Chapter 1 Physical Quantities and Units « keterehsky

5/30/12 Chapter 1 Physical Quantities and Units « keterehsky

1/13https://keterehsky.wordpress.com/2011/06/22/chapter-1-physical-quantities-and-units/

keterehsky

Categories

Astronomi (17)Berita (7)Fizik (29)My Nikon and Me (4)Uncategorized (100)

Archives

May 2012February 2012July 2011June 2011May 2011April 2011March 2011February 2011January 2011October 2010September 2010August 2010May 2010April 2010March 2010February 2010January 2010September 2009August 2009

Chapter 1 Physical Quantities and Units

Posted on 22/06/2011 by amimo5095Physic is simple

Simplicity is everything

Introduction

What is physic ?

• Definition of physics – derives from Greek word means nature.

• Each theory in physics involves:

(a) Concept of physical quantities.

(b) Assumption(andaian) to obtain mathematical model.

(c) Relationship between physical concepts.

- proportional (berkadar langsung)

(d) Procedures to relate mathematical models to actual measurements from experiments.

(e) Experimental proofs to devise explanation to nature phenomena.

1.1 Basic Quantities and International System of Units (SI units)

> Physical quantity

A physical quantity is a quantity that can be measured. Physical quantity consist of a



numerical magnitude and a unit.

Example:

250 ml (magnitude and unit)

> Basic quantity

Quantity that cannot be derived (terbit) from other quantities. This quantity is important because it

- can be easily produced

- does not change its magnitude

- is internationally accepted

> SI units

The unit of a physical quantity is the standard size used to compare different magnitudes

of the same physical quantity.

> Systems of units

Several systems of units have been in use. Example:

- The MKS (meter-kilogram-second) system

- The cgs (centimeter-gram-second) system

- British engineering system: foot for length, pound for mass and second for time.

Today the most important system of unit is the Systems International or Sl units.

Basic quantity and the SI base units

• Physical quantities can be divided into two categories:

1. basic quantities and

2. derived quantities.

The corresponding units for these quantities are called base units and derived units.

• In the interest of simplicity, seven basics quantities, consistent with a full description of the

physical world, have been choosen.

Basic quantitySI baseunit

Symbol

Dimension

(basequantitysymbol)

Definition

Length (l) Metre m L

The meter is the length of

the path travelled by light

in vacuum during a time

interval of (http://keterehsky.files.wordpress.com/2011/06/clip_image002.gif)

of a second.

Mass (m) Kilogram kg M

Mass of platinum-iridium

cylinder kept at International

Bureau of Weights and

Measures, Paris.

Time (t) Second s T

One second is the time taken

for 9 192 631 770 vibrations

of the light emitted by a

caesium-133 atom.

Constant current in two



Electric

current (I)Ampere A A

straight parallel conductors

of infinite length and

negligible cross-sectional

area that produces a force

of 2 x 10-7 newton per metre

length of the conductors.

Thermodynamic

temperature (T)Kelvin K q

1 kelvin is the fraction

(http://keterehsky.files.wordpress.com/2011/06/clip_image004.gif)of the

thermodynamic temperature

of triple point of water.

Quantity of

matter (n)Mole Mol N

Quantity of matter which

has the same number of

atoms as 0.012 kilogram of

carbon-12.

Light intensity Candela cd I

Prefixes(Imbuhan)

•For very large or very small numbers, we can use standard prefixes with the base units.

Prefix tera giga mega kilo deci centi mili micro nano pico

Factor 102 109 106 103 10-1 10-2 10-3 10-6 10-9 10-12

Symbol T G M k d c m µ n P

Example:

2 x 10-7 —- 2 x 10-6 x 10-1 —- 2m x 10-1 – 0.2m

Derived quantities and derived units

•Derived quantity

Quantity that derived from basic quantities through multiplication and division.

•For example,

Derived quantity Formula Derived unit

Area length x length m2

Volume length x length x length m3

Density

(http://keterehsky.files.wordpress.com/2011/06/clip_image006.gif)kg m-3

Velocity

(http://keterehsky.files.wordpress.com/2011/06/clip_image008.gif)m s-1

Acceleration

(http://keterehsky.files.wordpress.com/2011/06/clip_image010.gif)m s-2

Frequency

Momentum

Force

Pressure

Energy

Ø The derived unit change



Example :

7854 kg m-3 change into g cm-3

(http://keterehsky.files.wordpress.com/2011/06/clip_image012.gif) —-

(http://keterehsky.files.wordpress.com/2011/06/clip_image014.gif) —– (http://keterehsky.files.wordpress.com/2011/06/clip_image016.gif)–

7.854 g cm-3

1.2 Dimensions(Dimensi) and Physical Quantities

> The dimension of a physical quantities is the relation between the physical quantity and the base quantities.

‘[ ]’ The dimension of (pronounce its loudly)

Example [v] “the dimension of velocity” , this means that the base quantities in the velocity.

Example 1

Write the dimensions for the following physical quantity

(a) Acceleration(pecutan),a

[a],the dimension for the acceleration

[a]= (http://keterehsky.files.wordpress.com/2011/06/clip_image018.gif)= (http://keterehsky.files.wordpress.com/2011/06/clip_image020.gif)=LT-2

(b) Density

[r]= (http://keterehsky.files.wordpress.com/2011/06/clip_image022.gif)=ML-3

(c) Force

[F]=M LT-2

Use of dimensions

•To check the homogeneity of physical equations

Concept of homogeneous

•The dimensions on both sides of an equation are the same.

•Those equations which are not homogeneous are definitely wrong.

•However, the homogeneous equation could be wrong due to the incomplete or has extra terms.

•The validity of a physical equation can only be confirmed experimentally.

•In experiment, graphs have to be drawn then. A straight line graph shows the correct equation and the non linear graph is not the correct equation.

•Deriving a physical equation

•An equation can be derived to relate a physical quantity to the variables that the quantity depends on.

Example 2

Determine the homogeneous of the equation.

v2 =u2 +2as

Left Right

[v]2 [u]2 +2[a][s]

L2 T-2 L2 T-2 + L2 T-2

L2 T-2 L2 T-2

homogenous

Example 2

(a)Diberi persamaan yang berikut tentang aliran cecair dalam sebatang paip mendatar

(a) (http://keterehsky.files.wordpress.com/2011/06/clip_image024.gif)



(b) (http://keterehsky.files.wordpress.com/2011/06/clip_image026.gif)

(c) (http://keterehsky.files.wordpress.com/2011/06/clip_image028.gif)

Dimana W,X,Y berdimensi sama dengan dimensi tekanan, A,B,C mewakili pemalar tak berdimensi

g mewakili pecutan graviti

T mewakili tegangan permukaan cecair (dimensinya MT-2)

r mewakili ketumpatan cecair

v mewakili halaju cecair

p mewakili perubahan tekanan

Tentukan kehomogenan persamaan di atas.

(b) Dibawah ialah bacaan-bacaan bagi p dan v :

p (Nm-2) 2.0 ´ 103 2.0 ´ 103 2.0 ´ 103 2.0 ´ 103 2.0 ´ 103

v (m s-2) 1.0 1.4 1.6 1.9 2.1

Dengan menggunakan bacaan-bacaan ini,

(i) cari persamaan yang betul

(ii) cari pemalar bagi persamaan yang betul itu dengan menggunakan maklumat berkenaan

[r = 1.0 ´ 103 kg m-3, T = 7.4 ´ 10-2 N m-1]

Example 3

Use the dimension analysis to obtain an expression which shows how the momentum p depends ‘ on the force; F, mass; m and the length, l.

1.3 Scalar and Vectors

Rujukan: http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l1a.html(http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l1a.html)

> A scalar quantity is a physical quantity which has only magnitude. For example, mass, speed (laju), density, pressure, ….

> A vector quantity is a physical quantity which has magnitude and direction. For example, force, momentum, velocity (halaju), acceleration ….

Graphical representation of vectors

(http://keterehsky.files.wordpress.com/2011/06/clip_image029.gif)•A vector can be represented by a straight arrow,

The length of the arrow represents the magnitude of the vector.

The vector points in the direction of the arrow.

Basic principle of vectors

• Two vectors P and Q are equal if:

a) Magnitude of P = magnitude of Q (b) Direction of P = direction of Q

• When a vector P is multiplied by a scalar k, the product is k P and the direction remains the same as P.

The vector -P has same magnitude with P but comes in the opposite direction.

Sum of vectors

Method 1: Parallelogram of vectors

It two vectors (http://keterehsky.files.wordpress.com/2011/06/clip_image031.gif) and (http://keterehsky.files.wordpress.com/2011/06/clip_image033.gif) are represented in magnitude and direction by the adjacent sides OA and OB of a

parallelogram OABC, then OC represents their resultant(paduan).

(http://keterehsky.files.wordpress.com/2011/06/clip_image035.jpg)

Method 2: Triangle of vectors

•Use a suitable scale to draw the first vector.

•From the end of first vector, draw a line to represent the second vector.



•Complete the triangle. The line from the beginning of the first vector to the end of the second vector represents the sum in magnitude and direction.


Example 4

A kite flies in still air is 4.0 ms-1. Find the magnitude and direction of the resultant velocity of the kite when the air flows across perpendicularly(serenjang) is

2.5 ms-1. If the distance of the kite is 30 m,

what is the time taken for the kite to fly? Calculate the height of the kite from the ground.

(http://keterehsky.files.wordpress.com/2011/06/clip_image0381.gif)

Vector 1- direction (yes)

2- magnitude (c2=a2+b2)

c= 4.7m s-1

Principles of vectors

Relative velocity

Let us look at two cases: VA = 10 ms-1 VB = 3 ms-1.

Case one

The velocity of A relative to B = (VA – VB)

= (10- 3) ms

= 7 ms -1 (in forward direction).

Case two

The velocity of B relative to A = (VB – VA)

= (3 – 10) ms

= -7 ms -1 (in backwards direction).

We observe that(VB – VA) and (VA – VB) are same magnitude but different direction.

Method 3 : Mathematical Method

•Resolving(leraian) vector

A vector R can be considered as the two vectors. R refers to the resultant vectors. There are two mutually perpendicular component Rx and Ry


Example 5

The figure shows 3 forces F1, F2 and F3 acting on a point O. Calculate the resultant force and the direction of resultant.




F1 F2 F3

magnitude 3N 5N 4N

Direction

(http://keterehsky.files.wordpress.com/2011/06/clip_image042.gif) (http://keterehsky.files.wordpress.com/2011/06/clip_image043.gif)(http://keterehsky.files.wordpress.com/2011/06/clip_image0441.gif)

degree 0° 150° 240°

Resolving X-axis

F1x=+3N

Y-axis

F1y = 0

X-axis

F2x=-4.3N

Y-axis

F2y =2.5N

1.4 Errors (Uncertainty)

> There are two main types of errors: systematic error and random error.

> Systematic error

Characteristics of systematic error in the measurement of a particular physical quantity:

-Its magnitude is constant.

-It causes the measured value to be always greater or always less than the true value.

Corrected reading = direct reading – systematic error

Sources of systematic error:

- Zero error of instrument.

- Incorrectly calibrated scale of instrument.

- Personal error of observer, for example reaction time of observer.

- Error due to certain assumption of physical conditions of surrounding

for example, g = 9.81 ms-2

Systematic error cannot be reduced or eliminated by taking repeated readings using the same method, instrument and by the same observer.

> Random error

Characteristics of random error:

- It’s magnitude is not constant.

- It causes the measured value to be sometimes greater and sometimes less than the true value.

Corrected reading = direct reading ± random error

The main source of random error is the observer.

The surroundings and the instruments used are also sources of random errors.

Example of random error:

- Parallax error due to incorrect position of the eye when taking reading

Parallax error can be reduced by having the line of sight perpendicular to the scale reading.

- Error due to the inability to read an instrument beyond some fraction of the smallest division

Reading are recorded to a precision of half the smallest division of the scale.

Random errors can be reduced by taking several readings and calculating the mean.

1.4.1 The uncertainty of the instrument is shown in the table below

Instruments

Uncertainty

(Absolute/actual)Example of readings

Millimeter rule 0.1 cm (50.1 ± 0.1)cm

Vernier caliper 0.01 cm (3.23 ± 0.01)cm



Micrometer screwgauge

0.01 mm (2.63 ± 0.01)mm

Stopwatch(analogue)

0.1 s (1.4 ± 0. 1 )s

Thermometer 0.5 °C (28.0 ± 0.5)°C

Ammeter (0 – 3A) 0.05 A (1.70 ± 0.05)A

Voltmeter (0 – 5V) 0.05 V (0.65 ± 0.05)V

Primary data and secondary data

• Primary data are raw data or readings taken in an experiment. Primary data obtained using the same instrument have to be recorded to the same degree ofprecision i.e to the same number of decimal places.

• Secondary data are derived from primary data. Secondary data have to be recorded to the correct number of significant figures. The number of significantfigures for secondary data may be the same (or one more than) the least number of significant figures in the primary data. Measurement play a crucial role inphysics, but can never be perfectly precise.

It is important to specify the uncertainty or error of a measurement either by stating it directly using the ± notation, and / or by keeping only correct numberof significant figures.

Example: 51.2 ± 0.1

Processing significant figures

• Addition and subtraction

When two or more measured values are added or subtracted, the final calculated value must have the same number of decimal places as that measured valuewhich has the least number , of decimal places.

Example

1. a = 1.35 cm + 1.325 cm

= 2.675 cm

= 2.68 cm

2. b = 3.2 cm – 0.3545 cm

= 2.8465 cm

= 2.8 cm

3. c = (http://keterehsky.files.wordpress.com/2011/06/clip_image046.gif)

= 1.142 cm

= 1.14 cm

· Multiplication and division

• When two or more measured values are multiplied and/or divided, the final calculated value must have as many significant figures as that measured valuewhich has the least number of significant figures.

Example

1. Volume of a wooden block = 9.5 cm x 2.36 cm x 0.515 cm

= 11.5463 cm3

= 12 cm3

2. If the time for 50 oscillations of a simple pendulum is 43.7 s, then the period of oscillation = 43.7 ÷ 50 = 0.874 s

3. The gradient of a graph (http://keterehsky.files.wordpress.com/2011/06/clip_image0481.gif)

Note: Sometimes the final answer may be obtained only after performing several intermediate calculations. In this case, results produced in intermediatecalculations need not be rounded off. Round only the final answer.

1.4.2 Analysing error/uncertainty of a mean value

- specifically error analysing is refer to error that cause by repetition and a combining measurement to produce a derive quantity.

- Meaning that if we want to measure a volume of cube, of course we cannot just used a single measurement then we will get the answer. First we have tomeasure the length with the ruler together with the width and the height. The we need to feed in the formula length x width x height to get the volume.

- While doing the measurement and caculating the answer actually we have continually increasing the error.

- It is a good idea to mention the uncertainty for every measurement and calculation.

- In this subtopic we deal with the repetition reading or data. It’s known that if we have more than one reading so the true value is the mean of the reading.



- Mean value for a is (http://keterehsky.files.wordpress.com/2011/06/clip_image0501.gif)

- Mean value of uncertainty of a, (http://keterehsky.files.wordpress.com/2011/06/clip_image0521.gif)should be caculated this way

1. Calculated the deviation of every data given:







2. Find the sum of deviation


3. find the mean of deviation (http://keterehsky.files.wordpress.com/2011/06/clip_image066.gif)

It’s known that the mean deviataion is equally the same as the uncertainty of the mean value(true value).

Or


Working example:

Diameter ,d of a wire was measured several time to reduce the uncertainty and the reading is given in the table below. Find the true value(mean value) andthe uncertainty of the diameter.

d/mm 1.55 1.52 1.54 1.53 1.54 1.53




1.4.3 Analysing error/uncertainty combining measurement or equation.

1. Actual Value

- is in the scale reading (pointer reading) of an instrument.(single reading)

Or

- is in the mean value.(of the repetition reading)

2. Fractional and percentage error,

(a) The fractional error of R : (http://keterehsky.files.wordpress.com/2011/06/clip_image0761.gif)

(b) The percentage error of R : (http://keterehsky.files.wordpress.com/2011/06/clip_image078.gif)

3. Consequential Uncertianties/Error- to state the error of a derive quantities

Given

R 1 ± DR1 = Data ± Absolute Data Error = 51.2 ± 0.1

R 2 ± DR2 = Data ± Absolute Data Error = 30.1 ± 0.1

(a) Addition

W = R1 + R2 = 51.2 + 30.3 = 81.3

DW = DR1 + DR2 = 0.1 + 0.1 = 0.2

So W ± DW = 81.3 ± 0.2



(b) Subtraction

S = R1 – R2 = 51.2 – 30.3 = 21.1

DS = DR1 + DR2 = 0.1 + 0.1 = 0.2

So S ± DS = 21.1 ± 0.2

(c) Product

P = R1 ´ R2 = 51.2 ´ 30.3 =1541.12

From (http://keterehsky.files.wordpress.com/2011/06/clip_image0801.gif)




P ± DP = 1541.12 ± 7.71

(d) Quotient


From (http://keterehsky.files.wordpress.com/2011/06/clip_image0901.gif)




Q ± DQ = 1.70 ± 0.01

Working example:

Find v and the uncertainty in v


and a =(1.83±0.01)m, b=(1.65 ±0.01) m, d=(0.00106±0.00003)m ,

q = (4.28 ± 0.05) s and T = (3.7 ± 0.1) x 103 s.

solution

First calculate the percentage uncertainties in each of the 4 terms:

(a – b) = (0.18±0.02)m 11%

d = (0.001 06 ± 0.000 03) m 3%

q = (4.28 ± 0.05) s . 1.2%

T = (3.7±0.1) x 103 s 3%

The uncertainty in (a – b) is now very large, although the readings themselves have been taken carefully. This is always the effect when subtracting two nearlyequal numbers.

The percentage uncertainty in d2 will be twice the percentage uncertainty in d;

The percentage uncertainty in (http://keterehsky.files.wordpress.com/2011/06/clip_image100.gif) will be half the percentage uncertainty in T because a

square root is a power of (http://keterehsky.files.wordpress.com/2011/06/clip_image102.gif).

This gives:



Uncertainty percentage in v = 11% + 2(3%) + 1.2% + (http://keterehsky.files.wordpress.com/2011/06/clip_image104.gif)(3%) = 19.7% ≈ 20%

This gives v = (7.8 ± 1.6) x 10-11 m3 s-1, a rather uncertain result which would be better expressed as:

v = (8 ± 2) x 10-11 m3 s-1

the rules for uncertainties therefore :

addition and subtraction ADD absolute uncertainties

multiplication and division ADD percentage uncertainties

powers Multiply the percentage uncertainty by the power

Note : There are some circumstances where the uncertainty in the final value is best found by working the problem through twice , once with the readings astaken and once with the limiting values which will give the maximum result. Equations containing trigonometrical ratios, or exponentials, or equations inwhich some of the terms appear both on the top and the bottom of the expression, such as

(http://keterehsky.files.wordpress.com/2011/06/clip_image106.gif) are best dealt with this way.

Example 5

The diameter of a cone is (98 ± 1)mm and the height is (224 ± 1 )mm. What is:

(a) The absolute error of the diameter.

(b) The percentage error of the diameter.

(c) The volume of the cone. Give your answer to the correct number of significant number.

Example 6

Discuss the ways of minimizing systematic and random errors

Example 7

The period of a spring is determined by measuring the time for 10 oscillations using a stopwatch. State a source of:

(a) Systematic error

(b) Random error

1.4.4. Method to find uncertainty/error from a graph


Figure 1

where n is the number of points plotted.

1. The usual quantities that are deduced from a straight line graph are

(a) the gradient of the graph m, and the intercept on the y-axis or the x-axis

(b) the intercepts on the axes.

First calculate the coordinates of the centroid using the formula

(http://keterehsky.files.wordpress.com/2011/06/clip_image110.gif) where n is the number of sets of readings.

2. The straight line graph that is drawn must pass through the centroid Figure . The best line is the straight line which has the plotted points closest to it. Thisline will give (http://keterehsky.files.wordpress.com/2011/06/clip_image112.gif)the best gradient together with c.

3. Two other straight lines, one with the maximum gradient (http://keterehsky.files.wordpress.com/2011/06/clip_image114.gif) and another with the

least gradient (http://keterehsky.files.wordpress.com/2011/06/clip_image116.gif), are then drawn. For a straight line graph where the intercept is not the

origin , the three lines drawn must all pass through the centroid. Here also we can find (http://keterehsky.files.wordpress.com/2011/06/clip_image118.gif)

and (http://keterehsky.files.wordpress.com/2011/06/clip_image120.gif)

4. To find the uncertainty for the gradient and intercept used this equation



(http://keterehsky.files.wordpress.com/2011/06/clip_image122.gif) and (http://keterehsky.files.wordpress.com/2011/06/clip_image124.gif)

Working Example

Table 1.7 shows the data collected in an experiment to determine the acceleration due to gravity using a simple pendulum. The time t for 50 oscillations of thependulum is measured for different lengths l of the pendulum. The period T is calculated using


from the theory of the simple pendulum, the period T is related to the length l, and the acceleration due to gravity g by the equation


Hence, the acceleration due to gravity, (http://keterehsky.files.wordpress.com/2011/06/clip_image130.gif)

A straight line graph would be obtained if a graph of (http://keterehsky.files.wordpress.com/2011/06/clip_image132.gif) against (http://keterehsky.files.wordpress.com/2011/06/clip_image134.gif) is plotted.

Note the various important characteristics when tabulating the data as shown in Table


Table 1

(a) Name or symbol of each quantity and its unit are stated in the heading of each column. Example: Length and cm, and T(s). The uncertainty for theprimary data, such as length and t time for 50 oscillations, is also written. Example: (l ± 0.05) cm and (t ± 0.1)s.

(b) All primary data, such as length and time, should be recorded to reflect the precision of the instrument used.

For example, the length of the pendulum l is measured using a metre rule. hence it should be recorded to two decimal places of a cm, that is 10.00 cm, and not10 cm or 10.0 cm.

The time for 50 oscillations t is recorded to 0.1 s, that is 32.0 s and not 32 s.

The average value of t is also calculated to 0.1 s. The average value of 31.9 s and 32.0 s is recorded as 32.0 s and not 31.95 s.

(c) The secondary data such as T and T2, are calculated from the primary data. Secondary data should be calculated to the same number of significant figures

as I hat in the least accurate measurement. For example, T and T2, are calculated to three significant figures, the same number of significant figures as thereadings of t.

(d) For a straight line graph, there should be at least six point plotted. If the graph is a curve, then more points should be plotted, especially near themaximum and minimum points.


Table 2

Note that the graph is plotted with the assumption that the origin (0, 0) is a point.

The x-coordinate of the centroid = (http://keterehsky.files.wordpress.com/2011/06/clip_image140.gif)

= (http://keterehsky.files.wordpress.com/2011/06/clip_image142.gif)

= 45.0cm

The y-coordinate of the centroid = (http://keterehsky.files.wordpress.com/2011/06/clip_image144.gif)



= (http://keterehsky.files.wordpress.com/2011/06/clip_image146.gif)

= 1.80s2

from the equation (http://keterehsky.files.wordpress.com/2011/06/clip_image1301.gif)


Hence a graph of T 2 against l is a straight line, passing through the origin, and gradient,


From the graph,

gradient of best line, (http://keterehsky.files.wordpress.com/2011/06/clip_image150.gif)

Maximum gradient, (http://keterehsky.files.wordpress.com/2011/06/clip_image152.gif)

Minimum gradient, (http://keterehsky.files.wordpress.com/2011/06/clip_image154.gif)

Absolute uncertainty in the gradient,


Fractional uncertainty in the gradient


percentage uncertainty in gradient


Acceleration due to gravity, (http://keterehsky.files.wordpress.com/2011/06/clip_image162.gif)

Hence the percentage uncertainty in g is the sum of the percentage error in m only because 4p2 is a constant.

Therefore percentage uncertainty in gravity,Dg = S uncertainty percentage = 1.88% according to above equation

Hence acceleration due to gravity,

Written in percentage uncertainty

g = (9.870±1.88%) m s2

also can be write in absolute uncertainty


g = (9.9 ± 0.2) m s2 Since there is error in the second significant figure, the value of g is given to two significant figures.

Filed under: Fizik

« chapter 20 Direct current circuits Chapter 21 Magnetic fields »

Blog at WordPress.com. Theme: Digg 3 Column by WP Designer.

Documents

Chapter 1 Physical Quantities and Units « keterehsky