chapter 1 physical quantities and

7/28/2019 Chapter 1 Physical Quantities And

1/12

Chapter 1 Physical Quantities and Units

1.1 Introduction

Definition of physics derives from Greek word means nature.

Each theory in physics involves:

(a) Concept of physical quantities.

(b) Assumption obtain mathematical model.(c) Relationship between physical concepts.

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

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

1.2 Physical quantities dan SI units

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

numerical magnitude and a unit.

Physical quantities can be divided into two categories:

1. basic quantities and

2. derived quantities.

(a) Basic quantity

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

- can be easily produced

- does not change its magnitude

- is internationally accepted

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

physical world, have been chosen.

SI units

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

of the same physical quantity.

Basic quantity SI base unit SymbolDimension (base

quantity symbol)

Length (l) Metre m L

Mass (m) Kilogram kg M

Time (t) Second s T

Electric current (I) Ampere A A

Thermodynamic temperature (T) Kelvin K q

Quantity of matter (n) Mole Mol NLight intensity Candela cd I

(b) 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 kg m-3

Velocity m s-1
https://keterehsky.wordpress.com/2011/06/22/chapter-1-physical-quantities-and-units/http://keterehsky.files.wordpress.com/2011/06/clip_image008.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image006.gifhttps://keterehsky.wordpress.com/2011/06/22/chapter-1-physical-quantities-and-units/


2/12

Acceleration m s-2

Prefixes

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 n P

1.2 Dimensions and Physical Quantities

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

quantities.

The symbol for dimensions is [ ]

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] =LT-2

(b) Density

[] =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

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

=
http://keterehsky.files.wordpress.com/2011/06/clip_image018.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image022.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image020.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image018.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image010.gif


3/12

homogenous

Example

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

(a)

(b)

(c)

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)

rmewakili ketumpatan cecair

v mewakili halaju cecair

p mewakili perubahan tekanan

Tentukan kehomogenan persamaan di atas.

(b) Dibawah ialah bacaan-bacaan bagip 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

[p = 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 onthe force; F, mass; m and the length, l.

1.3 Scalar and Vectors

> 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

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 and are represented in magnitude and direction by the adjacent sides OA andOB of a parallelogram OABC, then OC represents their resultant(paduan).
http://keterehsky.files.wordpress.com/2011/06/clip_image033.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image031.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image029.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image028.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image026.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image024.gif


4/12

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.

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 oneThe velocity ofA relative to B = (VA VB)

= (10- 3) ms

= 7 ms -1 (in forward direction).

Case two

The velocity ofB 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 twomutually perpendicular component Rx and Ry
http://keterehsky.files.wordpress.com/2011/06/clip_image0381.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image037.jpghttp://keterehsky.files.wordpress.com/2011/06/clip_image035.jpg


5/12

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

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.
http://keterehsky.files.wordpress.com/2011/06/clip_image0441.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image043.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image042.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image041.jpghttp://keterehsky.files.wordpress.com/2011/06/clip_image039.gif


6/12

> Random error

Characteristics of random error:

- Its 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

InstrumentsUncertainty

(Absolute/actual)Example of readings

Millimeter rule 0.1 cm (50.1 0.1)cmVernier caliper 0.01 cm (3.23 0.01)cm

Micrometer screw

gauge0.01 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 of precision 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 significant figures 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 in physics, 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 number of 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 value which 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


7/12

= 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 value which 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

Note: Sometimes the final answer may be obtained only after performing several intermediate

calculations. In this case, results produced in intermediate calculations 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 to measure 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. Its known that if we have more thanone reading so the true value is the mean of the reading.

- Mean value for a is

- Mean value of uncertainty of a, 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

Its 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) and the uncertainty of the diameter.
http://keterehsky.files.wordpress.com/2011/06/clip_image0681.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image066.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0641.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0621.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0581.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image056.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0541.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0521.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0501.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0481.gif


8/12

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 :

(b) The percentage error of R :

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

R1 DR1 = Data Absolute Data Error = 51.2 0.1

R2 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

P DP = 1541.12 7.71

(d) Quotient

From
http://keterehsky.files.wordpress.com/2011/06/clip_image096.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0941.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0921.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0901.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0881.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image086.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0841.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0821.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0801.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image078.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0761.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image074.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0721.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image0701.gif


9/12

Q DQ = 1.70 0.01

Working example:

Find v and the uncertainty in v

and a =(1.830.01)m, b=(1.65 0.01) m, d=(0.001060.00003)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) = (0.180.02)m 11%

d = (0.001 06 0.000 03) m 3%

q = (4.28 0.05) s . 1.2%

T = (3.70.1) 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 nearly equal numbers.The percentage uncertainty in d2 will be twice the percentage uncertainty in d;

The percentage uncertainty in will be half the percentage uncertainty in T because a square root

is a power of .

This gives:

Uncertainty percentage in v = 11% + 2(3%) + 1.2% + (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 as taken and once with the limiting

values which will give the maximum result. Equations containing trigonometrical ratios, or

exponentials, or equations in which some of the terms appear both on the top and the bottom of the

expression, such as

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
http://keterehsky.files.wordpress.com/2011/06/clip_image106.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image104.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image102.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image100.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image098.gif


10/12

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

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 thestraight line which has the plotted points closest to it. This line will give the best gradient together

with c.

3. Two other straight lines, one with the maximum gradient and another with the least gradient

, 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 and

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

and

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 the pendulum 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,

A straight line graph would be obtained if a graph of against is plotted.

Note the various important characteristics when tabulating the data as shown in Table
http://keterehsky.files.wordpress.com/2011/06/clip_image112.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image118.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image130.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image134.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image132.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image130.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image128.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image126.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image124.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image122.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image120.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image118.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image116.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image114.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image112.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image110.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image108.jpg


11/12

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 the primary 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 theinstrument 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 not 10 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 the readings of t.

(d) For a straight line graph, there should be at least six point plotted. If the graph is a curve, thenmore points should be plotted, especially near the maximum 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 =

=

= 45.0cm
http://keterehsky.files.wordpress.com/2011/06/clip_image142.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image142.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image140.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image138.jpghttp://keterehsky.files.wordpress.com/2011/06/clip_image136.jpg


12/12

The y-coordinate of the centroid =

=

= 1.80s2

from the equation

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,

Maximum gradient,

Minimum gradient,

Absolute uncertainty in the gradient,

Fractional uncertainty in the gradient

percentage uncertainty in gradient

Acceleration due to gravity,

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

is aconstant.

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 = (9.8701.88%) 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.
http://keterehsky.files.wordpress.com/2011/06/clip_image146.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image164.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image162.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image160.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image158.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image156.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image154.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image152.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image150.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image148.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image1281.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image1301.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image146.gifhttp://keterehsky.files.wordpress.com/2011/06/clip_image144.gif

chapter 1 physical quantities and

Documents