Summary
The main purpose of writing this book was to provide a complete and precise knowledge of basic physics. The author’s aim was to provide a comprehensive material with an easy language, so that students can easily understand the concepts of physics. This book does not only help the students of F.Sc, physics diploma but general physics students will also find a lot of helpful information in it. As the author has used easy and clear concepts, therefore, he feels confident that students will appreciate it.
Excerpt
Table Of Contents
5
PHYSICS AND MEASUREMENT
PHYSICS
It is a branch of science which deals the properties of matter and energy, and also their
correlation.
Physics enlightens the natural phenomena in terms of fundamental principles and laws.
BRANCHES
(i)
Mechanics: It is the branch of physics which deals the motion of bodies under the action
of forces.
(ii)
Field Theory: It is the branch of physics which deals the properties, nature and origin of
different fields. Such as gravitational field, electric field and magnetic field etc.
(iii)
Thermodynamics: It is the branch of physics which deals the conversion of heat energy
into mechanical energy.
(iv)
Acoustics: It is the branch of physics which deals the effect of sound on the buildings.
(v)
Optics: It is the branch of physics which deals the nature of light and also deals optical
instruments, such as micro scope, telescope, and spectrometer etc.
(vi)
Electricity and magnetism: It is the branch of physics which deals the properties of
static as well as moving charges and also the properties related to these moving charges.
(vii) Atomic physics: It is the branch of physics which deals the structure and characteristics
of an atom with in region of 10
-10
m.
(viii) Nuclear Physics: It is branch of physics which deals the nucleolus of an atom with in
the region of 10
-15
m.
(ix)
Elementary Particle Physics: Branch of physics which deals the properties of
elementary particle such as mesons, bosons, quarks etc.
(x)
Plasma Physics: Branch of physics which deals the properties of ionized gasses is called
plasma physics.
(xi)
Astrophysics: Branch of physics which deals the properties of heavenly bodies and also
the matter and energy interaction in these bodies are also discussed.
(xii) Solutionid State Physics: It is the branch of Physics which deals the properties of
Solutionid form of matter.
(xiii) Geo Physics: It is the branch of physics which deals the earth structure and its
atmosphere.
(xiv) Bio Physics: It is the branch of physics which deals Biology in term of physics.
6
UNIT
The standard with which things are compared is called Unit. For example Kg, Meter,
second, etc.
MAGNITUDE
The number and unit are collectively called magnitude. For example 5 kg,10 sec, and
15C
o
etc.
PHYSICAL QUANTITIES
The quantities which can be measured are called physical quantities.
TYPES
Physical quantities are of two types.
(1) Fundamental or base units
(2) Derived units.
1. Base Units:
Those physical quantities which are simplest and other physical quantities can be defined
with the help of these quantities are called base units. For example length, mass, time etc.
2. Derived Units:
Those physical quantities which obtained from base quantities are called derived units.
For example Speed, Torque, work etc.
Speed =
Time
Length
SYSTEM OF UNITS
A complete set of units both base and derived units are called system of units.
There are many systems of units are present. For example
(i) M.K.S
(meter kg sec)
(ii) C.G.S
(centimeter gm sec)
(iii) F.P.S
(foot pound sec)
(iv) S.I. System
7
S.I SYSTEM:
This system is adopted by international committee in 1960. This system has three types
of units.
(1)
Base Units
(2)
Supplementary Units.
(3)
Derived Units.
BASE UNITS:
The S.I system consists of seven base units.
1. Length 2. Mass 3. Time 4 electric current
5. Temperature 6. Light intensity
7. Mole (amount of substance)
(1) Length (L):
It is measured in meters.
1 meter: It is the length equal to the 165076373 wave length of orange red radiation emitted by
Krypton-86 atom from transition 2P
10
5d
5
orbital. OR one meter is the distance traveled by
light in vacuum during a time of 1/299792458 sec.
2. Mass (M):
It is measured in kilogram. Its symbol is Kg.
1Kg: the mass of a platinum iridium alloy cylinder ratio 9:1 having diameter 3.9cm and height
3.9 cm placed in international bureau of weights is called 1kg.
3. Time (T):
It is measured in second. Its symbol is sec.
1 sec: 1/86400 the part of mean Solutionar day is called one second. OR the duration of
9192631770 vibration of Cs-133 atom is called 1 second.
4. Electric Current (A):
It is measured in amperes. Its symbol is A.
1A= It is the current which produce a force of 2x10
-7
N/m in two parallel wires of infinite length
and negligible circular cross section placed one meter apart from each other in vacuum.
5. Temperature (T):
It is measured in Kalvin. Its symbol is K.
8
1K= The triple point of water in thermodynamic scale of temperature is 273.16. The 1/273.16
th
part of the triple point of water is called 1K.
6. Light intensity (I):
It is measured in candela. Its symbol is Cd.
1 Cd=The amount of radiation emitted normally from 1/600000 m
2
of surface of black body in
one second. The temperature of black body is kept at Solutionidification of platinum under
standard atmospheric pressure.
7. Amount of substance (n):
Ii is measured in mole. Its symbol is mol.
1 mol = The amount of a substance which contain atoms or molecules equal to molecule or
atom contain 0.012 kg of corbon-12 atom. 1 mole contain atom = 6.0225x10
23
atoms.
SUPLEMANTRY UNITS
There are two supplementary units in S.I. System of units.
(1) Plane Angle
(2) Solutionid Angle
1. Plane Angle()
It is measured in radian. Its symbol is rad.
1 rad = Plane angle is one radian, if arc length covered in a circle is equal to radius of circle.
In Fig:
r
AB
=
Plane angle = 1 rad.
2. Solutionid Angle (
)
It is measured in steradian. Its symbol is Sr.
1 Sr = Solutionid angle is called one steradian,if the angle subtended at centre of sphere by an
area of its surface.
1 Sr =
2
4 r
3. Derived Unit
The units which are composed from base and supplementary units are called derived
units. For example:
(i)
Force: Its units is Newton and symbol is N and in term of base units Kg m/sec
2
.
A
o
1rad
B
r
o r
o
r
1sr
9
(ii)
Work: Its unit is Joule and symbol is J and in term of base units Kg m
2
/sec
2
.
(iii)
Pressure: Its unit is Pascal and symbol is Pa and in term of base units Kg m
-1
sec
-2
.
PREFIXS
A very large or small numbers are some time expressed in power of ten. This power of ten's has
given some name called prefix.
For example 1 centimeter = 1 cm = 10
-2
m
1 kilometer = 1 Km = 10
3
m
Some prefix are.
(iv)
More than base units are written one space apart.
For example Nm.
(v)
More than one prefix are not allowed. For example
10MMm is not allowed. We write it 10Tm and
5x10
5
cm is written as 5 x 10
3
m.
(vi)
When a number with base unit has some power
then power is applied on number as well as base
unit. For example
(6 km)
2
= (6x10
3
m)
2
= (6)
2
x (10
3
)
2
(m)
2
= 36 x 10
6
m
2
(vii) In practical work measurements would be recorded
in a convenient way. For example reading of screw
gauge should be in mm and mass of calorimeter should be in grams.
ERRORS
All physical measurements can never be 100% accurate. The difference between
measured value and actual value is called error. Error is due to following reasons.
(i)
Negligence or inexperience of a person.
(ii)
Faulty apparatus.
(iii)
Improper method.
TYPES OF ERROR
There are three types of errors.
Power
Prefix
Symbol
10
-18
Atto
A
10
-15
Femto
f
10
-12
Pico
p
10
-9
Nano
n
10
-6
micro
µ
10
-3
Milli
m
10
-2
centi
c
10
-1
Deci
d
10
1
Deca
da
10
3
Kilo
k
10
6
mega
M
10
9
Giga
G
10
12
Tera
T
10
15
Peta
P
10
18
Exa
E
10
(i)
Personal Error: It is an error which is produce due to natural deficiency of person. This
error is due to incorrect method of reading a scale. This error can be reduced by placing
the object on line of scale.
(ii)
Systematic Error: This error is due to the fault in instrument. This error is also
produced due to poor calibration, zero error of the instrument or incorrect marking. This
error can be reduced by changing the instrument or by applying correction factor such as
zero error.
(iii)
Random Error: This type of error is produced when different results of same
experiments are come under same condition. This error also arise due to the accidental
changes such as temperature humidity, line voltage or other unknown causes. The
random error can be reduced by taking the average of different results of same
experiments.
PRECISION AND ACCURACY
We know that a measurement is never abSolutionutely correct; there will always be an
error or uncertainty in it. The magnitude of error in a measurement is called precision.
Measurements having smaller error will be more precise.
Suppose we measure the width of a book with the help of a meter ruler and our reading is w =
15.4 cm. The meter ruler which is calibrated in mm has minimum error is
0.5 mm = 0.05 cm.
In the other hand we go from D.I.Khan to Peshawar let measured distance is 300 km. we
measure the distance from speedometer. The minimum distance on the dial of speedometer is 1
km. so maximum possible error should be 0.5 km.
The magnitude of error in first reading is 0.05 cm while in other reading is 0.5km.
As 0.05 < 0.5 km
Thus first measurement is more precise than second measurement.
ACCURACY
The magnitude of relative error is called accuracy. OR in other word we can say that
smaller is the relative error in a measurement greater is the accuracy. Where relative error is:
Relative error =
measured
quantity
error
For example in case of book measurements
11
Relative error =
cm
15.4
cm
0.05
= 3.24 x 10
-3
Relative error of D.I.Khan to Peshawar trip is
Relative error =
km
300
km
0.05
= 1.6 x 10
-4
The relative error of second measurement is less than first measurement. Thus second
measurement is more accurate than first one.
UNCERTAINITY
We know that measurements are not abSolutionutely correct. There will always be an
error in it called uncertainty. These uncertainties due to the instruments used but other factors
also.
TYPES
(i)
AbSolutionute Uncertainty: Precision of a measurement is called abSolutionute
uncertainty. The precision or abSolutionute uncertainty is equal to the least count of
instrument. In case of meter ruler the least count is 0.1 cm. Therefore abSolutionute
uncertainty = ± 0.1 cm. for example length of a book is this meter ruler is 15.2 cm. the
correct length is now written as 15.2 cm ± 0.1 cm.
(ii)
Relative or Fractional Uncertainty: It is the ratio of abSolutionute uncertainty to the
observed reading.
Relative or fractional uncertainty =
reading
observed
y
uncertaint
absolute
Fractional uncertainty tells us about the degree of accuracy. For example fractional
uncertainty of above example is.
F. uncertainty =
cm
15.2
cm
0.1
±
(iii)
Percentage Uncertainty: Fractional uncertainty describe in term of percentage is called
percentage uncertainty.
Percentage Uncertainty =
x100
reading
actual
y
uncertaint
absolute
For example in above case!
Percentage Uncertainty =
100
x
cm
15.2
cm
0.1
±
12
ASSESSMENTS OF UNCERTAINTY IN RESULTS
When we do an experiment its result is not final result. In calculating the final result we have to
consider the uncertainty effects on the final results following the results for assessments
(1) Sum or Difference:
When two measured quantities are added or subtracted then their abSolutionute
uncertainties are added. a and b are two measured quantities. If we want to add or subtract them
then
Q = a ± b
Where Q is final result.
Let
x
1
= (4.5 ± 0.1) cm
x
2
= (14.6 ± 0.1) cm
Then x = x
2
x
1
= (14.6 ± 0.1) cm (14.6 ± 0.1) cm
= (10.1 ± 0.2) cm
(2) Product and quotient:
When two or more quantities are multiplied or divided then their percentage
uncertainties are added.
For example from ohm's law
R =
I
V
If V = (5.2 ± 0.1) volt
I = (0.841 ± 0.05) amp
Percentage uncertainty of v =
%
2
100
5.2
1
.
0
=
x
Percentage uncertainty of I =
%
6
100
0.841
05
.
0
=
x
Total uncertainty = 2% + 6% = 8%
Thus final result is
R =
0.841
2
.
5
= 6.19 ± 8%
= 6.2 ± 8%
R = 6.2 ±
100
8
x 6.2 = (6.2 ± 0.5)
13
(3) Power of a quantity:
In this case the percentage uncertainty is multiplied with power. For example the volume
of sphere is
V =
3
r
3
4
---------- (1)
If radius is measured by Vernier Caliper is 2.25cm which has least count is 0.01 cm.
Then abSolutionute uncertainty = ± 0.01 cm
Percentage uncertainty in r =
100
x
2.25
0.01
= 0.4 %
as in volume power of r is 3
So total uncertainty in volume V = 3 x 0.4% = 1.2%.
Now putting the value of r in eq (1)
V =
3
)
25
.
2
)(
14
.
3
(
3
4
V = 47.689 cm
3
with 1.2% uncertainty.
Final result
V = 47.689 ± 47.689 x 1.2%
= 477 ± 47.7 x 1.2%
= (47.7 ± 0.6) cm
3
(4) Uncertainty in the average value:
If we have many measurements then following steps are used for final results
(i) Find the average value.
(ii) Find the deviation of each value.
(iii) The mean deviation is the uncertainty in the average value.
For example the diameter of a wire is measured with the help of screw gauge in mm is
for six time as
1.20, 1.22, 1.23, 1.19, 1.22, 1.21
Average =
6
21
.
1
22
.
1
19
.
1
23
.
1
22
.
1
20
.
1
+
+
+
+
+
= 1.21mm
Deviation of each value is
0.01, 0.01,0.02,0.02,0.01,0
14
mean deviation =
6
0
01
.
0
02
.
0
02
.
0
01
.
0
01
.
0
+
+
+
+
+
= 0.01mm
The uncertainty in the average value
mm is 0.01 mm
Then the final result is d = (1.21 ± 0.01) mm
(5) Uncertainty in timing experiment:
The uncertainty in the time period of a vibrating body can be found by dividing the least
count of timing device by number of vibrations.
For example if time of 30 vibrations of a simple pendulum is 54.6 sec which is measured
by stop watch whose least count is 0.15
Time period T =
825
.
1
30
6
.
54
n
t
=
=
Uncertainty =
vibrations
of
number
count
least
=
sec
003
.
0
30
0.1
=
Thus final time period is = T = (1.825 ± 0.003) s
SCIENTIFIC NOTATION
When a number is expressed in term of negative or positive power of 10 then it is called
scientific notation.
If N is the number then scientifically it is written as
N = M x 10
n
Where M is a number whose first digit is non zero from left to decimal point and n is
+ve or ve power
For example N = 1500000
in scientific notation.
N = 1.5 x 10
6
CONVENTIONS FOR S.I UNITS
(i)
Full name of the unit cannot starts from capital letter. For example newton.
(ii)
The unit which is the name of scientist is written with capital and first alphabet. For
example Newton = N
15
(iii)
Prefix is written before unit. For example
5000000 C
o
= 5 x 10
6
C
o
= 5 MC
o
SIGNIFICANT FIGURES
In any measurement the accurately known digits and first doubtful digit are called significant
figures.A significant figure is reasonably reliable. For example we want to measure the length of
a pencil with the help of a meter ruler. The meter ruler is calibrated in millimeters,
so least count is 1mm and error is of 0.5mm 0.05cm.
Let the end point of pencil placed between 10.3 cm and 10.4cm.
If pencil end is placed before mid point then we consider our reading 10.3cm and if pencil end is
placed after mid point then we consider 10.4 cm our reading.
As the maximum uncertainty in the length is ± 0.05cm and abSolutionute uncertainty of two
ends 2(± 0.05) = ± 0.1 cm = ± 1mm
This is equal to least count of meter ruler. Suppose we take 10.3 cm reading then first two digits
are (1,0) are 100% accurate while third digit 3 is doubtful so all digits (1,0,3) are significant
figures.
GENERAL RULES FOR SCIENTIFIC FIGURES
(i)
All non zero digits 1, 2, 3,4,5,6,7,8,9 are significant. For example in 622.4mm. There are
four significant figures.
(ii)
The zero between two non-zero digits are significant figures. For example 501.6mm has
four significant figures.
(iii)
A zero right to decimal point and left of a non- zero digits is not significant. For example
in 0.000234m only three significant figures (2, 3, 4) zero on the right of decimal point is
not significant it is used only to locate the position of decimal point.
(iv)
All zeros to the right of decimal point that appears after a non zero digit are significant.
For example in 0.07080cm and 20.00cm each reading has four significant figures.
(v)
In scientific notation a number is written in the power of 10. i.e in the form of
N = M x 10
n
Here M consists of all significant figures. In M decimal point occurs after first non zero
digit. For example 8.70x10
3
kg has three significant figures (8, 7, and 0)
(vi)
When two or more numbers are multiplied or divided then result is obtained in least
significant figures.
16
For example
x
1
= 5.8m
x
2
= 4.12m
Then x
1
X x
2
= 5.8 x 4.12 = 23.89 m
2
= 23.9 m
2
and
4
.
1
40
.
1
4.12
5.8
x
x
2
1
=
=
=
(vii) In addition and subtraction process is again in least significant figures number are taken.
For example
x
1
= 5.8m
x
2
= 4.12m
Then x
1
+ x
2
= 5.8 + 4.12 = 9.92 m = 9.9 m
and x
1
x
2
= 5.8 4.12 = 1.68m= 1.6m
RULES FOR ROUNDING OF NUMBERS
The dropping of insignificant figures from quantity is called rounding of numbers.
Following are the rules for rounding a number.
(1)
If first digit to be dropped is less than 5 then last digit should remain same
for example 65.523 is rounded of 65.5
(2)
If first digit is more than 5 then digit is increased by 1
for example 56.8546 is written as 56.9
(3)
If digit which is to be dropped is 5 then precious digit is if odd then it is increased by 1
and if it is even then it remain same.
For example 43.75 is rounded off 43.8
And 73.650 is rounded off as 73.6
(4)
In addition or subtraction process the answer is rounded off to smallest decimal places.
In this case significant figures have no importance.
For example
(i) 72.1
3.42
0.003
--------
75.523
it is rounded off 75.5
17
(ii) 2.7543
4.10
1.273
--------
8.1273
it is rounded off 8.13
(iii) 88.9
44.32
--------
44.58
it is rounded off 44.6
(iv) 50.5
3.2
--------
47.3
it is rounded off 47
DIMENSIONS OF A PHYSICAL QUANTITY
When a physical quantity is represented by basic fundamental units symbols enclosed in
a square bracket is called dimension of physical quantity.
For example dimension of distance [ L ]
Dimension of speed = [ LT
-1
]
Dimension of force = dimension of mass x
dimension of acceleration
= [ M ] [ LT
-2
]
= [MLT
-2
]
Dimension is used to check the correctness of a formula also dimension is used to derive the
formula or equation.
PRINCIPLE OF HOMOGENETY
If the dimension of physical quantities on both sides of equation is same then this is
called principle of homogeneity. For example we check the corrections of the formula of
velocity of wave in a string.
i.e.
V =
m
x
T l
Where
V = speed of wave
18
T = tension in string
l = length of string
m = mass of string
As dimension of V = [ LT
-1
] on L.H.S
On R.H.S
dimension T = [MLT
-2
]
dimension of l = [ L ]
dimension of m = [ M ]
So dimension of
V =
m
x
T l
=
2
1
2
-
2
M
T
ML
=
[ ]
2
1
2
-
2
T
L
=
[ ]
-1
LT
So R.H.S = L.H.S
So above equation is dimensionally correct
DERIVING A FORMULA
We can derive a possible formula from the dimension.
For example, we find a formula for time period of simple pendulum by the use of dimensions.
The various possible factors on which the time period T may depend are.
(i) Length l of pendulum.
(ii) Mass m of the bob
(iii) Angle at mean position
(iv) Acceleration due to gravity (g)
Now we can write
T m
a
x l
b
x
c
x g
d
---------- (1)
Now we find the value of a, b, c d
In the form of dimension
[ T ] = constant [ M ]
a
[ L ]
b
[ ]
c
[ g ]
d
or
[ M ]
0
[ L ]
0
[ T ] = constant [ M ]
a
[ L ]
b
[ ]
c
[ g ]
d
-- (2)
Since [ M ]
0
= [ L ]
0
= 1
As is small we can write
S = r
B
l
s
A
19
S = l
=
l
S
So dimension of =
L
L
= [ L L
-1
]
And dimension of g =
[ L T
-2
]
So equation (2) becomes
[ M ]
0
[ L ]
0
[ T ]
1
= constant [ M ]
a
[ L ]
b
[L L
-1
]
c
[L T
-2
]
d
comparing the coefficients on both sides
[ T
1
] = [ T
-2
]
d
= [ T ]
-2d
[ M ]
0
=
[ M ]
a
[ L ]
0
= [ L ]
b
[ LL
-1
]
c
[ L ]
d
or [ L ]
o
= [ L ]
b+d
Equating powers
1 = 2d
=>
d =
2
1
-------- (i)
a = 0
-------- (ii)
b + d = 0
=>
b = d
-------- (iii)
b= -(
2
1
)
b =
2
1
And
c
= [ LL
-1
]
c
=
[ L
0
]
c
= 1
Putting the value of a b, c and d in eq. (1
T = constant m
o
x l
½
x 1 x g
½
= constant
1/2
2
/
1
g
l
= constant
g
l
T =
g
l
2
Where
2 is constant.
20
SHORT ANSWER QUESTIONS
1. What is a system of unit? List the basic SI units.
A complete set of units both fundamental and derived for physical quantities is called
system of units. The basic or fundamental units of S.I. are:
(i) meter
(m)
used for length.
(ii) kilogram (g)
used for mass
(iii) second (s)
used for time
(iv) kalvin
(k)
used for temperature
(v) ampere
(A)
used for current
(vi) mole
(mol)
used for amount of substance
(vii) candela (cd)
used for luminous intensity
2. Define the number
and show that 2
radian=360
o
The ratio of the circumference of a circle divided by its diameter is called pi.
As circumference of circle = 2
r
Where r is the radius of circle
And diameter = d = 2r
2r
r
2
d
2
diameter
ce
circumfren
=
=
r
: d = 2 r
=
we know that when a circle completes the angular displacement = 360
o
we know that S = r
where S = arc length
r = radius
now
=
r
S
if S is circumference then
S = 2
r
= 2
Putting the value of
360
o
= 2
radius
o
S
r
21
3. How the units of length, mass and time are presently defined?
Length:
The unit of length is meter and defined as "the distance covered by light in
vacuum in 1/299792458 second"
Mass:
The unit of mass is Kg and is defined as "the mass of platinum iridium cylinder
kept at the international bureau of weights and measures at France.
Length:
The unit of time is second and is defined as a time during which 9192631770
vibrations of Cs-133 atom execute.
4. Distinguish between base and derived physical quantities.
Base Units:
The minimum number of physical quantities in term of which we can define
other quantities are called base units. For example length, mass, time etc.
Derived Quantities:
The units which are formed due to the combinations of base units is called
derived units or derived quantities.
5. Explain with examples the scientific notation for writing numbers.
To represent a large or small numbers in term of power of 10 is called scientific notation
i.e N = M x 10
n
Where N is the number and M is also a number whose first digit from
left to right is non zero. It is followed by decimal point and n is power of 10.
For Example
1.
N = 150000000000 = 1.5 x 10
11
2.
N = 0.0000525 = 5.25x10
-5
6. Define the terms: error, uncertainty, precision and accuracy in measurements.
Error:
The difference between measured value and actual value is called error.
Uncertainty:
The estimated possible range of error is called uncertainty.
Precision:
It is the agreement among various observations in an experiment.
Accuracy:
The measurement of correctness of a experiments results is called accuracy.
22
7. Explain the principle of the dimensional homogeneity of physical equations.
According to principle of homogeneity the dimensions on both side of a equation should
be same. This principle is very useful to check the correctness of formula.
For example
v =
t
S
Or
S = vt
--------- (1)
Dimension of S = [ L ]
And dimension of v = [ LT
-1
]
And dimension of t = [ T ]
Therefore equation (1) become.
[ L ] = [ LT
-1
] [ T ]
[ L ] = [ L ]
so above equation is dimensionally correct.
8. An old saying is that "A chain is only as strong as its weakest link." What
analogous statement can you make regarding experimental data in calculation?
The analogous of old saying "A chain is only as strong as its weakest link" in physics is
that after a scientific method the result obtained at final is more reliable.
9. Write the dimensions of (i) pressure (ii) power (iii) density (iv) revolutions.
(i) PRESSURE
P =
A
F
=
2
-2
L
MLT
= [ ML
-1
T
-2
]
(ii) POWER
P =
t
w
=
T
T
ML
-2
2
= [ ML
2
T
-3
]
(iii) DENSITY
D =
V
m
=
3
L
M
= [ ML
-3
]
23
(iv) REVOLUTION/SEC
=
t
=
T
1
: has no dimension
= [ T
-1
]
10. The time period of a simple pendulum is measured by a stop watch. What types of
errors are possible in the time period?
Possible errors during a time period of a simple pendulums are
(i)
systematic error
(ii)
Random error
(iii)
Personal error
11. A Circle has a diameter of 0.400m. What is its area?
d = 0.400 m
A = ?
As
A =
r
2
A =
2
2
d
: r =
2
d
=
4
)
400
.
0
(
14
.
3
2
= 0.1256 m
2
According to significant figures rule of least
A = 0.125 m
2
12. Write five units used in Pakistan for measuring mass.
(i) Kg (ii) gm (iii) Ton (iv) Tela
(v) pound (vi) ounce
SAMPLE PROBLEMS
1.
Express the following quantities using prefixes:
(a) 3.0 x 10
-4
m; (b) 5.0 x 10
-5
s; (c) 72.0 x 10
2
g;.
Solution
(a) 3.0 x 10
-4
m
Multiply and divide by 10
24
3.0 x 10
-4
x
10
10
m
10
3
x 10
-3
m
0.3 mm
as 10
-3
= mili
(b)
5.0 x 10
-5
s
Multiply and divide by 10
5.0 x 10
-5
x
10
10
sec
(5x10) 10
-5
x 10
-1
sec
50 x 10
-6
sec
50 sec
as 10
-6
= micro =
(c) 72.0 x 10
2
gm
Multiply and divide by 10
72 x 10
2
x
10
10
gm
10
72
x 10
3
gm
7.2 Kg
as 10
3
= kilo = k
2. Estimate your:
(a) Age in seconds (b) Mass in grams
(c) Weight in newtons
(d) Height in millimeters
Solution
(a)
Age in seconds
Suppose your age is 20 years.
As 1 year = 365 day
1 hour = 60 min
And 1 min = 60 sec
Age in seconds
= 20 x 365 x 24 x 60 x 60
= 26280000 sec
= 2.6 x 10
7
sec
25
(b) Mass in grams
Suppose your mass is 70 kg
As 1000 gm = 1 kg
70 kg = 70 x 1000 gm
= 70000 gm
= 7 x 10
4
gm
(c) Weight in newton
As our mass = m = 70 kg
As weight = w = mg
= 70 x 9.8
= 686 N
(d) Height in millimeters
Suppose your height is 5 feet
h = 5 feet
As 1 meter = 3.4 feet
1 feet = m/3.4
Height in meter
h =
4
.
3
5m
= 1.4 m
as 1 m = 1000 mm
height in millimeters
h = 1.4 x 1000 mm
h = 1400mm
3. If there are N
o
= 6.02x10
23
atoms in 4 gm of helium, what is the mass of helium
atom?
Solution
Number of atoms = N
o
= 6.02x10
23
Mass of helium = M = 4 gm
Mass of helium atom = m = ?
As mass of N
o
atoms = M
Mass of one atom = m =
o
N
M
=
23
6.02x10
4
26
= 0.664 x 10
-23
gm
= 6.64 x 10
-24
gm
4. Rest mass of electron is 9.11 x 10
-31
kg
(a) Write it without use of powers of 10.
(b) Convert it to grams.
Solution
(a) m
o
= 9.11 x 10
-31
kg
now rest mass m
o
without power of 10
m
o
= 0.000,000,000,000,000,000,000,000,000,000,000911kg.
(b)
mass in gm
as 1 kg = 1000 gm = 10
3
gm
mass in grams
m
o
= 0.11 x 10
-31
x 10
3
gm
= 9.11 x 10
-28
gm
5. Density of air is 1.2 kg m
-3
. Change it into g cm
-3
.
Solution
Density = d = 1.2 kg /m
3
As 1 kg = 1000 gm
And 1 m = 100 cm
d in gm /cm
3
is
d =
3
cm
100
x
100
x
100
gm
1000
x
1.2
d = 0.0012 gm/cm
3
6. Density of water is 1 g m
-3
. change it into kg cm
-3
.
Solution
Density of water = d = 1g /cm
3
As 1 gm =
1000
kg
and 1 cm = m/100
density d in kg/m
3
is
d =
100
m
100
m
100
m
1000
kg
x
x
27
d =
1000000
m
1000
kg
3
d =
3
m
1000
kg
1000000
= 1000 kg /m
3
7. Express the following in terms of power of 10:
(a) 5 picofarad (b) 12 megawatt (c) 100 million volts
Solution
(a) C = 5 picofarad
As pico = 10
-12
C = 5 x 10
-12
farad
let P = 12 megawatt
As 1 mega = 10
6
P = 12 x 10
6
watt
(c) Let V = 100 million volt
As 1 million = 10
6
V = 100 x 10
6
volt
V = 10
8
volts
8. A light year is the distance light travels in one year. How many meters are there in
one light year? The velocity of light c = 3 x 10
8
ms
-1
.
Solution
d = 1 light year
d = in meters = ?
c = 3 x 10
8
m/s
as c =
t
d
d = ct
-------- (1)
As time taken by light
t = 365 days x 24 hours x 60 min x 60 sec
= 3.15 x 10
7
sec
Put in (1)
d = 3 x 10
8
x 3.15 x 10
7
28
= 9.46 x 10
15
m
9. Compute the following to correct significant digits:
(a) 3.85 m x 3.19 m (b) 1023 kg + 8.5489 kg
(c)
7
22
(d)
kg
10
x
9.1096
kg
1.67x10
m
m
31
-
-27
e
p
=
Solution
3.85 m x 3.19 m = 12.2815 m
2
According to rule of significant figures we can take answer in term of least
significant number. As least significant number is 3.
3.85 m x 3.19 m = 12.2 m.
Let m = 1023 kg + 8.5489 kg.
= 1031.5489 kg.
According to rule of significant figures.
m = 1031 kg.
(c)
7
22
= 3.142857143...
By least rule of significant figures.
7
22
= 3
(d)
kg
10
x
9.1096
kg
1.67x10
m
m
31
-
-27
e
p
=
=
31
27
-
10
10
9.1096
1.67
+
x
x
= 0.18332 x 10
+4
Using the rules of significant figures.
= 0.183 x 10
4
10. The length and width of a rectangular plate are measured to be 15.3 cm and 12.80
cm, respectively. Find the area of the plate.
Solution
L = 15.3 cm
W = 12.80 cm
A = ?
A = L x W
29
= 15.3 x 12.80
= 195.84 cm
2
According to rule of significant figures (least significant)
A = 195 cm
2
11. A rectangular metallic piece is 3.70±0.01cm long and 2.30±0.01cm wide.
(a) Find the area of the rectangle and the uncertainty in the area.
(b)Verify that percentage uncertainty in the area is equal to the sum of percentage
uncertainties in the length and in the width.
Solution
L = 3.70±0.01cm
W = 2.30±0.01cm
(a) A = ?
Now percentage uncertainty of L =
100
70
.
3
01
.
0
x
= 0.27%
Percentage of uncertainty in W = =
100
30
.
2
01
.
0
x
= 0.43%
total percentage uncertainty in area = 0.27% + 0.43%
= 0.70%
Now area
A = L x W
= 3.70 x 2.30 = 8.51 ± 0.70%
A = 8.51 ± 0.70% x 8.51
= 8.51 ±
51
.
8
100
70
.
0
x
= (8.51 ± 0.05) cm
2
(b)
As percentage uncertainty in area = 0.70%
Percentage uncertainty in length L = 0.27%
Percentage uncertainty in width W = 0.43%
Percentage uncertainty in L + W = 0.27% + 0.43%
= 0.70%
So percentage uncertainty in A = percentage uncertainty (L+W).
30
12. Find the mass of air in 3.00 m x 8.00 m x 6.00 m room. Density of air is 1.29 kg/m
3
.
Solution
m = ?
v = 3.00 m x 8.00 m x 6.00 m = 144 m
3
d = 1.29 kg/m3
As d =
v
m
m = d x v
= 1.29 x 144
= 185.76 kg.
According to the rule of significant figures we take least significant numbers.
d = 185 kg.
13. Prove the following equations are homogenous with respect to dimensions
(a) Kinetic energy = ½ mv
2
(b) Centripetal acceleration = (velocity)
2
/radius, that is a =v
2
/r.
Solution
(a) K.E = ½ mv
2
Take L.H.S dimensions
K.E = J = Nm
As N = Kg m/sec
2
K.E = kg m/sec
2
x m
= kg m
2
/sec
2
= kg m
2
sec
-2
Now in case of dimension
K.E = [ ML
2
T
-2
]
Now we check the dimension of R.H.S
R.H.S = ½ mv
2
= ½ kg (m/sec)
2
= ½ kg m
2
/sec
2
= ½ kg m
2
sec
-2
Dimension
R.H.S = [ ML
2
T
-2
]
As dimension of L.H.S = dimension of R.H.S
So equation
K.E = ½ mv
2
is dimensionally homogeneous
31
(b)
a
c
=
r
v
2
Take L.H.S
a
c
= m / sec
2
= m sec
-2
Dimension of L.H.S
a
c
= [ LT
-2
]
Take R.H.S
R.H.S =
r
v
2
In case of dimension
R.H.S =
m
(m/sec)
2
as v = m/sec, r = m
=
m
sec
m
2
2
=
m
sec
m
2
2
-
= m sec
-2
R.H.S = [ LT
-2
]
As dimension of L.H.S = dimension of R.H.S
So equation
a
c
=
r
v
2
is dimensionally homogeneous.
14. Find the dimensions and hence the unit of the coefficient of viscosity in the Stoke's
law for the drag force F on a spherical object of radius moving with velocity v given
as F = 6 r v
Solution
Dimension of = ?
Unit = ?
According to Stoke's law
F = 6 r v
rv
6
F
=
------- (1)
32
As dimension of F = [ MLT
-2
]
and dimension of v = [ LT
-1
]
and dimension of r = [ L ]
By putting the dimension of F, v and r in equation (1)
][L]
[LT
]
[MLT
1
-2
-
=
=
1
2
-2
T
L
MLT
-
= MLT
-2
L
-2
T
+1
= [ML
-1
T
-1
]
unit of = kg/m. sec
15. Find the dimensions and hence the unit of the universal gravitational constant G in
the Newton's law for the force F between two masses separated by distance r given
as:
2
2
m
m
1
r
G
F
=
Solution
Dimension of G = ?
Unit of G = ?
As according to Newtons law of gravitation
2
2
1
m
m
r
G
F
=
From above eq.
2
1
2
m
m
Fr
G
=
As Dimension of F = MLT
-2
Dimension of r
2
= L
2
dimension of m
1
= M
dimension of m
2
= M
so dimension of G =
MM
L
MLT
2
-2
= [ L
3
T
-3
M
-1
]
Unit of G = Nm
2
/Kg
2
33
16. Show that the famous Einstein's equation E = mc
2
is dimensionally correct.
Solution
Dimension of Einstein's eq. = ?
Einstein's equation.is
E = mc
2
As the unit of energy E = joule
=>
as joule = F.d = N-m
dimension of F = MLT
-2
dimension of meter = L
dimension of E = MLT
-2
.L
= [ML
2
T
-2
] ----- (1)
Take R.H.S = mc
2
Dimension of mass m = M
Dimension of speed of light c = LT
-1
Dimension of R.H.S = M (LT
-1
)
2
= [ML
2
T
-2
]
---- (2)
From eq. (1) and (2)
Dimension of L.H.S = dimension of R.H.S
So Einstein's Equation is Dimensionally Correct.
34
DIRECTED QUANTITIES: VECTOR
NON PHYSICAL QUANTITIES
The quantities which cannot be measured are called non physical quantities. For
example, love, hate, faith and emotions etc.
PHYSICAL QUANTITIES
The quantities which can be measured are called physical quantities. For example
distance, speed, velocity and force etc.
TYPES
1. SCALAR QUANTITIES
Those quantities which required two things for their representation (i) Number (ii) Unit
are called scalar quantities. OR those quantities which required only magnitude for their
representation are called scalar quantities.
For example temperature, volume, density, distance etc. Scalar quantities are not represented
graphically.
scalar are +, , X and
÷ by simple algebra.
2. VECTOR QUANTITIES
Those quantities which required three things for their representation
(i) Number (ii) Unit (iii) direction are called vector quantities OR those quantities which
required magnitude as well as direction for their representation are called vector quantities.
For example velocity, displacement, torque, force etc. Vectors are drawn graphically by
an arrow. Vectors are +, , X and
÷ by vector algebra or graphical methods or vector methods.
REPRESENTATION OF VECTORS
Vector are represented by two methods
1.SYMBOLIC REPRESENTATION
In this method vector are represented by bold face letter with an arrow above or below
the letter as
B
,
A
,
B
,
A
v
r
35
2.GRAPHICAL METHOD
In this method vector is represented by a line with an arrow head. Line gives magnitude
while arrow head gives direction. Line is taken according to suitable scale, while for direction
we use NEWS diagram
For example
A displacement of 1000 km east
Symbolic representation
d
= 1000 km east
Graphically representation
NEWS diagram
Scale
100 km = 1 cm
d
1000 km = 100 cm
TYPES OF VECTOR
NULL OR ZERO VECTOR
A vector whose magnitude is zero but direction is arbitrary is called Null Vector. It is
represented by
0
r
. This vector is located at origin. Null vector can be obtained by many
methods. For example by adding two vectors having same magnitude but opposite in direction.
A
r
and
A
r
are such vectors whose magnitude is same but direction is opposite by
addition
A
r
+ (
A
r
) = 0
Other examples are
A
r
+
B
r
+
C
r
=
0
r
A
r
+
B
r
+
C
r
+
d
r
=
0
r
N
E
S
W
origin
A
r
A
-
r
A
r
A
r
A
r
A
r
B
r
B
r
B
r
B
r
C
r
C
r
C
r
C
r
d
r
d
r
origin
origin
36
UNIT VECTOR
A vector whose magnitude is one but direction is arbitrary is called unit vector. Unit
vector is used to find the direction of any vector. Unit vector is represented by a latter with a cap
above it.
For example â, Â,
Let
A
r
is a vector then mathematically
A
r
is written as
A
r
= |
A
r
| Â
Where |
A
r
| = A = magnitude of
A
r
And  is unit vector.
From above e.q
So by dividing a vector to its magnitude, we get a vector called unit vector.
i^
,
j^
,
k^
,
n^
are famous unit vectors.
i^
= points x-axis
j^
= points y-axis
k^
= points z-axis
n^
= points normal
EQUAL VECTORS
Two vectors
A
r
and
B
r
are said to be equal if both vectors has
same magnitude and same direction. If
A
r
and
B
r
are equal then
A
r
=
B
r
PARALLEL VECTORS
Two vectors
A
r
and
B
r
are said to parallel, if they have same direction although their
magnitude is not same.
If
A
r
//
B
r
then angle between them will be 0
o
.
ANTIPARALLEA VECTORS
Two vectors
A
r
and
B
r
are said to antiparallel, if they have opposite direction, although
their magnitude is not same.
If
A
r
//
B
r
then angle between them will be 180
o
.
A
A
A^
r
=
A
r
B
r
A
r
B
r
A
r
B
r
37
PERPENDICULAR VECTORS OR RECTANGULAR VECTORS
Two vectors are said to be perpendicular, if they are at right
angle to each other if
A
r
B
r
then angle b/w them is 90
o
.
POSITION VECTOR
It is a vector which gives the position of a point or an object with respect to origin.
It is represented by
r^
where
i
^
x
r
=
r
When object is placed on x-axis
j
^
y
r
=
r
When object is placed on y-axis
k^
z
r
=
r
When object is placed on z-axis
j^
y
i^
x
r
+
=
r
When object is placed inside x & y axes
k^
z
j^
y
i^
x
r
+
+
=
r
When object is placed inside x ,y,z axes.
NEGATIVE OF A VECTOR
The negative of a vector
A
r
is symbolically written as
A
r
. Graphically the magnitude of
A
r
and
A
r
is same but direction is opposite.
Negative of
A
r
=
A
r
But
|
A
r
| = |
A
r
|
Or
Multiplication of a vector with 1, gives a vector called ve of that vector.
For example
1 x
A
r
=
A
r
A
r
A
-
r
0
y
x
rr
z
0
y
x
rr
0
y
x
rr
p
z
0
y
x
rr
p
z
0
y
x
rr
p
z
B
r
A
r
o
90
38
ADDITION OF VECTORS
Vectors are added by two methods
(i)
Mathematical method
(ii)
Graphical method
MATHEMATICAL METHOD
Let
1
A
r
,
2
A
r
, - - - - - ,
n
A
r
are vectors. Mathematically their addition is
R
r
=
1
A
r
+
2
A
r
+ - - - - - +
n
A
r
R
r
=
i
A
n
i
1
i
r
=
=
Where
R
r
is the resultant vector, also called vector sum.
Where |
R
r
| |
1
A
r
| + |
2
A
r
|+ - - - - - + |
n
A
r
|
GRAPHICAL METHOD
In this method vectors are added by head to tail rule. Following the steps for this rule
(i)
Draw the vectors graphically by suitable scale.
(ii)
Connect the head of 1
st
vector with tail of 2
nd
vector and head of 2
nd
vector with tail of
3
rd
vector and so on.
(iii)
The resulted vector is the displacement vector, which connects the tail of 1
st
vector with
head of last vector.
R
r
=
1
A
r
+
2
A
r
+
3
A
r
+
n
A
r
=
i
A
n
i
1
i
r
=
=
1
A
r
2
A
r
3
A
r
n
A
r
1
A
r
2
A
r
3
A
r
n
A
r
R
r
39
ADDITION OF TWO VECTORS BY PARALLELOGRAM METHOD AND
COMMOTATIVE PROPERTY
Let
A
r
and
B
r
are two vectors inclined at angle . In order to add
these vectors, we use a method called parallelogram method. For this
method,
Draw a vector
A
r
from head of
B
r
parallel to
A
r
Draw a vector
B
r
from head of
A
r
parallel to
B
r
In this way we get a parallelogram. The diagonal of this parallelogram is the resultant of
A
r
and
B
r
R
r
=
A
r
+
B
r
---------------- (1)
Also from above Figure
R
r
=
B
r
+
A
r
---------------- (2)
From equation 1 and 2, we see that
A
r
+
B
r
=
B
r
+
A
r
This property is called commutative property of addition.
SUBTRACTION OF VECTORS
There is no direct method for subtraction of vectors, but we can adopt following methods
for vector subtraction.
(i)
Mathematical Method
(ii)
Graphical Method
MATHEMATICAL METHOD
Let
A
r
and
B
r
are two vectors, if we want to subtract
B
r
from
A
r
then mathematically
R
r
=
A
r
B
r
is not allowed in vector algebra.
While
R
r
=
A
r
+ (
B
r
) is allowed in vector algebra.
Where
R
r
is resultant vector.
GRAPHICAL METHOD
Graphically vectors are also subtracted by head to tail rule. Following the steps for this
rule
(i) Draw the vectors graphically by suitable scale.
(ii) Draw the ve vector of subtracting vector.
(iii) Connect the head of 1
st
vector with tail of ve vector of subtracting
vector.
A
r
B
r
A
r
B
r
R
r
A
r
B
-
r
B
r
A
r
B
r
R
r
40
(iv) Resultant vector is a displacement vector, which connects the tail of 1
st
vector with head of last.
R
r
=
A
r
+ (
B
r
)
RESOLUTIONUTION OF VECTOR
Splitting of a vector into its components is called reSolutionution of vector. Consider a
vector
F
r
which makes an angle with x-axis.
In Fig
0P =
F
r
We draw a perpendicular PQ on x-axis.
We get two components
0 Q =
x
F
r
= called horizontal component
Q P =
y
F
r
= called vertical component
Now
F
r
=
x
F
r
+
y
F
r
or
F
r
=
j^
Fx
i
^
Fx
+
DETERMINATION OF F
X
In fig from OPQ
F
F
hyp
Base
Cos
x
=
=
F
x
= F Cos
(i)
DETERMINATION OF F
Y
In fig from OPQ
F
F
hyp
Per
Sin
y
=
=
F
y
= Fsin
(ii)
So we can write
j^
FSin
i^
FCos
F
+
=
r
DETERMINATION OF MAGNITUDE OF
F
r
Squaring and adding eq. (i) and (ii)
F
x
2
= F
2
cos
2
F
y
2
= F
2
sin
2
F
x
2
+ F
y
2
= F
2
cos
2
+ F
2
sin
2
y
F
r
x
0
x
F
r
Q
P
y
F
r
41
= F
2
(cos
2
+ sin
2
)
:cos
2
+ sin
2
= 1
F
x
2
+ F
y
2
= F
2
2
y
2
x
F
F
F
+
=
DETERMINATION OF DIRECTION OF
F
r
Divide eq (i) and (ii)
Fcos
Fsin
F
F
x
y
=
=>
tan
F
F
x
y
=
:
tan
cos
sin
=
=>
=
-
x
y
1
F
F
tan
RECTANGULAR COMPONENT OF A VECTOR
The component of a vector which are at right angle to each other are called rectangular
component of a vector. In figure angle between F
x
and F
y
is 90
o
, so these are the rectangular
components of
F
r
.
ADDITION OF TWO VECTORS USING RECTANGULAR COMPONENTS
OR ADDITION OF TWO VECTORS USING ANALYTICAL METHOD.
Consider two vectors
A
r
and
B
r
making angle
1
and
2
with x-axis.
such that
OP =
A
r
B
r
OQ =
B
r
Now mathematically their addition is
R
r
=
A
r
+
B
r
(1)
where
R
r
is vector sum of
A
r
and
B
r
According to head to tail rule in fig:
Where OS =
R
r
Draw perpendicular
PT
and
SM
We get rectangular components of
A
r
,
B
r
&
R
r
Where OT = A
x
PT = A
y
PL = B
x
SL = B
y
SM = R
y
OM = R
x
x
O
A
r
1
P
Q
2
y
y
x
O
A
r
P
R
r
B
r
S
y
R
x
R
L
y
B
y
A
x
B
T M
x
A
42
In component form
R
r
= R
x
i^
+ R
x
j^
A
r
= A
x
i^
+ A
y
j^
B
r
= B
x
i^
+ B
y
j^
Putting the values of
R
r
,
A
r
and
B
r
in eq (i)
R
x
i^
+ R
y
j^
= A
x
i^
+ A
y
j^
+ B
x
i^
+ B
y
j^
Compare the coefficient of
i^
and
j^
on both sides
R
x
= A
x
+ B
x
(2)
R
y
= A
y
+ B
y
(3)
Magnitude of
R
r
as
2
2
Ry
Rx
R
+
=
2
2
By)
(Ay
Bx)
(Ax
+
+
+
=
(4)
Direction of
R
r
as
+
+
=
=
-
Bx
Ax
Bx
Ay
Tan
Rx
Ry
Tan
1
1
-
(5)
Now
2
1
Bcos
Bx
,
Acos
Ax
,
Rcos
Rx
=
=
=
2
1
Bsin
By
,
Asin
Ay
,
Rsin
Ry
=
=
=
putting in eq. (4)
2
2
1
2
2
1
)
Bsin
,
(Asin
)
Bcos
,
(Acos
R
+
+
+
=
2
1
2
2
2
1
2
2
2
1
2
2
2
1
2
2
sin
2ABsin
sin
B
,
sin
A
cos
2ABcos
cos
B
,
cos
A
+
+
+
+
+
=
)
sin
sin
cos
2AB(cos
)
sin
(cos
B
)
Sin
,
(cos
A
R
2
1
2
1
2
2
2
2
2
1
2
1
2
2
+
+
+
+
+
=
as
1
sin
cos
2
2
=
+
)
(
cos
-
=
sin
sin
cos
cos
+
above eq. become
)
2ABcos(
B
A
R
1
2
2
2
-
+
+
=
which is another form of resultant
R
r
43
Now another form for direction of
R
r
is
+
+
=
2
1
2
1
1
-
Bcos
Acos
Bsin
Asin
Tan
if we have N-vectors
then
A
-
-
-
-
-
A
,
A
,
A
n
3
2
1
r
r
r
r
,
Magnitude of resultant
R
r
2
3
2
1
2
3
2
1
-)
-
-
y
A
y
A
y
A
(
-)
-
-
x
A
x
A
x
(A
R
+
+
+
+
+
+
=
2
2
2
1
1
2
2
2
1
1
-)
-
-
sin
A
sino
(A
-)
-
-
cos
A
cos
(A
R
+
+
+
+
+
=
2
n
j
1
j
j
j
2
n
i
1
i
i
i
sin
A
cos
A
R
+
=
=
=
=
=
and
Direction of resultant
R
r
+
+
=
-
-
-
-
-
-
cos
A
cos
A
-
-
-
-
-
-
sin
A
sin
A
Tan
2
2
1
1
2
2
1
1
1
-
=
=
=
=
=
cos
A
sin
A
Tan
n
i
1
i
i
i
n
j
1
j
j
j
1
-
MULTIPLICATION OF VECTORS
Vectors are multiplied by two methods
(1) Scalar Product
(2) Vector Product
1. SCALAR PRODUCT
If we multiply two vectors such that their resultant is a scalar quantity then
multiplication is called scalar product.
We know that force
F
r
and displacement
d
r
are vector quantities. The product of force of force
and displacement is called work.
W =
F
r
·
d
r
Where, work is scalar quantity. This Product is called scalar product.
Let
A
r
and
B
r
are two vectors making angle with each other.
44
Mathematically scalar product of
A
r
and
B
r
is
A
r
·
B
r
= |
A
r
| |
B
r
| cos
= A B cos
= A(B cos
)
= AB
x
= A (projection of
B
r
on
A
r
)
= (Acos
) B
= A
x
B = (projection of
A
r
on
B
r
) B
Where symbol "·" Is called dot.
Therefore scalar product is also called dot product.
Properties of Scalar Product.
(i)
Commutative property
A
r
·
B
r
=
B
r
·
A
r
proof
L.H.S
A
r
·
B
r
= A B cos
(i)
R.H.S
B
r
·
A
r
= B A cos
Since A and B are numbers so they can be interchanged.
B
r
·
A
r
= A B cos
(ii)
from eq. (i) and (ii)
L.H.S = R.H.S
(ii)
Parallelism property
(a)
If
A
r
·
B
r
= A B
then
A
r
//
B
r
Proof
A
r
·
B
r
= A B Cos
when
A
r
//
B
r
then
=0
o
A
r
·
B
r
= A B Cos0
o
: cos0
o
= 1
A
r
·
B
r
= A B
(b)
A
r
·
A
r
= A
2
as
A
r
·
A
r
= A A cos
A
r
B
r
A
r
A
r
A
r
A
r
B
r
x
B
A
r
B
r
x
A
B
r
Details
- Pages
- Type of Edition
- Erstausgabe
- Publication Year
- 2017
- ISBN (PDF)
- 9783960676683
- File size
- 2 MB
- Language
- English
- Publication date
- 2017 (June)
- Keywords
- Basic physics Easy Undergraduate Translatory motion Physical optics Physical quantities Newton Guidebook Newton's law Law of Motion
- Product Safety
- Anchor Academic Publishing
