Chapter 1 Physical Quantities and Units
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.
Basic Quantities
• In the interest of simplicity, seven basics quantities1, consistent with a full description of the
physical world, have been chosen.
Basic quantity
Symbol
Dimension Definition2
SI units
(base
quantity
symbol)
l
Length
L
length most commonly refers to the
longest dimension of an object
Meter
Mass , more specificallyinertial
Mass
m
M
mass, can be defined as a
quantitative measure of an object’s
Kilogram
resistance to acceleration
Time is a dimension in which
events can be ordered from the
Time
t
T
past through the present into the
future, and also the measure of
Second
durations of events and the
intervals between them
Electric current is a flow of electric
Electric
I
current
A
charge through a conductive
Ampere
medium
Temperature is a physical property
Thermodynamic
temperature
T
q
of matter that quantitatively
expresses the common notions of
Kelvin
hot and cold.
Amount of substance is a
Amount of
standards-defined quantity that
substances,
n
Quantity of
N
matter
measures the size of an ensemble
of elementary entities, such as
Mole
atoms, molecules, electrons, and
other particles
luminous intensity is a measure
of the wavelength-weighted power
Luminous
Iv
intensity
J
emitted by a light source in a
Candela
particular direction per unit solid
angle
Base Units
There are only seven base unit3 in SI system.
SI Base
units
Symbol Definition1
"The metre is the length of the path travelled by light in vacuum during a
Metre
m
time interval of 1/299 792 458 of a second."
17th CGPM (1983, Resolution 1, CR, 97)
Kilogram
kg
"The kilogram is the unit of mass; it is equal to the mass of the international
prototype of the kilogram."
3rd CGPM (1901, CR, 70)
"The second is the duration of 9 192 631 770 periods of the radiation
corresponding to the transition between the two hyperfine levels of the
Second
ground state of the caesium 133 atom."
13th CGPM (1967/68, Resolution 1; CR, 103)
s
"This definition refers to a caesium atom at rest at a temperature of 0 K."
(Added by CIPM in 1997)
"The ampere is that constant current which, if maintained in two straight
parallel conductors of infinite length, of negligible circular cross-section, and
Ampere
A
placed 1 metre apart in vacuum, would produce between these conductors a
force equal to 2 × 10−7 newton per metre of length."
9th CGPM (1948)
"The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of
the thermodynamic temperature of the triple point of water."
13th CGPM (1967/68, Resolution 4; CR, 104)
Kelvin
"This definition refers to water having the isotopic composition defined
K
exactly by the following amount of substance ratios: 0.000 155 76 mole of 2H
per mole of 1H, 0.000 379 9 mole of 17O per mole of 16O, and 0.002 005 2
mole of 18O per mole of 16O."
(Added by CIPM in 2005)
"1. The mole is the amount of substance of a system which contains as
many elementary entities as there are atoms in 0.012 kilogram of carbon 12;
its symbol is ‘mol.’
2. When the mole is used, the elementary entities must be specified and
Mole
Mol
may be atoms, molecules, ions, electrons, other particles, or specified
groups of such particles."
14th CGPM (1971, Resolution 3; CR, 78)
"In this definition, it is understood that unbound atoms of carbon 12, at rest
and in their ground state, are referred to."
(Added by CIPM in 1980)
"The candela is the luminous intensity, in a given direction, of a source that
Candela
emits monochromatic radiation of frequency 540 × 1012 hertz and that has a
cd
radiant intensity in that direction of 1/683 watt per steradian."
16th CGPM (1979, Resolution 3; CR, 100)
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
Derive from base quntity of
Derived unit
Area
length x length
m2
Volume
length x length x length
m3
Density
kg m-3
Velocity
m s-1
Acceleration
m s-2
Frequency
s-1/hz
Momentum
Mass x velocity
Kg ms-1
Force
Mass x acceleration
Kg ms-2
Pressure
N m-2
Energy
Kg m2 s-2
Ø The derived unit change
Example :
7854 kg m-3 change into g cm -3
—-
—–
– 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) or the base quantity of
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
(b) Density
(c) Force
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
Example 3
(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)
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
1.5 ´ 103
1.2 ´ 103
0.7 ´ 103
0.3 ´ 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 4
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
> 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 ….
In most cases in physic, the physic quantity is express in vector. If the number(magnitude) can be
operated through Subtract, Add, multiplication and fraction. Then the vector also can be threat the same
way except fraction, but it’s have to follow the rule that govern them.
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.
Principles of vectors
(a) Substitute of Vector (Relative of)
Relative velocity
Let us look at two cases: VA = 10 ms-1 (faster) VB = 3 ms-1. (slower)
Case one
The velocity of A relative to B = (VA – VB) (comparing faster toward slower)
= (10- 3) ms
= 7 ms -1 (in forward direction).(mean that A is 7 ms
-1
faster than B)
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.
(b) Sum of vectors (Resultant of)
If there are two or more vector , these vector can be add to form a single vector called
a Resultant vector.
To solve the problem involving vectors in two dimension, we usually used any one of these method
depend on the information given.
Method 1: Parallelogram of vectors
It’s the drawing method. The drawing of the parallelogram need to be draw according scale and angle
given in the question. The instrument used for this drawing are:
(a) ruler
(b) protractor
(c) sharp pencil
It two vectors
and
are represented in magnitude and direction by the adjacent sides OA and
OB of a parallelogram OABC, then OC represents their resultant(paduan).
This method is used when there are information about angle and magnitudes of the vector.
Method 2: Triangle of vectors and polygon of vector
It’s the drawing method. The drawing of the vectors need to be draw according scale and angle given in
the question. The instrument used for this drawing are:
(d) ruler
(e) protractor
(f) sharp pencil
•Use a suitable scale to draw the first vector.
•From the end of first vector, draw a line to represent the second vector. (attaching the head with the it’s
tail)
•Complete the triangle/polygon. The line from the beginning of the first vector to the end of the second
vector represents the sum in magnitude and direction.
Example 5
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.
Example 6
Figure 1.8.1
Five coplanar forces act on a particle, as drown in Figure 1.8.1. Draw a scaled force polygon for these
forces. State the magnitude and direction of the resultant of these forces.
Method 3 : Component Method
It’s is a calculation method , because every vector can be replace into x-component and y-component.
Replacing a single vector into it’s components is called Resolving.
To determine the resultant of the vector using this method, it’s need to follow these four keyword
carefully.
1. Axis
2. Resolve vector
3. add vector component
4. Resultant
Axis
Need to be determine before resolving the vector.
•Resolving(leraian) vector
The vector that is not on any axis have to be resolve into it’s component. Resolving vector mean resolving
:
(a) magnitude
(b) Direction
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
Add Vector Component
and
Only the same axis component can be added.
Resultant
Magnitude,
and Direction of R,
Example 7
The figure shows 3 forces F1, F2 and F3 acting on a point O. Calculate the resultant force and the direction
of resultant.
(d) Multiplication of vector
It’s have been discuss about subtraction and addition of the vector. From subtraction and addition of
vector we can explain most of the physical quantity. Now is about multiplication of vectors. When two
vectors were multiply the result is called product.
There are two kind of product produced :
1. Dot Product
2. Cross Product
Dot Product
The dot product is fundamentally a projection.
The dot product of a vector with a unit vector is the projection of that vector in the direction given by the
unit vector. This leads to the geometric formula
Furthermore,it follows immediately from the geometric definition that two vectors are orthogonal if and
only if their dot product vanishes, that is
Cross Product
The cross product is fundamentally a directed area.
whose magnitude is defined to be the area of the parallelogram?. The direction of the cross product is
given by the right-hand rule, so that in the example shown
points into the page.
In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation
conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist
John Ambrose Fleming in the late 19th century.
Unit vector
When comes into multiplying vector it’s easier to used component method. The basis for the coordinate
system used in vector notation is unit vector.
in mathematics, a unit vector in a normed vector space is a vector whose length is 1 (the unit length). A
unit vector is often denoted by a lowercase letter with a "hat", like this: (pronounced "i-hat"),
The ,
and
is use in 3D or cartesian coordinate and :
and
and
.
is use in Euclidean space.
The operation on the vector will be much more faster compared to the drawing method.
A vector can be represent in component method as
meaning that a vector A is stretch from
origin to point (2,3) in Euclidean space.
1.4 Metrology
Metrology is the science of measurement and its application.
Terminology related to measurement uncertainty is not used consistently among experts. To avoid further
confusions lets refer to BIPM-VIM(International Vocabulary of Basic and General Terms in Metrology) and
GUM (Guide to the expression of uncertainty in measurement).
1.4.1 Error
VIM define the error as below:
error (of measurement) [VIM 3, 2.16] – measured quantity value minus a reference quantity value
there are two type of error
(a) 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.
(b) 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 Uncertainty is the observer.
The surroundings and the instruments used are also sources of random error.
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 Error can be reduced by taking several readings and calculating the mean.
Error contributes to but is different from Uncertainty
1.4.2 The Uncertainty of the Instrumental
VIM define the Uncertainty as below
uncertainty of measurement [VIM 3, 2.6] non-negative parameter characterizing the dispersion of
the quantity values being attributed to a measurand (quantity intend to measure), based on the
information used and it’s have a statistical concept of standard deviation means.
Instrumental Measurement
When handling the experiment the reading is given by the apparatus used, these apparatus have their
own uncertainty.
instrumental measurement uncertainty(VIM 3, 4.24) – the amount (often stated in the form ±dx) that
along with the measured value, indicates the range in which the desired or true value most likely
lies. Instrumental measurement uncertainty is used in a Type B evaluation of measurement
uncertainty
Here the magnitude of ±dx is called the absolute Uncertainty. Absolute Uncertainty is the smallest scale
of the instrument or half of the smallest scale if it’s can be determine “easily”.
Instruments
Absolute Uncertainty
Example of readings
Millimeter ruler
0.1 cm
(50.1 ± 0.1)cm
Vernier caliper
0.01 cm
(3.23 ± 0.01)cm
0.01 mm
(2.63 ± 0.01)mm
Stopwatch (analogue)
0.1 s
(1.4 ± 0. 1 )s
Stopwatch(Digital)
0.01 s
(1.452 ± 0.01)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
Micrometer screw
gauge
The smaller absolute uncertainty of the instrument is contribute to the high accuracy, precision and
sensitivity of the measuring system of the experiment.
1.4.2 Analysing Uncertainty of the data
- specifically Uncertainty analysing is refer to Uncertainty that cause by repetition measurement to
produce more accurate data.
- Meaning that if we want to measure a mass of cube, of course we cannot just used a single
measurement then we will get the answer. We have to measure the mass with the triple balance beam
more than one time for example 3 time.
- While doing the measurement actually we have continually increasing the Uncertainty.
- 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
- 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
It’s known that the mean deviataion is equally the same as the Uncertainty of the mean value(true value).
Or
Working example on a single quantity :
1. Aim : to determine the diameter, d of a wire
2. Theory : used outer jaw of vernier caliper
3. Precaution : measure more than one reading
4. Choosing Apparatus and Determine the absolute uncertainty:
Uncertainty
Instruments
(Absolute/actual)
Vernier caliper
0.01 cm
5. Manage the reading/data:
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.
i
ii
iii
iv
v
vi
(d±0.01)/cm 1.55
1.52
1.54
1.53
1.54
1.53
6. Determine the quantity and it’s uncertainty
a. Calculating the true value of diameter (mean value) <d>:
b. Calculating the uncertainty of diameter:
So the diameter of a wire should be written (1.54 ± 0.01)cm
Note: calculating the uncertainty this way is refer to a single quantity and not involving with the graph.
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 Uncertainty 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 =
= 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.
1.4.3 Analysing Uncertainty of 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 Uncertainty,
(a) The fractional Uncertainty of R :
(b) The percentage Uncertainty of R :
3. Consequential Uncertianties/Uncertainty- to state the Uncertainty of a derive quantities
Given
R 1 ± DR1 = Data ± Absolute Data Uncertainty = 51.2 ± 0.1
R 2 ± DR2 = Data ± Absolute Data Uncertainty = 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
Q ± DQ = 1.70 ± 0.01
Working example:
1. Aim : to determine the value of B
2. Theory :
B is given by
3. Precaution : B have a combine uncertainty from various apparatus (quantity)
4. Choosing Apparatus and Determine the absolute uncertainty:
Uncertainty
Quantity
Instruments
a,b
meter ruler
1 cm
q
Stopwatch(Digital)
0.01 s
(Absolute/actual)
5. Manage the reading/data:
After the measuring and calculating the uncertainty of the quantity a,b,d,q and T(refer 1.4.2). The true
value (mean value) and the uncertainty of the quantities are witten as below :
a =(1.83±0.01)m,
b=(1.65 ±0.01) m,
d=(0.00106±0.00003)m,
q = (4.28 ± 0.05) s
T = (3.7 ± 0.1) x 103 s.T is
6. Determine the quantity and it’s uncertainty
(a) Find B use the equation given
B = 7.8 x 10-11 m3 s(b) Find the uncertainty of B
1. Fisrt check the equation for addition and subtraction, by applying 1.4.3 no 3 (b) , subtraction so (a – b)
= (0.18±0.02)m
2. Second calculate the percentage uncertainties in each of the 4 terms:
Term
Magnitude and
Fractional
Uncertainty
uncertainty
Uncertainty
percentage
(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 nearly equal numbers.
- The percentage Uncertainty in d2 will be twice the percentage Uncertainty in d;
- The percentage Uncertainty in
a power of
will be half the percentage Uncertainty in T because a square root is
.
This gives:
Uncertainty percentage in B = 11% + 2(3%) + 1.2% +
This gives B = (7.8 ± 1.6) x
10-11
m3
(3%) = 19.7% ≈ 20%
s-1.
the rules for uncertainties therefore :
Operator
Uncertainty
addition and subtraction
ADD absolute uncertainties
multiplication and division ADD percentage uncertainties
powers
Multiply the percentage Uncertainty by the power
Example 8
The diameter of a cone is (98 ± 1)mm and the height is (224 ± 1 )mm. What is:
(a) The absolute Uncertainty of the diameter.
(b) The percentage Uncertainty of the diameter.
(c) state the volume of the cone and it’s uncertainty. Give your answer to the correct number of significant
number.
Example 9
Discuss the ways of minimizing systematic and random Error
Example 10
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/Uncertainty 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 readings5,6.
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. This line will give
the best gradient together with
c.
3. Two other straight lines, one with the maximum gradient
gradient
and another with the least
, 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
1. Aim
To determine the acceleration due to gravity using a simple pendulum.
2. Theory : 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
3. Precaution :
against
is plotted.
The time t for 50 oscillations of the pendulum is measured for different lengths l of the pendulum. The
period T is calculated using
4. Choosing Apparatus and Determine the absolute uncertainty:
Instruments
Uncertainty
(Absolute/actual)
Millimeter ruler
0.1 cm
Stopwatch (analogue)
0.1 s
5. Manage the table
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 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 (absolute
uncetainty) 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 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, then more
points should be plotted, especially near the maximum and minimum points.
Note that the graph is plotted with the assumption that the origin (0, 0) is a point.
The x-coordinate of the centroid =
=
= 50 cm
The y-coordinate of the centroid =
=
= 2.00s2
The coordinate for the centroid is (50cm, 2.00s2)
Graph 1
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 Uncertainty in m only because 4p 2 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 Uncertainty in the second significant figure, the value of g is given to
two significant figures.