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Uncertainties in Measurement

2016-08-16T13:38:09Z

Angonik: /* ABOUT THE ACCURACY AND PRECISION */

''Kreshnik Angoni''
 

'''BRIEF SURVEY OF UNCERTAINTY IN PHYSICS LABS''' 
===FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA===
 
The drawing of graphs during lab measurements is a practical way to estimate quickly: 

a) Whether the measurements confirm the expected behaviour predicted by the theory 

b) If any of recorded data is measured in a wrong way and must be excluded from further data
treatments. 


'''Example''':

We drop an object from a window and, from free fall model calculations, we expect it to hit ground after 2sec. To verify our
prediction, we measure this time several times and record the following results: 

1.99s, 2.01s, 1.89s, 2.05s 1.96s, 1.99s, 2.68s, 1.97s, 2.03s, 1.95s 

::::::(Note: '''3-5 measurements is a minimum acceptable number of data for estimating a parameter during a lab session''', i.e. repeat the measurement 3-5 times) 

To check out those data we include them in a graph (fig.1). From this graph we can see that: 

a) The falling time seems to be constant and very likely ~2s. So, in general, we have acceptable data. 

b) The seventh measure seems too far from the others results and this might be due to an abnormal circumstance during its measurement. To eliminate any doubt, we exclude this value from the following data analysis. We have enough other data to work with. Our remaining data are: 

1.99s, 2.01s, 1.89s, 2.05s, 1.96s, 1.99s, 1.97s, 2.03s, 1.95s.
 

[[image:Graph1a_Uncertainty.PNG|top]]
 
*[[#FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA|TOP]] 

===SECOND STEP: ORGANIZING RECORDED DATA IN A TABLE===
 
Include all data in a table organized in such a way that some cells be ready to include the uncertainty
calculation results. In our example, we are looking to estimate a single parameter “T”, so we have to
predict (''at least'') two cells for its average and its absolute uncertainty. 

'''Table 1''' 

{| border = "2" cellpadding = "4"
|-

|T1 ||T2 ||T3 ||T4 ||T5 ||T6 ||T7 ||T8 ||T9 ||Tav ||ΔT
|-
|1.99s ||2.01s ||1.89s ||2.05s ||1.96s ||1.99s ||1.97s ||2.03s ||1.95s || ||

|}
 
*[[#FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA|TOP]] 

===THIRD STEP: CALCULATIONS OF UNCERTAINTIES===
 
The ''true value'' of measured parameter is unknown. We use the recorded data to find an '''estimation''' of the '''true value''' and the '''uncertainty''' of this '''estimation'''. 

====Three particular situations for uncertainty estimations====
 

A] - We measure a parameter several times and always get the same numerical value.
 


'''Example:'''

We measure the length of a table three times and we get '''L= 85cm''' and
''a little bit more or less''.

This happens because the smallest unit of the meter stick is '''1cm''' and we '''cannot be precise''' about what portion of 1cm is the quantity “'''''a little bit more or little bit less'''''”. 

In such situations we use “'''the half-scale rule'''” i.e.; the uncertainty is equal to '''the half of the smallest unit available for measurement'''.
 
In our example '''ΔL= ±0.5cm''' and the result of measurement is reported as '''L= (85.0 ± 0.5)cm'''. 

 '''-'''If we use a meter stick with '''''smallest unit available 1mm''''', we are going to have a more precise result but even in this case there is an uncertainty.
 
Suppose that we get always the length '''L= 853mm'''. 
Being aware that there is always a parallax error (eye position) on both sides reading, one may get '''ΔL= ± 1''', '''2mm''' (and even '''3mm''', ) depending on the measurement circumstances. 
'''In this situation, it is suggested to accept 1 or 2 units of measurement'''. 

The result is reported '''L= (853 ± 1)mm''' .
 
Our estimation for the table length is '''853mm'''. Also, our measurements show that the true length is '''between 852 and 854mm'''. 
The uncertainty interval is '''(852, 854)mm'''. 
The absolute uncertainty of estimation is '''ΔL= ± 1mm'''
 
 '''-'''Now, suppose that, using the same meter stick, we measure the length of a calculator and a room and find '''Lcalc= (14.0 ±0.5)cm''' and '''Lroom= (525.0 ±0.5)cm'''. 

In the two cases we have the same '''''absolute uncertainty''''' ΔL= ± 0.5 cm, but we are conscious that the length of room is measured more precisely. 

The '''precision''' of a measurement is estimated by the uncertainty portion that belongs to the unit of measured parameter. Actually, it is estimated by the '''relative error:''' 

<math>\varepsilon = \frac{\Delta L}{\bar{L}}*100%</math>. 

 '''-'''Smaller relative error means higher precision of measurement. In our length measurement, we have: 

<math>\varepsilon_{calc} = \frac {0.5}{14}*100% = 3.57%</math>

<math>\varepsilon_{room} = \frac {0.5}{525}*100% = 0.095%</math>

We see that the room length is measured much more precisely (about 38 times). 

'''Note''': Don't mix the ''precision'' with '''''accuracy'''''! A measurement is ''accurate'' if the '''''uncertainty interval''''' contains an expected ('''known''') value and ''non-accurate'' if it does not contain it. 


B] - We measure a parameter several times and always get different numerical values. 


'''Example:''' 

We drop an object from a window and we measure the time it takes to hit ground. We find the different values of time intervals inserted in table _1. In cases like this, we have to calculate the average value and the absolute uncertainty based on statistical methods. 

B.1) We estimate the value of parameter by the '''average''' of measured data.

In case of our example:

<math>\bar{T}= \frac{1}{n}\sum_{i=1}^nT_i = \frac{1}{9}\sum_{i=1}^9T_i = \frac{1}{9}[1.99+2.01+1.89+2.05+1.96+1.99+1.97+2.03+1.95] = 1.982 s</math> 

B.2) To estimate how far from the average can be the true value we use the spread of measured data.

A first way to estimate the spread is by use of '''mean deviation''' i.e. '''“average distance”''' of data from their average value. In case of our example we get: 

<math>\Delta T = \frac{1}{n}\sum_{i=1}^n \left |T_i - \bar{T}\right | = \frac{1}{n}\sum_{i=1}^9 \left |T_i - 1.982\right | = 0.035 s</math>
 
Now we can say that the '''real value''' of fall time is inside the uncertainty interval (1.947, 2.017)s or between <math>T_{max}</math> = 2.017s and <math>T_{min}</math> = 1.947s with average value 1.982s. 

Taking in account the rules of significant figures and rounding off: 

::::::::::The result is reported as '''T = (1.98 +/- 0.04)s''' 
 
Another (statistically better) estimation of spread is the '''standard deviation''' of data. 

Based on data for falling time (T) in our first example and the mathematical expression for the standard deviation we get: 

<math>\sigma T = \sqrt{\frac{\sum_{i=1}^n \left (T_i - \bar{T}\right )^2}{n-1}} = \sqrt{\frac{\sum_{i=1}^9 \left (T_i - 1.982\right )^2}{8}} = 0.047 s</math> 

The result is reported as '''T = (1.98 +/- 0.05)s''' 

B.3) For spread estimation, the standard deviation is a better estimation for the absolute uncertainty.

This is because '''a larger interval of uncertainty''' means a more '''“conservative estimation”''' but in the same time a more reliable estimation. 

Note that we get ΔT= +/- 0.05 s when using the standard deviation and ΔT= +/- 0.03 s when using the mean deviation. Also, the relative error (relative uncertainty) calculated from the standard deviation is bigger. In our example, the relative uncertainty of measurements is: 

<math>\varepsilon = \frac{\sigma T}{\bar{T}} *100% = \frac{0.047}{1.982} * 100% = 2.4%</math> (when using the '''''standard deviation''''')

<math>\varepsilon = \frac{\Delta T}{\bar{T}} *100% = \frac{0.035}{1.982} * 100% = 1.8%</math> (when using the '''''mean deviation''''')

'''Note''': We will accept that our measurement is enough precise if the '''''relative uncertainty''''' “<math>\varepsilon</math>” is smaller than 10%.

'''If the relative uncertainty is > 10%, we may proceed by:'''
<ul>
<li> '''Cancelling''' the data “shifted the most from the average value” </li>
<li> '''Increasing''' the number of data by repeating more times the measurement </li>
<li> '''Improving''' the measurement procedure </li>
</ul>
 


C] - Estimation of uncertainties for calculated quantities (uncertainty propagation).

Very often, we use the experimental data recorded for some parameters and a mathematical expression to estimate the value of a given ''parameter of interests'' ('''POI'''). As we estimate the measured parameters with an uncertainty, it is clear that the estimation of POI will have some uncertainty, too. Actually, the calculation of '''POI average''' is based on the ''averages of measured parameters'' and the formula that relates POI with measured parameters. Meanwhile, the uncertainty of '''''POI estimation''''' is calculated by using the '''Max-Min''' method. This method calculates the limits of uncertainty interval, <math>POI_{min}</math> and <math>POI_{max}</math> by using the formula relating POI with other parameters and the combination of their limit values in such a way that the result be the smallest or the largest possible. 


'''Example:''' 

To find the volume of a rectangular pool with constant depth , we measure its length, its width and its depth and then, we calculate the volume by using the formula '''V=L*W*D'''. 

Assume that our measurement results are '''L = (25.5 ± 0.5)m''', '''W = (12.0 ±0.5)m''', '''D = (3.5 ±0.5)m'''.
 
In this case, the '''average estimation''' for the volume is <math>V_av = 25.5 * 12.0 * 3.5 = 1071.0 m^3</math>.
 
This estimation of volume is associated by an uncertainty calculated by '''Max-Min''' method as follows:

<math>V_{min} = L_{min} * W_{min} * D_{min} = 25 * 11.5 * 3 = 862.5 m^3\,\!</math>
 
<math>V_{max} = L_{max} * W_{max} * D_{max} = 26 * 12.5 * 4 = 1300.0 m^3\,\!</math>

So, the uncertainty interval for volume is '''(862.5, 1300.0)''' and the absolute uncertainty is:

<math>\Delta V = \frac{V_{max} - V_{min}}{2} = \frac{1300.0 - 862.5}{2} = 218.7 m^3</math>

while the relative error is:

<math>\varepsilon_v = \frac{218.7}{1071.0} * 100% = 20.42%</math> 

'''Note''': When applying the Max-Min method to calculate the uncertainty, one must pay attention to the mathematical expression that relates POI to measured parameters.
 

'''Example:''' 

'''-'''You measure the period of an oscillation and you use it to calculate the frequency ('''POI''').

As <math>f = \frac{1}{T}</math>, <math>f_{av} = \frac{1}{T_{av}}</math>,
the Max-Min method gives 
<math>f_{min} = \frac{1}{T_{max}}</math> and <math>f_{max} = \frac{1}{T_{min}}</math>. 

'''-''' '''If''' <math>z = x - y\,\!</math>, '''then''' 
<math>z_{av} = x_{av} - y_{av} \,\!</math> '''and''' <math>z_{max} = x_{max} - y_{min} \,\!</math> '''and''' <math>z_{min} = x_{min} - y_{max} \,\!</math>. 

'''Note_2''': Another way to calculate <math>POI_{av}</math> is by use of the formula

<math>POI_{av} = \frac{POI_{max} + POI_{min}}{2}</math>

after finding the limits of its uncertainty interval. 

----
<math>^1</math> ''The standard deviation can be calculated direct in Excel and in many calculators''

 
*[[#FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA|TOP]] 

===HOW TO PRESENT THE RESULT OF UNCERTAINTY CALCULATIONS===
 
You must provide the '''average''', the '''absolute uncertainty''' and the '''relative uncertainty'''. So, for the last example, the result of uncertainty calculations should be presented as follows: 

'''<math>V = (1071.0 \pm 218.7) m^3</math> , <math>\varepsilon = 20.42%</math>'''. 
'''Note''': Uncertainties must be quoted to the same number of decimal digits as the average value. The use of [scientific notation] helps to prevent confusion about the number of significant figures.


'''Example:''' 

If calculations generate, say '''A = (0.03456789 ± 0.00245678)'''. This should be presented after being rounded off (leave '''''1,2 or at maximum 3''''' digits after decimal point):

<math>A = (3.5 \pm 0.2) \times 10^{-2}</math> or <math>A = (3.46 \pm 0.25) \times 10^{-2}</math> 

 
*[[#FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA|TOP]] 

===HOW TO CHECK IF TWO QUANTITIES ARE EQUAL===
 
This question appears essentially in two situations:

# We measure the same '''parameter''' by two different methods and want to verify if the results are equal.
# We use measurements to verify if a '''theoretical expression''' is right.

In the first case, we have to compare the estimations <math>A \pm \Delta A</math> and <math>B \pm \Delta B</math> of the “two parameters”. The second case can be transformed easily to the first case by noting the left side of expression '''A''' and the right side of expression '''B'''. Then, the procedure is the same. 

'''Example:''' 
We want to verify if the thins lens equation <math>\frac{1}{p} + \frac{1}{q} = \frac{1}{f}</math> is right. For this we note <math>\frac{1}{p} + \frac{1}{q} = A</math> and <math>\frac{1}{f} = B</math>.

'''Rule''': We will consider that '''the quantities A and B are equal'''<math>^2</math> '''if their uncertainty intervals overlap'''.

[[image:Graph2_Uncertainty.PNG|top]]

----
<math>^2</math> ''They should be in the same units.''

 
*[[#FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA|TOP]] 

===WORKING WITH GRAPHS===

We may use graphs to check the theoretical expressions or to find the values of physical quantities. 

'''Example:''' 

We find theoretically that the oscillation period of a simple pendulum is <math>T = 2\pi*\sqrt{L/g}</math> and we want to verify it experimentally. 

For this, as a first step, we prefer to get a linear relationship between two quantities we can measure; in our case period T and length L. So, we square both sides of the relationship and get: 

<math>T^2 = 4\pi^2 \frac{L}{g}\,\!</math> 

Next, after noting <math>T^2 = y\,\!</math> and <math>L = x\,\!</math> we get the linear expression 

Y = a*X where <math>a = \frac{4\pi^2}{g}</math>. 

So, we have to verify experimentally if there is such a relation between <math>T^2\,\!</math> and L. 

Note that, if this expression is confirmed, we may use the experimental value of '''''a''''' to calculate an estimation for the free fall constant '''g''' by expression: 

“<math>g = \frac{4 \pi^2}{a}</math>”. 

Assume that, after measuring several times the period for a given pendulum length, one calculates the average value and uncertainty for y(=T^2). By repeating this procedure for different values of length x(L=1,..,6m) one get the data shown in table No 1. 
'''Table 1''' 

[[image:Uncertainties_Table_1.png|left]]

At first, we graph the average data. We see that they are aligned on a straight line; this confirms the theoretic expression, as expected. Next, we use Excel to find the best linear fitting for our data and we request to this line to pass from (X = 0, Y = 0) because this is predicted from the theoretical formula. 

[[image:Uncertainties_Graph2.png|top]] 

 We get a straight line with: 

<math>a_{av} = 4.065\,\!</math>. 

'''Using our theoretical formula we calculate the estimation for''' 
<math>g_{av} = 4\pi^2/a_{av} = 4\pi^2/4.065= 9.70\,\!</math> 

'''which is not far from expected value 9.8.''' 
 
'''Next, we add the uncertainties in the graph and draw the best linear fitting with maximum /minimum slope that pass by origin.''' 

From these graphs we get: 
<math>a_{min}= 3.635\,\!</math> and 
<math>a_{max}= 4.202\,\!</math>. 

So, we get: 
<math>g_{min} = 4 \pi^2/a_{max} = 4 \pi^2/4.202= 9.38\,\!</math> and 
<math>g_{max} = 4 \pi^2/a_{min} = 4 \pi^2/3.635= 10.85\,\!</math>. 
 
'''This way, by using the graphs we''': 

* '''proved experimentally that our theoretical relation between T and L is right'''.

* '''found that our measurements are accurate because the uncertainty interval (9.38, 10.85) for “g” does include the officially accepted value''' <math> g = 9.8m/s^2\,\!</math>

*'''found the absolute error''' Δg = <math> (10.85-9.38)/2=0.735 m/s^2\,\!</math> 
 
The relative error is ε = (0.735/9.70)*100% = 7.6% which means an acceptable (ε < 10%) precision of measurement. 

*[[#FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA|TOP]] 



===ABOUT THE ACCURACY AND PRECISION===

- '''Understanding accuracy and precision by use of hits distribution in a Dart’s play'''. 

[[image:Uncertainties_last.png|top]] 

-The estimation of accuracy is essential during a calibration procedure. As a rule, before using a method (or device) for measurements, one should make sure by measurements that it does produce accurate results in the range of expected values for the parameter under study. During such a procedure one knows in advance the “officially accepted value” which is expected to be the measurement result. 
If the result of measurement is unknown previously, there is no sense to talk about the accuracy. Meanwhile, during any kind of measurement one must report the relative uncertainty i.e. the precision of measurement . 

-So, we will refer to accuracy only in those labs that deal with an officially accepted value for a given parameter like free fall acceleration "g", Planck constant "h", etc. In principle, there is an accurate experiment result if the “average of data” fits to the” officially accepted value”.
We will consider that our experiment is “'''enough accurate'''” if the ” '''officially accepted value'''” '''falls inside the interval of uncertainty for the estimated parameter'''; otherwise we will say that the result is inaccurate. 

*[[#FIRST STEP: VERIFYING THE VALIDITY OF RECORDED DATA|TOP]]

Basic Concepts

2012-06-05T21:07:59Z

Angonik: /* Velocity Versus Speed */

''Kreshnik Angoni''

Kinematics is that part of physics that studies the movement itself without paying attention to the reason why it happens. Simply put, it uses one set of physical quantities (<math>x, y, z</math> coordinates, velocity and acceleration) to describe how the movement happens without use of forces that cause it. 

'''Dynamics''' uses the forces to explain why the movement happens. '''Static''' uses “the forces” to explain why the object is at rest. In general, the Static is studied in the frame of Dynamics. 

==ONE DIMENSIONAL KINEMATICS==

===Introduction===

One may distinguish three basic ways of motion: '''translation''', '''rotation''' and '''vibration'''. Note that in translation and rotation there is no deformation; the object shape and dimensions remain the same. In a vibration motion the object suffers periodic changes in one dimensions or its whole shape.

'''Translation:''' All points in the object move in the same way. For each point of the object, the final position can be found by using the same displacement vector (fig.1). 

[[image: Basic Concepts_KIN_1.png|thumb|none|upright=3.0|The same displacement vector <math>\overrightarrow{D_1}</math> shifts each object point from position (0) to position (1). The same displacement vector <math>\overrightarrow{D_2}</math> shifts each object point from position (1) to position (2).]] 

'''Rotation:''' The object changes orientation in space. Each object point has a different displacement vector. Even if a rotation is followed by a translation the displacement vectors are different (fig 2.b). 

[[image: Basic Concepts_KIN_2.png|thumb|none|upright=4.5|Figure 2.a: Only rotation. Figure 2.b: Rotation followed by a translation or vice-versa.]] 

'''Vibration:''' A periodic change of one dimension or the whole shape. 

[[image: Basic Concepts_KIN_3.png|thumb|none|upright=5.0|In figure 3.a, the system (spring plus body) changes only one dimension in a periodic way. In figure 3.b, the system (an elastic ball) changes shape (three-dimensions).]] 

Often, in the real life, the three types of movement combine together. We study them separately, by building a physical model for each of them. In this chapter we will deal only with movement along a fixed direction in space (one dimension, 1-D translation).

As mentioned above, in a translation, all object points have the same displacement vector. This simplifies greatly our work; “We study the movement of just one point in the object and the results apply over the object as a whole”. So, one uses '''a particle motion''' as a model for the object motion. 

===Displacement Versus the Traveled Distance===

Consider an object (plane, car..) moving along a straight line. We model this motion by a particle in motion along a straight line. To define the position of this particle, we need only one axis. So, we select:

# Frame origin O
# Positive direction
# Length unit "m"
# Time unit "sec"
# t=0 at initial location

Figure 4 presents the 1-D motion of an object. The particle in the model starts its motion at time <math>t</math> = 0sec and at position <math>X_I</math> = +1m. It moves 4m right, turns and stops at <math>X_F</math> = -3m. The motion lasts for 2sec.
We define the particle '''displacement''' as <math>\Delta x = X_F - X_I</math> where <math>X_F</math> is the '''final''' location and <math>X_I</math> is the '''initial''' location. Note that if <math>\Delta x > 0</math> the particle is shifted along <math>Ox</math> direction and if <math>\Delta x < 0</math> it is shifted in opposite direction to <math>Ox</math> axis <math>(-Ox)</math>. Remember: The displacement is not the same as the '''travelled distance''' which is always positive. In the case of figure 4, the travelled distance is 4 + 8 = 12m while the displacement is <math>\Delta x</math> = (-3) - (1) = -4m. 

[[image: Basic Concepts_KIN_4.png|thumb|none|upright=4.5|1-D motion of an object.]] 

===Velocity Versus Speed===

How fast is the particle moving? In everyday vocabulary one uses the '''speed''' (positive scalar) to answer this question in the case of cars, planes etc. If one does not have specific information about the way the particle is moving in particular portions of its path, one has to refer to the '''average speed''' along the path:

'''average speed = traveled distance / time interval'''

The average speed along the path is always a positive scalar. 

*Example: In the upper mentioned example of 1-D motion, one would find: average speed = 12m / 2sec = 6m/sec. 

In physics, one uses the velocity (''vector'') to describe the way a particle is moving. For 1D motion along Ox one starts by defining the '''average velocity''':

<center>
<math>
\upsilon_{Av} = \frac{\Delta x}{\Delta t}
</math>
</center>

Note that the velocity may be '''negative''' or '''positive'''. 

*Example: In the 1-D motion example, Vav = -4m / 2sec = -2m/sec, which is different from the average speed (6m/sec). 

'''Remember''': Although the speed and velocity have the same units (m/s in SI) they are very different. 

'''Note''': The displacement and the velocity are both '''positive''' when the particle moves along <math>Ox</math> direction.
When the particle moves along opposite direction <math>(-Ox)</math>, they are both '''negative'''. 

Very often one uses the graph <math>x = x(t)</math> to present the history of a particle motion. The figures 5 & 6 present two such graphs. The '''average velocity''' is calculated easily from the slope at these graphs. In the graph of figure 5, it does not depend on initial moment or the length of time interval. The motion follows all time with the same velocity. In the graph of fig. 6, we see that average velocity increases for smaller time interval (<math>\Delta x</math> ~ equal, <math>\Delta t</math> smaller). So, <math>\upsilon_{Av}</math> does not offer good information for all situations. 

[[image: Basic Concepts_KIN_5&6.png|thumb|none|upright=5.0|Figure 5: The particle moves at a constant velocity. Figure 6: The particle moves at a changing velocity.]] 

===Instantaneous Velocity===

A simple observation of figure 7 shows that for the right information about the particle velocity at point P, one must refer to the smallest time interval counted from the moment when the particle is at point P.

[[image: Basic Concepts_KIN_7.png|thumb|none|upright=2.5|For the smallest time interval, the slope (average velocity) fits the curve more accurately.]] 

In fact, one may see that the best estimation about the velocity close to point P location is the limiting value of average velocity when the time interval goes to zero. This is the definition of '''instantaneous velocity''' at point P:

<center>
<math>
{\upsilon_x^P} = lim_{\Delta t \rightarrow 0} {\Delta x \over \Delta t} \mid_{t_P} = {dx \over dt} \mid^P
</math>
</center>

'''Note''': As this derivative is equal to the slope of curve tangent at point P, one may find the velocity straight from the graph <math>x = x(t)</math>. 

The graph in figure 8 presents a motion along <math>Ox</math> axis:

[[image: Basic Concepts_KIN_8.png|thumb|none|upright=3.5|
Point A: <math>\upsilon_x > 0</math>; Motion along <math>+Ox</math> 
Point B: <math>\upsilon_x > 0</math> (larger); Faster motion along <math>+Ox</math> 
Point C: <math>\upsilon_x = 0</math>; Instantaneous rest, turn back point 
Point D: <math>\upsilon_x < 0</math>; Motion along <math>-Ox</math> 
Point E: <math>\upsilon_x < 0</math> (smaller); Slower motion along <math>-Ox</math> 
Point F: <math>\upsilon_x = 0</math>; Long rest before turning back along <math>+Ox</math>]] 

===Acceleration===

In the popular vocabulary the word “acceleration” means "speed increase". In physics, it means simply a '''change of velocity vector''' (magnitude, direction or magnitude & direction simultaneously). Knowing the relation <math>x = x(t)</math> for the motion of a particle, one may build the graph of instantaneous velocity versus time. If it is a straight line, the average acceleration is a constant and is sufficient for the velocity change description. However, such graphs may have different curved parts and one uses '''instantaneous acceleration'''.

[[image: Basic Concepts_KIN_9.png|thumb|none|upright=2.5|]]

So, if the average acceleration is defined as

'''average acceleration = change of velocity / time interval'''

the instantaneous acceleration at point D is

<center>
<math>
{a_x^D} = lim_{\Delta t \rightarrow 0} {\Delta \upsilon_x \over \Delta t} \mid_{t_D} = {d\upsilon_x \over dt} \mid^D
</math>
</center> 

The acceleration at any point D on the graph <math>\upsilon = \upsilon(t)</math> is equal to the graph '''slope''' at that point. The acceleration may be '''positive''' or '''negative'''. It is important to mention that the acceleration sign alone is not sufficient to understand whether the particle is speeding up or slowing down. To get this information, one must compare the <math>a_x</math> sign to the <math>\upsilon_x</math> sign. The following table describes all four possible situations. 

{| border="3" style="margin-left: 3em;"
|-
! scope="col" |
! scope="col" | <math>\upsilon_x</math> is POSITIVE
! scope="col" | <math>\upsilon_x</math> is NEGATIVE
|-
! scope="row" | <math>a_x</math> is POSITIVE
| Velocity and acceleration have the '''same sign''' so the body is '''speeding up'''. Velocity is '''positive''' which means the motion is along <math>Ox</math>.
| Velocity and acceleration have '''opposite signs''' so the body is '''slowing down'''. Velocity is '''negative''' which means the motion is along <math>-Ox</math>.
|-
! scope="row" | <math>a_x</math> is NEGATIVE
| Velocity and acceleration have '''opposite signs''' so the body is '''slowing down'''. Velocity is '''positive''' which means the motion is along <math>Ox</math>.
| Velocity and acceleration have the '''same sign''' so the body is '''speeding up'''. Velocity is '''negative''' which means the motion is along <math>-Ox</math>.
|}
 

'''Remember''': The direction of motion is always shown by the sign of velocity. 

To get a better meaning of this, let's consider the <math>\upsilon</math> and <math>a</math> graphs in figure 10:

[[image: Basic Concepts_KIN_10.png|thumb|left|upright=3.0|]]

'''A'''-point: <math>\upsilon < 0</math>; motion along <math>-Ox</math>

Since <math>a > 0</math> (opposite sign), there is "slowing down" 

Between '''A''' & '''B''': <math>a</math> = max, (slope on <math>\upsilon</math>) = max; there is "slowing down" 

'''B'''-point: <math>\upsilon = 0</math>, but <math>a \neq 0</math>; this means '''instantaneous rest'''

Since <math>a > 0</math>, ready to move along <math>+Ox</math> 

Between '''B''' & '''C''': <math>\upsilon > 0</math>, <math>a > 0</math>; motion along <math>+Ox</math> 

'''C'''-point: max acceleration 

Between '''C''' & '''E''': <math>\upsilon > 0</math>; motion along <math>+Ox</math>

Since <math>a > 0</math> but decreasing, slight <math>\upsilon</math> increase 

'''I'''-point: zero acceleration; '''instantaneous constant velocity''' 

Between '''I''' & '''J''': <math>\upsilon > 0</math>; motion along <math>+Ox</math>

Since <math>a < 0</math>, there is "slowing down" 

Between '''G''' & '''H''': greater "slowing down" effect 

Beyond '''G'''-point: the stopping effect decreases until it becomes zero at point '''J''' ('''constant velocity''') 

'''Notes''': In real life the velocity cannot change instantaneously. This means that the acceleration has always a finite value (no infinite). At this level, we will study only motions with constant acceleration. To describe a 1-D motion with constant acceleration only "<math>x</math>, <math>\upsilon</math>, and <math>a</math>" parameters are needed. 

A simple way to distinguish an accelerated motion (<math>a \neq 0</math>) from one with constant velocity (<math>a = 0</math>): Fix an interval of time (say 1sec) and measure the travelled distance for several successive intervals of 1 second. If the distances are equal, there is a motion with '''constant velocity''' (fig 11.a). If the successive distances increase (or decrease) by the same quantity, there is a motion with '''constant acceleration''' (fig 11.b). If the successive distances change differently, there is a change of acceleration. 

[[image: Basic Concepts_KIN_11.png|thumb|none|upright=5.5|]]

Measurement and Data Analysis

2011-09-22T21:34:09Z

Angonik: /* ABOUT PHYSICS */

''Kreshnik Angoni''

===ABOUT PHYSICS===

'''Physics is one of the natural sciences'''; so it deals with the two nature components: '''matter and fields'''. 

The object of physics is '''the study of motion in nature:''' 

“'''HOW''' does evoluate and '''WHY''' a given '''MOVEMENT'''(of matter or field) is produced”. 

How does the study of physics proceed? It starts by defining the objective (say-''study of a soccer ball''motion) and the reference frame where the motion is studied (say- ''field sides''). 

Next, one follows by identifying the necessary parameters for description of the movement (example; '''position, velocity,
acceleration'''). 
Next, one records a set of data for each parameter. 
Then, one looks for any possible relationship pattern between the measured data. The easiest way to do this is by using a graph. 

Next, if a graphical pattern appears, one tries to get a mathematical expression for the relation between the
parameters in graph. 

Then, one builds a theoretical model to explain the observed relationship and gets
one (or several) equation that relates the considered parameters. 

After that, one uses the model and its
equations to predict the numerical values of these parameters in any similar situation ('''experiment'''). 

'''Example''': '''Object of study:''' The motion of a glider on an air track (select '''track as reference frame'''). 

'''Parameters :''' '''''velocity''''' and''' ''time'''''.
 
'''Measurement''':

[[image:Kreshnik_Measuremment_Table 1.png|left]] 

[[image:Kreshnik_Measuremment_Fig 1.png|left]] 

'''Getting a mathematical relation from the graph'''

Build a graph with recorded data. Find out a linear relationship 
v(m/sec) = 3.1352*t(sec) 

'''Building a model:''' 

One considers that the glider can be modeled by a material point with mass = mgl and considers the movement of material point with constant acceleration a = v/t. 

'''Result:''' The glider moves as a material point with acceleration <math>a = 3.1352m/s^2\,\!</math>. 

===UNITS===

A '''physical parameter''' is characterised by a '''numerical value''' and '''a unit of measurement'''. 

In physics, one discerns the '''basic units''' and the '''derived units'''. 

'''a)''' '''''A basic unit''''' is a unit used during a direct measurement. In this case the numerical value of the
physical parameter is equal to the number of times the basic units enters into the parameter in
consideration. 

(Example: When measuring the length of a table, one has to define first what unit will use - say meter. Then it verifies haw many times the meter unit enters into the length of the table - say1.6 times. So, one gets that the table length is 1.6meters.) 

The basic units are selected following the humanity experience and fit well with a wide range of
measurements. 

The '''basic unit''' of the physical parameter '''“length”''' is the '''meter (m)'''. 

The universal etalon of meter, kept in Sevres (France), is defined as the distance between two fine scratches on a
special material bar. All meter units used over the world must be equal to this etalon. 

The '''basic unit''' of the physical parameter '''”time”''' is the '''second (s)''' and it is the time light takes to
travel over 299 792 458m(~3*106m/s) in vacuum. 

All second units used over the world have to be equal to this etalon. 

The '''basic unit''' of physical parameter '''“mass”''' is the '''kilogram (kg)''' and it is defined as the mass of a particular metal cylinder kept in Sevres. 

All kilogram units used over the world have to be equal to this etalon. 

'''The meter, the sec. and kg are the basic units in SI (Système International) system of units.''' 

Although the etalons are selected to produce logical numerical values for everyday
measurements, the continuous increase of human activity requires dealing with numerical
values, which are pretty big or small if referred in basic units. To avoid this problematic, one
introduced subsequent definitions: 

[[image:Kreshnik_Measuremment_Table 2.png|TOP]] 

- For historical reasons there are several different units used for the same physical quantity. For
example the distance is expressed in miles (mi) or inches (in), too. 

('''Example:''' One may see the speed given in Km/hr and miles/hr in some cars). 

How to find the value of a physical propriety in a given unit when we know its value in another unit? This question is answered by the procedure of unit conversions (see next sections). 

'''Example:''' We know that the distance between two cities is 425km and we wants to find it in miles. 

From the manual we find that 1 mi =1.609Km. So, 1 km = (1/1.609) mi and (1mi/1.609km)=1. 

We multiple our expression by 425 km*1= 425km*(1mi/1.609km) = 264.14 mi. 

'''b)''' '''''A derived unit''''' is a unit that is expressed through the basic units. 

'''Example:''' the volume is expressed in <math>m^3</math>; the density of a liquid is expressed in kg/<math>m^3</math>. 
There are three basic units (in SI system m, sec, kg) and many derived units (<math>m^2</math>, <math>m^3</math>, N, m/s, <math>m/s^2</math>…) 

===DIMENSIONAL ANALYSIS===

- In physics, one uses often expressions of type ” this parameter has length dimensions”. So,
without being interested on the real unit (meter, mile, cm ) one confirms that the parameter is
expressed in length units. The three basic dimensions are “L for length”, “'''T''' for time” and “'''M''' for
mass”. 

When referring to the dimension of a given physical quantity one uses the square brackets. 

'''Ex'''.: [a] = <math>LT^{-2}\,\!</math> for the dimension of acceleration. 

Suppose that calculations produce an algebraic expression for the required physical quantity. As a
first step of verification, one may use the dimensional analysis. So, if the expression is '''Z = X + A''',
at first, one has to verify that ['''X'''] and ['''A'''] are equal because one can not add up different physical
quantities (Ex. position + acceleration!!) So, before proceeding to numerical calculations, one must
verify the dimensional consistency of the found expression to the dimension of required quantity. 

'''Ex'''.: If a calculation gives for acceleration the expression a = <math>\frac{m*t^2}{v}\,\!</math>, by verifying the dimensions; 

[a] = <math>\frac{[m]*[t^2]}{[v]} = \frac{M*T^2}{L/T} = M*T^3*L^{-1}\,\!</math> 

it is a wrong expression because [a] = <math>LT^{-2}\,\!</math>. 

===CONVERSION OF UNITS===

Often, one needs often to convert the result ( for a physical quantity) from a given unit to another unit. 

“How to convert the value of a physical propriety from a given unit to another unit”? 
To do this one has to use the conversion procedures. 

At first, one refers to conversion factors; Ex: 1 h = 60 min; 1 h = 3600 s; 1mi = 1.609km 

The ratio of all conversions is equal to 1; (1h/60min) =1 (1h/3600s) = 1; (1mi/1.609km) =1. 

Then, as the multiplication by 1 does not change the result one goes step by step ; 

'''Ex 1:''' The distance between two cities is 425km and we want to find it in miles. From the manual
we find that 1 mi =1.609Km 

So, <math>\frac{1 mi}{1.609 km} = 1</math> and so: 

<math>425 km = 425 *\frac{1 mi}{1.609 km} = 264.4 mi</math> 

'''Ex 2:''' Convert the speed 60km/h in m/s. As 1km = 1000m, 1000m/1km =1 and 1h/3600s =1, 

60km/h = <math>60 * \frac{1000 m}{1 km} * \frac{1 h}{3600 s}\,\!</math> = 16.67 m/s 

'''Ex 3:''' The area of a paper sheet is <math>6580 cm^2</math>. Convert it in <math>m^2</math>. Knowing that 1 m = 100cm and 

<math>6580 cm^2 = 6580 cm*cm = 6580 cm * (\frac{1 m}{100 cm})*cm * (\frac{1 m}{100 cm}) = 6580(\frac{1}{100})m (\frac{1}{100}) m = 0.658 m^2\,\!</math> 

'''Ex 4:'''

'''Area''' = <math>1 in^2= [1 in*(\frac{2.54 cm}{1 in})*(\frac{1m}{100cm})]^2 = [2.54m/100]^2 = [2.54*10^{-2}m]^2 = 6.4516*10^{-4}m^2</math> 

'''Vol''' = 1 litre(l) = <math>10cm*10cm*10cm=10^3 cm^3 = 10^3 (\frac{1}{100m})^3 = 10^3(10^{-2}m)^3 = 10^3*10^{-6}m^3 = 10^{-3}m^3</math> 

 
===SCIENTIFIC NOTATION===

Although the units are selected to produce “normal values” for physical quantities, the practice
shows that often one has to deal with very big or very small values of a physical quantity. 

In these situations the use of scientific notation avoids errors and simplifies the calculations. 

When written in scientific notation, the numerical value of considered quantity is presented as a number with one digit
before the decimal point multiplied by a factor of 10. (Ex. <math>1.253x10^{-8}\,\!</math>; <math>-8.253x10^{-38}\,\!</math>). 

'''The scientific notation of a number is known also as its exponential presentation with base 10'''. 

To convert a number to scientific notation move the decimal point to the right or the left until you
get one digit on the left of decimal point. 

To keep the same number value, for each left shift multiply by 10 and for each right shift divide by 10. 

'''Ex1:'''

<math>125.4 = 125.4 *10^0 = 12.54 *10^1 = 1.254 *10^2\,\!</math> 

<math>0.001254 = 0.001254 *10^0 = 0.01254 *10^{-1} = 0.1254 *10^{-2} = 1.254 *10^{-3}\,\!</math> 

Basic operations with scientific numbers. Given two numbers: 
<math>x_0 = a_0 10^{b_0}</math>, <math>x_1 = a_1 10^{b_1}</math> 

<math>x_0*x_1 = (a_0a_1)*10^{b_0 + b_1}</math> and <math>\frac{x_0}{x_1} = (\frac{a_0}{a_1})*10^{b_0 - b_1}\,\!</math> 

'''Ex2:''' 
<math>(5.67*10^{-5} )*( 2.34*10^2) = 13.2678*10^{-3} = 1.32678*10^{-2} ; (5.67*10^{-5} )/ ( 2.34*10^2) = 2.423*10^{-7}\,\!</math> 

Before performing addition or subtraction the numbers must be presented by the same exponent. 

Usually the smaller number is transformed before performing the addition or subtraction. 

'''Ex3:''' 
<math>3.17*10^{-5} + 1.34*10^{-4} = 0.317*10^{-4} + 1.34*10^{-4} = 1.657*10^{-4}\,\!</math> 

<math>2.13*10^6 - 5.34*10^2 = 2.13*10^6 - 0.000534*10^6 = 2.130534*10^6\,\!</math> ≅ <math>2.13*10^6\,\!</math> 
 

===SIGNIFICANT FIGURES===
 

In experimental measurements, one has to deal with a 
'''“minimal available unit”''' and this
defines the 
'''uncertainty of the measurement'''. 

For example, when measuring the length of an object by using a ruler with minimum unit scale 1mm, 
if you report the result of measurement L = 17.5mm, it means that the its length is estimated 
with '''three significant figures''' and the last digit '''“5”''' is '''uncertain'''. 

''The concept of significant figure(reliable digit) is related to measurable parameters.'' 

If the length is reported as 12.345 m, then this is a number with 5 significant figures 
where last figure “5” is not certain. 

''From the practical point of view, this means that in that measurement the minimum unit was 0.01m
= 1cm and it is accepted that the observer is able to distinguish without being sure a length 0.5 cm''. 


'''In majority of cases, by convention, during a measurement process, one assumes that the uncertainty
of measured value is 0.5 of minimum available unit.''' [[image:Kreshnik_Measuremment_Fig 1b.png|right]]

'''In the above example the uncertainty is 0.5mm and the true length value is inside the interval (17.5 ± 0.5) mm.''' 


'''Note:''' If the mass of an object is reported 15.5g, the uncertainty is 0.5g but if reported 15.55g, it is 0.05g.
 

* Considering a data treatment situation, one has to remember that the number of digits at absolute
uncertainty (or absolute error) defines the number of digits at reported values. 


'''Ex''': One uses a meter stick with 1 cm unit (Δ = 0.5cm) to measure the length of a set of wood beams
and gets values 17.5, 18.5, 18.5…. 

[''Absolute uncertainty is 0.5cm (or smaller, i.e. 0.2cm or 0.1cm depending on observer ability)''] 

Then, one calculates the average length of the set and gets the average value 18.311cm. 

As the absolute uncertainty is Δ= ±0.5cm, there is no sense to keep more
than one digit after decimal point at the average. 

The second and the third digits after the decimal point of average value have no reliability because they are much smaller that accepted value for the uncertainty. 

So, one must round off the number to one decimal digit and the average length of the set must be reported as (18.3± 0.5)cm. 

How to find the number of significant figures: 

<ol>
<li>''' All nonzero digits are significant: 1.324 g has 4 sign. fig., 1.5 g has 2 sign. figures.'''</li> 

<li>''' Zeroes between nonzero digits are significant: 3002 kg has 4 sign. fig.; 1.02 L has 3 sign. figures'''.</li> 

<li>'''Leading zeros to the left of the first nonzero digits are not significant; 0.001g has 1 sign. figure; 0.012 g has 2 sign.figures.'''

Such zeroes merely indicate place holders and they do not contain any information about the uncertainty of estimated parameter. 

Often one uses the scientific notation to obtain quickly the sig. figure. 

Ex: <math>0.0001035 = 1.035*10^-5\,\!</math> (4 sign.fig)</li> 

<li>''' Trailing zeroes (to the right end) are significant only if there is a decimal point.''' 

'''0.0230 m has 3 sign. figures. 0.20 g has 2 sign. figures.''' 
'''but <math>100 =1*10^2\,\!</math> has 1 sig.figure.''' 
'''while <math>100.0 = 1.000*10^2</math> has 4 sign. figures.'''</li> 

<li> ''' In scientific notation the significant figures are counted at the coefficient.''' 

'''Ex.''' The length 5.5mm has 2 sig. figures even if converted in meters; 

It is 0.0055m which is written <math>5.5*10^{-3}</math> m(2 sig. figure). 
<math>2.02 * 10^4</math> kg has 3 sign. figures, 
<math>2.020 * 10^4</math> ft has 4 sign.figures 
and <math>2.0200 * 10^4</math> N has 5 sign. figures.</li> 

<li>''' In addition and subtraction, the result is rounded off to the smallest number''' 
'''of decimal places occurring in all components.''' 
200(no decimals) + 25.643 (5 sign. figures) = 225.643 
which should be rounded off to 226 (no decimals). 

'''In multiplication and division, the result should be rounded off''' 
'''so as to have the same number of significant figures as in the component''' 
'''with the smallest number of significant figures.''' 

4.0 (2 sign. figures) × 13.60 (4 sign. figures) = 54.400 
which should be rounded off to 54 (2 sign. figures). 

12.589(5 sign. figures) x 2.0312(5 sign. figures) / 4.0 (2 sign. figures) = 6.3926942 
which should be rounded off to 6.4 (2 sign. figures). </li> 

<li>'''Rounding off rules: When rounding off, use only the first number to the right''' 
'''of the assumed uncertain digit(ignore the following digits)''' 

'''Example:''' 
For two significant figures, 18.71 is rounded to 19 and 18.46 to 18. 

Similarly, 5.249 is rounded to 5.2 not 5.3; 

18.51 is rounded to 19 (if the digit to be dropped is 5 
followed by a 0 digit leave it as it is, otherwise increase it by 1)</li> 

</ol>
 

'''Basic principle in numerical calculations:''' 

'''''The result cannot be more precise than parameters included in calculations. ''''' 

'''Step-by-step procedure:''' 
<ol>

<li>Identify the sig.figures and number of decimals for each number before calculations. </li> 

<li>Identify the smallest sig.fig in multipl./div. and smallest number of decimals in substr./add.</li> 

<li>Do first the mathematical calculations in normal mathematical way; get a mathematical result.</li> 

<li>At the end, round off the found mathematical result by referring to: 

::a) The smallest number of digits after decimal point for substraction /addition 
::b) The smallest sig.fig. for division/multiplication 
</li>

</ol>
 

'''Ex:''' 124.4 + 2.345 –11.005 = 115.74 and by rounding of (to 1 digit after dec. point) we get 115.7
(5.345 +12.3005) / 2.2 = 8.02068 and by rounding off (to 2 sig.fig) we get 8.0
(2.365*10-15 * 0.0287) / 1.23*10-12 = (2.365*10-15 * 2.87*10-2) / 1.23*10-12 =
= 5.518333*10-5 and by rounding of (to 3 sig.fig) we get 5.52*10-5 

'''Important note''': Pay special attention to the EXACT number like 4(cars) or 10(students). They have “no uncertain digits” and one must add infinite number of zeroes after decimal point to get to the uncertain digit;exact numbers have INFINITE SIG. FIGURE and INFINITE DIGITS after decimal point.

Ex: 4.795 / 145 = 0.033068966 must to be rounded to 0.03307 because Sig.Figmin = 4 (at 4.795)
while if “145 =145.)” is not an exact number the result is 0.0331 because Sig.Figmin = 3 (at 145.)

Ex: 156.3 - 11 = 145.3 if “11” stands for an exact number because of infinite zeros after dec. point
while it has to be rounded off to 145 if “11” is not an exact number because of 0 decimals.

To avoid any possible ambiguity, one prefers to note the decimal point when dealing with measurable (containing uncertainty) quantities even if there are no digits after the decimal point.

'''PRACTICAL RULE:''' If you are using a formula to calculate a physical quantity but the values for parameters in expression are given without decimal point, keep just 1 or 2 digits after the decimal point at the result. 

===RELATIONSHIPS===

As mentioned in section 1, to check the presence of a relation between two parameters X and Y, one
must record a set of data. After including them in a table one builds a graph and verify whether the
experimental points are distributed around a line. 

If this is the case, one can affirm the existence of a relationship between the two considered physical quantities. Otherwise, one says that there is no observable relation between them. In case of observable relation, in general, one may discern between two major situations: linear or power fitting function for the line on the graph. 

====''LINEAR RELATIONSHIPS''====

If the experimental points are distributed around a straight line, one says that the quantity Y varies
linearly with quantity X. In the particular case when the straight line passes through the origin one
says that the quantity Y is proportional to quantity X. 
In proportional relations, the ratio Y/X is

[[image: Kreshnik_Measuremment_Fig 2.png|left]]

a constant and is called proportionality constant (4 in the case of figure). 

In case of proportional relationships, if one of the quantities is multiplied by a factor, the other one is multiplied by the same factor, too. 

('''''Ex. if X is increases 3 times, Y increases 3 times, too, because their ratio must remain unchanged, 4'''''). 

This is not true for linear relationships that are not proportional (verify for the linear relation in figure y = 2x + 1.5). 

The general mathematical expression for linear relations is Y = a*X + b ...(1). 

By coordinates of two points in data set one gets <math>Y_1 = a*X_1 + b\,\!</math> and <math>Y_2 = a*X_2 + b \,\!</math> ...(2) 
Then, by subtracting the first equation from the second one <math>Y_2 - Y_1\,\!</math> = ΔY = <math>a*(X_2 - X_1)\,\!</math> = a*ΔX 

So, one gets that 

ΔY = a*ΔX ...(3) 

For all linear relations (proportional included), if the quantity X changes by ΔX the quantity Y
changes by ΔY= a* ΔX . 

Note: ΔY= <math>Y_2 - Y_1\,\!</math> not ΔY = <math>Y_1 - Y_2\,\!</math> (same for ΔX; see directions in fig1). 

Note that a - coefficient is equal to the slope of straight line while b - coefficient is equal to the y
value at the point where the straight line touches Y-axis. 

The slope of a linear relation may be positive or negative (+2,+ 4 in fig.2 and +2 or –3 in fig.3). 

[[image: Kreshnik_Measuremment_Fig 3.png|left]]

In physics there are many linear and proportional relationships. Here are some of them: 

'''Proportional:''' 

Force to acceleration F = m*a 

Elastic force to extension <math>F_{el}\,\!</math> = k*Δx 

Electric tension to current U = R*I 

'''Linear:''' 

Velocity to time <math>v = a*t +v_0\,\!</math> 

Metal bars Length to temperature <math>l = c*T+l_0\,\!</math> 
 

 

====''POWER FORM RELATIONSHIPS''====

If the experimental points are distributed around a line which slope changes from one to another
region of X-values, one says that the relationship between quantities X and Y is described by a curve.
In general, a curve can be fitted a power expressions. The following simpler forms are often met. 

'''Parabolic relationship''' 
<math>Y = a * X^2 \,\! </math> 
is widespread in different physics branches. 

'''Example:''' We know that the acceleration of a object close to earth surface is <math>g = 9.8m/s^2\,\!</math> and if it is left to fall from rest at a given point O, the distance from O will increase in time as <math>y = a* t^2/2\,\!</math>. So, if one does experimental measurements of distance y at moment t and builds a graph, one will see a curved line. 

[[image: Kreshnik_Measuremment_Fig 4.png|left]]

Then, comparing the ratios (<math>y/t^2\,\!</math>) one will find “almost the same value” a ~ 4.9 (= 9.8/2). 
If one would repeat this experiment on Moon surface, one would get a ~ 0.82. On Jupiter surface
one would get a ~ 13. 
So, one may assert the following general physics’ law: 

During the free fall, the distance of object from starting location increases in parabolic way with time. Note that the
a- value depends on the planet. 

Question: What is the gravitational acceleration on Moon and Jupiter surface (3rd curve)?…………………

[[image: Kreshnik_Measuremment_Fig 5.png|left]]

In dependence of the sign of the coefficient “a” one discerns positive and negative parabolas. 
Their graph forms are presented in figure 4. 

'''Note:''' 
The general mathematical expression for a 
parabola is <math>y = a*x^2 + b\,\!</math>. 

If b ≠ 0 the parabola top is not located at origin of coordinative system. 
 

'''Inverse relationship''' 
Y = const/X . 

This type of relationship must be considered when one of the measured quantities decreases while the other one increases. Verifying whether the product Y*X is almost constant for all measured data constitutes the first step trial. 

If this is true one can confirm that the Y and X are related by an inverse relationship. 

A physics experiment that would produce this kind of relationship: By compressing a gas its pressure P increases while its volume V decreases. 

Another important inverse relationship in physics is the '''INVERSE SQUARE''' 
Y = const / <math>X^2\,\!</math>. 

This is the mathematical form of gravitational and electrostatic forces F = const / <math>R^2</math> exerted between two
objects at distance R. 

As shown in figure 6, the real difference between Inverse and Inverse Square relationships appears clearly only for small values of argument X. 

So, when looking for the precise relationship one must compare X*Y to <math>X^2*Y\,\!</math> for small values of X. 

[[image: Kreshnik_Measuremment_Fig 6.png|TOP]]

[[image: Kreshnik_Measuremment_Fig 7.png|left]]

'''Exponential relationship''' 
Y = a*Exp[<math>b*(X- c)^d\,\!</math>]. 

It is very common in physics. Its graph takes different forms in dependence on the values of parameters a, b, c, d. 

One very known example is that of radioactive nuclei which number decreases in time following the exponential expression N = <math>N_0\,\!</math> Exp(−λ*t) . This expression is taken from the general one for a = <math>N_0\,\!</math>, b = -λ, X= t, c = 0 and d =1. 
 
The figure 7 presents the graph of function 
Y = 100 * Exp[−0.1*t] 
 

===FINDING THE ORDER OF MAGNITUDE OF THE CALCULATION RESULT===

This is a kind of “estimation” on the size of the result within an order of (x10). One uses the
approximations to get result estimations. It is very useful in answering questions like “find daily
consumption of fruits in a city with 3 millions of habitants”. One starts by estimating say ~0.5kg
fruits/day/person and follows <math>3*10^6*0.5 = 1.5*10^6\,\!</math> i.e. essentially (as an order of magnitude) <math>10^6\,\!</math> kg. 

The basic technique for finding the order of magnitude requires approximating each of input numbers by the closest number containing only one significant figure and then performs the calculations. 

'''Example:'''
2.135 + <math>\frac{898.475*10.812}{7.891}\,\!</math> ≈ 2 + <math>\frac{9 *10^2 *10^1}{8}\,\!</math> ≈ 2 + <math>\frac{9 *10^3}{8}\,\!</math> ≈ 1000 

The precise answer is '''1233.19''' and it has the same order of magnitude (one thousand) as 1000. 

Estimating first the order of magnitude many times prevent from small arithmetical mistakes. Say, if the normal step-by-step calculations give 12331.9 one sees quickly the existence of a mistake.

Concept of Force

2011-07-31T15:56:20Z

Angonik:

''Kreshnik Angoni''
 
'''Dynamics''' answers the question: 
'''“Why does an object move or stay at rest?”'''. 

It operates with two main concepts:” the '''force''' and the '''mass''' ”. 

The force describes the action which attempts to change the object velocity while its mass describes the object reaction versus “forced change of its velocity”. 

The combination of these two major concepts makes possible to explain why the considered object is in a given status of motion (i.e. a given acceleration and velocity).
 
====FORCE====
- This word was introduced initially to describe a '''“push or pull”''' action exerted on a body. People observed that the effect of “push or pull” force is to “deform” the body shape or ”change its velocity ” <math>^1</math> . 

- The “push or pull” forces are exerted when one uses ropes, springs, collisions...; i.e. in all situations where there is a contact between the source of action and the object that undergoes the force. 

The physics development showed the existence of forces exerted without contact (Ex: Magnetic, electrostatic, gravitational) between the source and the object that undergoes the force.
 
Meanwhile, no matter what type it is (contact or no contact), the effect of a force exerted on a body is the same: deformation, velocity change or both. So, a best definition of force would be related with its effects. One says that a force is exerted on a body when it is deformed, changes its motion or both of them.
Note that this definition of force is seen only from the point of view of the body that suffers it. 
Important Note: Dynamics neglects the deformations
and deals only with changes in the motion. 

 
- Physics deals with measurable quantities. How to measure force magnitude? By use of the normal method: a) define a force unit b) define a procedure for comparison of different forces. 
A simple procedure is by use of strings elasticity; spring extension is proportional to the exerted force at its end. One may select a standard spring and define the force unit by referring to spring extension by a unit length, say 1m. Due to proportionality, a two units force would produce a 2m extension on the standard spring and so on. So, in principle, there is no problem to define forces numerically. Unfortunately the described method has two relevant drawbacks. 
1-''It is impossible to produce a standard spring that keeps its elasticity unchanged forever''. 
2-''One cannot use springs for measuring forces on atomic or interplanetary scales.''
''So, one decided to define the unit of force through the second law of Newton''. This way it is a derived physical unit.

[[image:Kresh_Dyn_1.PNG|right]] 

Is the force a vector or scalar quantity?
One can verify easily that the force is a vector quantity by using three spring scales with tied tiles and oriented along different angles and using the vectors’ rules to explain the fact that the point P is at rest.
 
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<math>^1</math> ''The special case of a body at rest corresponds to velocity equal to zero''. 
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====MASS====

We know that the inertia is the propriety of bodies to resist to the changes on their velocity.
The mass is the physical quantity that measures the inertia of a body.
So, the mass of a body tells how difficult is to change its velocity (its magnitude, its direction or both). To achieve the same velocity change, a bigger effort is required in case of objects with bigger mass.

From the definition, it comes out that the mass is an intrinsic characteristic
of the whole body and it does not depend on direction. 
So, it is a scalar physical quantity.

How to measure the magnitude of mass?

By use of the normal method: 

a) define a mass unit.

 
''The standard in the International Office for Measurement Units in Paris''
''is chosen as the unit mass 1kg.'' 

b) Define a procedure for comparison of different masses.
 
The following is a direct procedure for mass measurement.
 One fixes two objects with masses <math>m_A</math>, <math>m_B</math> on two equal massless pucks<math>^2</math> at rest on two sides of a string kept compressed by use of a string. This set is placed on an air table. The air cushion avoids the friction and balances the gravity. Once one cuts the string, the spring extends and produces two forces with the same magnitude on its sides.
[[image:Kresh_Dyn_2.PNG|left]] 
The pucks, that were initially at rest (vo=0) will start to move with velocities <math>\bar v_A</math>, <math>\bar v_B</math> .

One can measure easily these velocities that are equal to the change of velocity itself. As the change of velocity is proportional to the inverse of mass, one gets
 


<math>\left | \bar v_A \right \vert \backsim \frac {1}{m_A}</math>

<math>\left | \bar v_B \right \vert \backsim \frac {1}{m_B}</math> 

and 

<math>\frac{m_A}{m_B} = \frac{\left | \bar v_B \right \vert}{\left | \bar v_A\right \vert}</math> 

and
 

<math>{m_A}= {m_B}\frac{\left | \bar v_B \right \vert}{\left | \bar v_A \right \vert}</math> 

In principle, one can measure the mass of different objects by this method using a standard unit mass <math>m_B</math> = 1kg and the last expression 

''This method for mass measurements is perfect but it is not simple for practical use. Actually, one measures the mass of objects through weight measurements by use of a balance. The mass estimation via weight measurement is very simple and the precision is good enough for general-purpose measurements. ''
 
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<math>^2</math> Insignificant with respect to <math>m_1</math>, <math>m_2</math>