the magnitude of the parallel component of the force at Δri.

If you now would add up all those small products, we get:

Approximation of the Work over [P1,P2] = ∑ Fparallel, i · Δri

By letting Δri going to "dr" and thus the summation then will get infinite,
we actually get an Integral:

Work over [P1,P2] = ∫P1P2 F(r) · dr       (equation 19)

Believe me: it's actually not even so hard to perform real calculations "a la" equation 19,
if we have a "reasonable" force vectorfield, and a "reasonable" trajectory.

However, this subsection was optional reading, and I like to keep this note relatively simple, and not clutter it
too much with too much detail. It's much more important that you get the general idea of the concepts discussed in this note.

4.2. Short remark on conservative force fields.

In fact, in a socalled "Conservative" force field, the work done by some force F(r) by moving an object
from point P1 to P2, is independent from the path taken.

Even if F(r) has all sorts of "angles" to the path dr, along the trajectory, we know that the contribution to the work,
is only due to the "parallel component" of F(r).

This means that the total sum of the "parallel component" times dr, will always add up to the same value.

The forces we have seen in this note, like the gravitational force on Earth, are all "Conservative".

It also implies, that if you must calculate the work done moving an object from P1 to P2, along some difficult trajectory,
and there also exists another "easier" path, you may choose that easier path in your calculation,
as long as the begin- and end points P1 and P2 are fixed.

We are not going to make "hard" calculations in this note. However, it would be great if you get the general idea
behind the concept of a conservative force field.

Figure 11. A conservative force field: the work done moving from P1 to P2, is independent of the path taken.



Now, let's go to a short discussion on Kinetic- and Potential Energy.

4.3. Short discussion on Kinetic- and Potential Energy.

4.3.1 What is "Potential Energy" Ep?

Please read the following two examples carefully:

Example 24: Potential Energy Ep in Earth's gravity.

Suppose you have an object of 2kg. Next, with that object, you climb a tower of a height of 30m (against gravity).
Then the work done by you is:

W = |Fg| * s = mg * h = 2*10*30 = 600 J.

In a way, "all that energy" is not "lost"!

Physicists say that the Potential Energy Ep of that object has increased by "mgh".
It's exactly the same as the work you invested by bringing it up, to a height of 30m.

If you would now drop that object, the object ofcourse falls down, with an increasing velocity
due to the constant acceleration "g".

While going down, at each point of the path, the Potential Energy drops while something called the "Kinetic Energy" grows.
This latter energy is dependent on the mass and velocity of the object.
The faster it goes, the more Kinetic Energy we can associate with that object.

In this example the "Potential Energy" Ep increased with "mgh". The rate and the amount by which the "Potential Energy" increases
is dependent on the type of force field. In this case, it's Earth's gravity, so the increase is always "mg Δh".

So, in Earth's gravity:

Ep = mg Δh

Example 25: Potential Energy of a charged particle in a static Electric Field.

This example is very similar to the case of Potential Energy due to Earth's gravity.

However, this time we take a look at an elementary particle with a positive charge "+q",
in an Electric field.

Maybe you are not so familiar with Electrostatics and charges. In short:

First of all, electric charges of the same "sign" repel each other. So, "+q1" and "+q2" repel each other.
Secondly, electric charges of opposite "sign", attract each other. So, "+q1" and "-q2" attract each other.

It's possible to have a condensator like "device", where two plates are opposite each other, and where each
plate has an opposite charge. So, say the upper is charged positively, while the lower one is negatively charged.
Then there also exists a certain voltage difference "V", between those plates.

Figure 12. Two opposite charged plates. Then there exists a "voltage difference" between those plates.



Take a look at figure 12. In the left figure, the positive test charge is near the lower plate.
This test charge, absolutely does not want to go to the upper posively charged plate.
If we move it, it will cost us energy, expressed by "work". We need to apply a force to move it
towards the posively charged plate.
It's possible that you are not familiar yet with Electrostatics, but there exists also a potential difference,
or "voltage difference" between the lower- and upperplate. Suppose that Voltage difference is "V" (or better: ΔV).

Then the work we must invest is:

W = q * V

This invested energy is not lost. The test charge aquired a Potential Energy of "qV".
Suppose we had "some way" to hold the particle near the upper plate. At the moment we let it go,
it will immediately accelerate towards the negatively charged plate.

Indeed, it is true that between the two plates a constant "Electric Force" exists, that will act
on any charged particle between the plates. In this case, when a positive test charge is involved,
the direction of the force is "downwards".

Thus, as the test charge accelerates toward the lower plate, it aquires "Kinetic Energy", while in an equal
amount the Potential Energy lowers.

Although here different physical entities and forces are involved, it is quite similar to example 24.
So, in an electrostatic field, the Potential Energy that a positive testcharge aquires (when moving to the + plate) is:

Ep = qV or, Ep = q ΔV       (equation 20)

The latter expression is valid, if we would call the voltage difference between the plates "ΔV" instead of "V".

4.3.2 What is "Kinetic Energy" Ek?

Examples 24 and 25, also illustrated the concept of "Kinetic Energy" Ek.

In general, when you invest "work" to move an object or particle against a force field,
then the object or particle aquires "Potential Energy".

If you then "let it go", the object or particle immediately accelerates in opposite direction (due to the force).
When it is back at it's original position, all "Potential Energy" is converted into "Kinectic Energy".

Yes, a fundamental "law" was expressed by that latter sentence:

Conservation of Energy applied to mechanical systems using Ep and Ek:

When "work" W is invested to increase Ep, then
when the constraint is lifted, Ep is fully converted to Ek

When you decide that you have invested enough "work", then Ep=W and Ek=0

when the constraint is lifted, the object aquires an increasing Kinetic Energy Ek, up to the point
where Ep=0 and Ek=W

At any point "in between" (for example during the fall of an object), Ek and Ep are "balanced",
meaning "Ek + Ep = W".

Although the amount of the Potential Energy depends on the type of Physical system (e.g like Earth's gravity, where Ep = mgh),
there indeed exists a uniform equation for the "Kinetic Energy" Ek.

Using "plain logic" dictates that Ek must depend on the mass "m". The higher the mass is, the more "Kinetic Energy" is involved.
Secondly, Ek must also depend on the speed "|v|", since the higher the speed is, again the more "Kinetic Energy" is involved.

There exists a simple derivation to find the formula of Ek. In principle, for the general case, I should use an "integral" again.
However, if we take example 24 is an "archetype", then we may say:

Total Work done = W = F * s (where s is for example the height).

Then we let the object go, and it accelerates downwards.

Then:

W = Fs = ma * s = ma * ½ at2.

At any moment during the fall, "v=at", so a=v/t. Substituting that into the equation above, gives us:

W = F * s = mas = ma * ½ a t2 = ½ m a2 t2 = ½ m (v2 / t2) t2 = ½ m v2

Although not "nicely" derived for the general case, I think that it is made plausible that in mechanical systems, it holds that:

Ek = ½ m v2       (equation 21)

Example 26:

Suppose an object of 2kg, is lifted to a height of 30m.
Then, we drop the object.

Question: what is the speed of the object when the Potential Energy has dropped to half of it's max value?

Answer:

When at height of 30m, then:

Ep = mgh = 2 * 10 * 30 = 600 J.

When Ep is lowered to 300 J, then Ek has increased to 300 J.

Thus, according to Ek = ½ m v2

300 = ½ m v2       =>

300 = ½ 2 v2       =>

300 = v2       =>

v = √ 300 = (about) 17.3 m/s

Another Question: what is Ek when the object hits the floor?

Since the object aquired an Ep of 600 J at 30m, this then will be the
Kinetic Energy when the object hits the floor again.

Best practice:

It's probably best if you deal with this sort of problems in the following way.
The total energy is constant, so we may write an energy "balance":

Ebegin = Eend

So, in the example where a mass is at the top of a tower, and we push it over the edge,
we can write an energy "balance". Suppose the mass is 0.5 kg and the height of the tower is 30 m.

Ebegin = Epot + Ekin = mgh + 0 = 0.5 * 9.81 * 30 = 147.15 Joule.

Eend = Epot + Ekin = 0 + ½mv2= 147.15 Joule.

From this, you can easily calculate the endspeed when the object hits the ground.

Now, suppose we may not neglect the friction with the air, when the object is falling down.
In that case, the energy equation will be:

Ebegin = Epot + Ekin = mgh + 0.

Eend = Epot + Ekin + Q = 0 + ½mv2 + Q.

Here, "Q" is the heat which arose due to the friction, and ofcourse this time the endspeed
will be somewhat lower compared to the example before.

Sometimes, in some problem (say an exam question), the force of friction is known, and you may
further assume that it is constant during the fall of the object.
It may help you to calculate "Q", using W = Ffric * s.

Often such exam questions provide some extra details by which you are able to calculate the endspeed
of the object in such situations.

Anyway, my main objective here was to make you aware of the "energy balance".

Chapter 5. Harmonic oscillators in classical Mechanics.

In "math note 10", I tried to discuss several harmonic oscillators, from the perspective of "differential equations".

This time, I like to use a typical Highschool approach. This mainly means that the material below, often treat
many physical quantities as scalars. However, this approach works, and allows you to solve all sorts of problems.

There are countless examples of harmonic ocillators, like the "pendulum", "particle in a box", etc...
However, the "uniform circular motion" is (sort of ) the standard example. Let's discuss that one.

In this case, an object (or particle), performs a circular motion (with radius "R") around an origin,
with a constand orbital speed "vT".
Using the subscript "T", I like to make clear that this "vT", is always "tangent" to the path of motion.

Figure 13. Uniform circular motion. Note also the projections on the x- and y-axes.


About the figure: from "wikimedia commons" (public domain).

Orbital speed vT:

Suppose the radius of the circle is "R". Then the circumference of the circle is 2πR.
Also, suppose that the time to complete one full circle is T seconds.
Then, since vT is constant, and based on equation 1, we find:

vT   =   S
--
t
  =   2πR
--
T
      (equation 22)

Angular speed ω:

In T seconds (also called "the Period"), our particle completes a full circle of 2π (rad).
So, it is possible to define an "angular speed", meaning "how much" angle has the radius R completed in time "t"?
This angle then, is the angle between R and the x-axis.

One thing is for sure: in T it does 2π (rad).
Different cicular motions, will have a different T and a different vT ofcourse.
So, T indeed characterizes the "angular speed" (ω) of a certain circular motion.

Since ω also is the answer to the question "how much angle has the radius R completed in time t"?
we may define the following equation for ω:

ω   =  
--
T
  =   Δ θ
--
Δ t
      (equation 23)

Ofcourse, in the case of a uniform circular motion, ω is constant, as can be seen from equation 23.
If T (the constant Period) is known, then ω is known too, by using equation 23.

However, θ changes by time. If we know the constant ω, then in a certain timeslot "t", θ has increased by "ω t".

Thus we also have:

θ = ω t       (equation 23' - note the accent in 23')

Often in figures like figure 14, the angle between R and the x-axis is denoted by α. However, in most textbooks
and articles about harmonic oscillators, that same angle is often called θ, and I think I better to conform to conventions.
(Ofcourse, it does not matter at all which letter one uses for denoting an angle).

Relation of vT and ω:

We can also express vT in terms of ω and R. Simply take a look again at equations 22 and 23.
It then immediately follows that
vT   =   R
--
T
  =   ωR       (equation 24)

Projections of R on the x-axis and y-axis:

Actually, this was all already in "math note 4". If your memory on sin() and cos() are a bit hazy,
then please consult "math note 4" again.

Figure 14. The projections on the x- and y-axes expressed in sin(θ) and cos(θ).



In fact, by looking at figure 14, and memorizing the pure definitions of sin() and cos(), we know that at any specific time "t":

projection of R on the x-axis:     a = R cos(θ)
projection of R on the y-axis:     b = R sin(θ)     (equations 25)

But the object is moving in a circular motion. So, θ changes all the time, or more exactly, by time,
at the constant rate of ω

This means that the "time-dependent" equations for the projections on the x- and y- axes, are:

time dependent projection of R on the x-axis: x(t):     x(t) = R cos(ωt)
time dependent projection of R on the y-axis: y(t):     y(t) = R sin(ωt)     (equations 26)

This also means that we can easily find the vx and vx components of the
speed vT (or |vT|), at any time "t". We simply differentiate x(t) and y(t) with respect to t.

vx(t) = d/dt R cos(ωt) = -ωR sin(ωt)
vy(t) = d/dt R sin(ωt) = ωR cos(ωt)     (equations 27)


Movement of the dot on the y-axis:

Figure 13 is quite nice indeed. But also note the "dot" moving up and down on the y-axis.
Actually that dot (and the dot on the x-axis, which is not visible), performs a harmonic oscillation.
It's displacement is characterized by a "sin(t)" or "cos(t)" like function.

If you would attach a small pen on such dot, and roll a piece of paper with a constant speed under that pen,
you really would get a "sin(t)" or "cos(t)" function on paper.

Note that throughout section 5.1, all equations are in scalar form. It can all also be done using a vectorial approach.
However, the theory sofar holds ok.

For example, if you know T and R, it's easy to calculate ω or vT as scalars,
but that is really good enough.

Where is the Force ?:

Something is still missing. Yes, there has to be a "centripetal force", that is always perpendicular to vT,
and pointed towards the Origin at all times.

Only then, it is possible that an object turns around an origin in a circular fashion, with a constant speed vT.

You probably have experimented with such scenario (as a kid?), with a mass on a rope, and turn it around and around,
and you yourself in fact created (or excerted) that force (through the tension of the rope).

A "strange" thing might be, that the work done by the force is "0".
As we have seen in chapter 4, in the end, only the parallel component contributes to work, and in this case,
the force is always perpendicular.

Also, since vT stays constant, that may help to convince you too, that no work is done.

Note: the term "centripetal force" is questioned by some folks, and they rather only like to speak of "centripetal acceleration" as a better term.

So, what are the expressions for the "centripetal force" Fc and "centripetal acceleration" ac?

By the way, the tangent components are absent. The speed remains constant, thus there is no "tangent acceleration" aT.
Likewise, there is not tangent (or parallel) component of the Force present: The force is always perpendicular to vT.

Here I will only list the magnitudes of the "centripetal force" and "centripetal acceleration":

aT   =   vT2
--
R
      (equation 28)

FT   =   mvT2
--
R
      (equation 29)

Note that the equations 26 and 27, really describes the motions of the "dots" on the x-axis and y-axis,
and form the "heart" of the harmonic motion example of this section.

5.2 Other harmonic oscillators.

The "Pendulum":

Since the Pendulum is also a subject of "math note 10: differential equations Part 1, section 2.3".
I may say that you check that section in "math note 10".
However, it is based on deriving the "differential equation" first, and next find the solutions to that equation.

Mass (damped or undamped) on a spring:

This too is also a subject of "math note 10: differential equations Part 1, section 2.2".
But again, here too I first try to find the "differential equations" and next I try to find solutions.

Many other examples are available.

My text in "math note 10", using differential equations, might be percieved as pretty difficult....
However...:

Maybe it's a nice task to "Google" on keywords as the "Pendulum" or "equations of the Pendulum",
and try to find a nice easy article on the motion of the Pendulum.
It's never a bad idea to see some other perspectives too.

Or, you can also rightfully say, that you first want to see some completely other subjects of Classical Physics.
Indeed, lots of stuff is not in this note, like "angular momentum", "classical waves" etc.. etc..
There is a lot more you can study, which is a great thing.

Chapter 6. Gravitation and potential Energy.

6.1. Newton's Law of Gravity.

According to Newton, any two masses M1 and M2, exert an attractive gravitational force between them.

However, in the microscopic realm (atoms, elementary particles), the force is so increadably weak, that it is unmeasurable
for any practical purposes. Mind you: here we discuss the "classical" view on Gravitation.

But on a macroscopic scale, say in effect in the Solar system (Planets, Sun, comets etc..), Newton's law
works amazingly well. Newton's law of gravitation is:

F = G M1 * M2
--------
    r2

Here, M1 and M2 are the masses of the two objects (in kg), "r" is the distance between them (in m),
F is the force (in N), and G is the socalled "gravitational constant".
G has been determined to be 6.673 * 10-11 N m2/kg2.

Newton arrived to this equation, by deduction, and interpreting observations. But the equation sounds absolutely logical.
We may indeed assume, that the higher M1 and/or M2 are, the larger the Force will be.
Also, if the distance increases, we may expect that the Force will decrease, which is accounted for by "r
" in the denominator of the equation. So, the equation "looks" sound enough.

In Newton's framework, it is indeed so that the Force that M1 exerts on M2, is exactly the same
as M2 exerts on M1.

If you look at the Solar system, the mass of the Sun is so large, that the Planets orbit the Sun.

You can use Newton's equation for calculations involving the Sun and Planets, or sattelites orbiting Earth etc..

Example:

Here is an example. Suppose a Person of 70kg is at the surface of Earth. What is the force that the Earth excerts on this person
(and ofcourse the other way around)?

If we take:

G = 6.673 * 10-11 N m2/kg2, r=6.38 x 106 m (radius Earth), M1 =70kg, M2=6*1024 kg (mass Earth),

Then we have:

F = 6.673 * 10-11 70 * 6*1024
--------------
  (6.38 x 106)2

which is about 688 N.

6.2. Gravitational potential energy.

If two masses are (relatively) close, one can address the Gravitational potential energy of one system
in the gravitational field of the other system.

Usually, one system (M1) is substantially smaller than the other (M2), like a Planet in the field of a Star, or
a sattelite in the gravitational field of Earth etc..
Suppose, the distance between the objects is "r". Then the potential Energy of M1 is:

Epot, M1 = - G M1 * M2
--------
    r

You may have two questions. Why the "-" sign, and what happened to the quadratic power of "r"?

To start with the latter: You know that W = F * s. So, Energy is equal to Force x Distance.
The Force has a quadratic term in the denominator, and we have a multiplication with "distance" (r),
which leads to the equation shown above.

I immediately admit that this is a bit of a "Jip and Janneke" explanation.
Better is, to review the amount of work needed, to bring an object of mass M1, located at infinity,
thus where it's potential Energy with respect to M2 must be "0", and then move it nearer and nearer,
all the way up to distance "r" from Mq2. This is an integral, and it looks a bit like:

Work of F from infinity to r =

infr G

M1 * M2
--------
    r2

dr


If you know math, then you might know that the primitive of r-2, is -r-1,
which then explains the formula of the gravitational potential, and the negative sign.

However, if you are like me, the negative sign is not explained adequately enough.

Ok...

If I would say that only relative values matter, you might start laughing, although it's really so.
I mean, the temp difference between 1000K and 1100K, is not different as in 10000K and 10100K.

But as you carry a brik upwards a tower, it's potential energy might increase from 0 to e.g. 100 J.
Indeed, the higher the altitude, the more positive the potential energy becomes. Yes? Yes.

But that's the same as if it would increase from -300 J to -200 J, which is a positive increment of +100 J.

But in Highshool questions/tasks, it is often presented as being absolute values. Ofcourse, if we take
a brick upwards a tower, it's potential energy goes up with 100 J.

Since Epot, M1 resembles a y=1/x function (since r is in the denominator), a rough
graph of Epot, M1 is shown below.
At a larger distances, Epot, M1 is getting nearer and nearer to "0".

Figure 15.



6.3. The escape velocity.

What velocity do you need, to "just" escape the gravity of a larger mass? In such cases, we assume that the
object that needs to escape (like a rocket), must have a minimal energy, or a minimal speed,
to "just" get out of the attractive gravitational field (the force) of a large mass (like a planet as
Jupiter, Mars, Earth, or any other larger mass).

It's the "minimum" speed, so, at a large enough distance, the speed may even go to "near 0".
And Ekin is going to 0 as well. So, we are looking for that minimum amount of speed (or energy).

Usually, one calculates that, when the smaller object (the rocket) is on the surface of the larger mass,
but it may also already be in orbit (although then obviously the numbers will be different).

There are several approaches to find the escape velocity. We know the expression for the "centripal Force",
as was shown in equation 29. You can equal that to the gravitational force, as shown above.
With a bit of rearrangements, you can find the formula.

It's even easier to use an energy balance. We can write down the general equation for the initial kinetic energy
required, and equal that to the potential energy. The total energy at the start (on the surface) is 0,
that is, Ekin + Epot = 0. So:

½ m * vesc2 + - G m * M
-----
    r
= 0

So:

½ m * vesc2 = G m * M
-----
    r

On both sides of the equation, we have "m", so we can eliminate that. It's amazing that we can do that,
because then it turns out that vesc is independant of "m". That's amazing. Did you expect that?

½ vesc2 = G M
--
r

From here, it is quite easy to find that:

vesc = √ ( 2GM
----
  r
)

Additional exercises:

Exercise 1:

Suppose we have a track in the form of a circle. A small ball of 31 gram
is mounted at the top. The radius of the track is 42 cm.
The angle α is 50 degrees.

At t=0, we release the ball, and the ball leaves the track at point C.
You may neglect any form of friction.
See the figure below.



1. What is the amount of work which gravity performed up to point C ?

2. Suppose the track delivers 2.5 N of force Fw due to friction.
Which total amount of work is performed by Fw?

3. What is the velocity of the ball at point C (no friction) ?


Part 1.

At the top, Ep + Ek = mgh + 0 = 0.031*0.42*9.81 = 0.128 J
At the bottom Ep + Ek = 0 + ½mv2 = 0.128 J

The above facts are nice to know, but we can do the calculation as follows:

We only need the height difference between A and C. This way, we can find Ep,
which will is exactly the positive amount of work performed by gravity.

cos(α) = MX/MC.
cos(50) = MX/MC
0.643 = MX /0.42 => MX= 0.643 * 0.42 = 0.27 m.

=> XB = 0.42-0.27= 0.15.

Ep at height X = 0.15 * 0.031 * 9.81 = 0.045 J. This amount is "negative".

Ep difference point A to X = 0.128 - 0.045 = 0.083 J = 8.3 * 10-2
The work done by gravity at point C is netto 8.3 * 10-2 J.

Can it be analyzed in a much simpler fashion?

You only need to know the height difference between A and C.

Ep=W=mgh=m * g * r*sin(90-40) = 0.031 * 9.81 * (0.42 * sin(90-40)) = 8.3 * 10-2 J.

Part 2.

We need to know the length of the arc ABC. Call this length "S".
Then we can apply W = F * S.

Circumference circle = 2 * π * r = 2 * π * 0.42.

The arc ABC is the part (90+50) degrees / 360 degrees of the total Circumference circle.

S = 2 * π * 0.42 * (140/360) = 1.02m.

W = 2.5 * 1.02 = 2.55 J.

Part 3.

Try this yourself.

Exercise 2:

Suppose a Person of 70kg is at the surface of Earth. What is the force that the Earth excerts on this person
(and ofcourse the other way around)?

We must use:

F = G M1 * M2
--------
    r2

If we take:

G = 6.673 * 10-11 N m2/kg2,
r=6.38 x 106 m (radius Earth),
M1 =70kg,
M2=6*1024 kg (mass Earth),

Then we have:

F = 6.673 * 10-11 70 * 6*1024
--------------
  (6.38 x 106)2

which is about 688 N.


Exercise 3:

A steel bal rests on the floor. Someone, with hammer, hits the bal, and this ball
rolls along a road, untill it slowly stops due to friction.

- The ball is 1.5kg.
- The distance "d" covered is 25m, before it comes to rest.
- The force of Friction Fw is 15 N.

What was the initial velocity, immediately after the hammer hits this ball?


½ m v2 = Fw * d

½ 1.5 v2 = 15 * 25

v2 = (15 * 25) * 2 / 1.5 = 500 => v= √500 m/s (is about 22.4 m/s)

Exercise 4:

At t=0, an object is traveling with a constnt speed of 10 m/s.
At the same time, an unknown but constant friction sets in, and after 10 seconds,
the object comes to a halt.
See the figure below.



What is the distance S covered by the object?

There are multiple ways to calculate this:

Method 1.

The initial speed = 10 m/s. The end speed is 0 m/s.
So, the average speed is 5 m/s.

S = vaverage * t = 5 * 10 = 50m.

Method 2.

S = v * t.

Just like with a rectangular, the Surface is length * height. This principle works here too.

So, calculate the Surface of the triangle:

S = ½ * v * t = ½ * 10 * 10 = 50.

Question:

Are you happy with the shape of the function, which is a line?

Yes. It's a constant accelertion or decelleration, so v=at is a linear function, and a is constant.

Here we would have:

v(t) = v0 + at

v(t) = 10 -at


That's it. Hope you liked it.