Chapter 1(Physics 1st Year) Measurements
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Q # 1. Define Physics? Describe its main areas of research.
Ans. Physics is the branch of science that deals with matter, energy and the relationship between
them. The study of physics involves laws of motion, the structure of space and time, the nature and
types of forces, the interaction between different particles, the interaction of radiation with matter etc.
Q # 2. What do you know about the natural philosophy?
Ans. Initially, the observations of man about the world around him give birth to the single discipline
of science, called natural philosophy.
Q # 3. Differentiate among the physical and biological sciences.
Physical Sciences Biological Sciences
i) Physical sciences deal with non-living things.
ii) Examples: Physics, Chemistry, Astronomy
i) Biological sciences deal with living things.
ii) Examples: Zoology, Botany etc.
Q # 4. Define the following branches of modern physics.
(i) Nuclear Physics (ii) Particle Physics (iii) Relativistic Mechanics (iv) Solid State Physics
Ans. (i) Nuclear Physics: The nuclear physics deals with the atomic nuclei.
(ii) Particle Physics: It deals with the ultimate particles with which the matter is
composed.
(iii) Relativistic Mechanics: It deals with motion of bodies which moves with very large
velocities (approaching that of light).
(iv) Solid State Physics: The solid state physics deals with structure and properties of matter.
Q # 5. Write down the significance of science and technology. Also describe the role of physics in
the development of science and technology?
Ans. Modern tools of science and technology have brought all parts of world in close contact. The
information media and fast means of communications have made the world a global village. The
computer networks play pivotal role in the development of science and technology. The computer
networks are the products of chips developed from basic ideas of physics.
Q # 6. What do you know about physical quantities? Also describe their significance.
Ans. The quantities that can be measured and are used to describe the properties of matter are called
physical quantities.
Significance: The foundation of physics rest upon physical quantities in terms of which the laws of
physics are expressed.
Q # 7. Differentiate among the base and derived quantities.
Base Quantities Derived Quantities
(i) The base quantities are those physical
quantities in terms of which other physical
quantities are defined.
(ii) Examples: Mass, length, time
(i) The quantities that are derived from the base
quantities are called derived quantities.
(ii) Examples: Velocity, acceleration, force
Q # 8. How the base quantities are measured?
Ans. The measurement of base quantity involves two steps:
(i) The choice of a standard.
(ii) The establishment of a procedure for comparing the quantity to be measure with standard.
Q # 9. What are the characteristics of an ideal standard?
Ans. An ideal standard has two principle characteristics.
(i) It is accessible
(ii)It is invariable
Q # 10. What do you know about international system of units? Describe its significance.
Ans. In 1960, an international committee agreed on a set of definitions and standards to describe the
physical quantities. The system that was established is called System International of units.
Significance: Due to simplicity and convenience with which the units in this system are amenable to
arithmetic manipulation, it is in universal use by the world’s scientific community.
Q # 11. Define following?
(i) Base Units (ii) Derived Units (iii) Supplementary Units (iv) Radian (v) Steradian
Ans. (i) Base Units: The units associated with the base quantities are called base units.
(ii) Derived Units: The units associated with the derived quantities are called derived units.
(iii) Supplementary Units: The General Conference on Weights and Measures has not yet
classified certain unit of SI under either base or derived units. These SI units are called
derived supplementary units. Radian and steradian are supplementary units.
(iv) Radian: The angle between two radii of a circle corresponding to the arc length of one
radius on its circumference is called radian.
(v) Steradian: The 3D angle subtended at the center of the sphere corresponding to the
surface area of one square radius is called steradian.
Q # 12. What do you mean by scientific notation? Describe the following numbers in scientific
notation.
(i) . (ii) . (iii) . ×
Ans. The standard form to represent numbers using power of ten is called scientific notation. In
scientific notation of any measurement, there should be only one non-zero digit at the left of the
decimal point. The measurements expressed in scientific notation are as follows:
(i) The scientific notation of measurement 134.7 is 1.347 × 10
(ii) The scientific notation of measurement 0.0023 is 2.3 × 10
(iii) The scientific notation of measurement 43.94 × 10
is 4.394 × 10
Q # 13. Define error. Also describe possible causes of error.
Ans. The difference between the observed and calculated value of any measurement is called error.
The errors may occur due to following reasons.
(i) Negligence or inexperience of a person
(ii) The faulty apparatus
(iii) Inappropriate method or technique
Q # 14. What types of errors are possible in measuring the time period of pendulum by stop
watch? [BISE Sargodha 2008, 2009]
Ans. The possible errors that might occur are the personal error and systematic error. The personal
error occurs due to negligence or inexperience of a person, while the systematic error may be due to
the poor calibration of equipment or incorrect marking etc.
Q # 15. Differentiate among the random and systematic error.
Random Error Systematic Error
(i) If the repeated measurements of a quantity
give different values under same conditions,
then the error is called random error.
(ii) The random error occurs due to some
unknown causes
(iii) Repeating the measurement several times
and taking an average can reduce the effect
of random error.
(i) Systematic error refers to the effect that
influences all measurement of a particular
quantity equally.
(ii) It may occur due to zero error of the
instrument, poor calibration or incorrect
marking etc.
(iii) The systematic error can be reduced by
comparing the instrument with another
which is known to be more accurate.
Q # 16. What are the significant figures? Describe their significance.
Ans. In any measurement, the accurately known digits and the first doubtful digit are called the
significant figures. The uncertainty or accuracy in the value of a measured quantity is indicated by
significant figures.
Q # 17. How many significant figures are there in following measurements?
(i) (ii) . (iii) . (iv) .
(v) . (vi) . × (vii) with least count of
(viii) with least count of
Ans. (i) The number of significant figures in the measurement 1007 are 4.
(ii) The number of significant figures in the measurement 0.00467 are 3.
(iii) The number of significant figures in the measurement 02.59 are 3.
(iv) The number of significant figures in the measurement 7.4000 are 5.
(v) The number of significant figures in the measurement 3.570 are 4.
(vi) The number of significant figures in the measurement 8.70 × 10 are 3.
(vii) The number of significant figures in the measurement 8000, with least count of 10,
are 3.
(viii) The number of significant figures in the measurement 8000, with least count of
1000, is 1.
Q # 18. Write down the final result of following computation up to appropriate precision.
(i) . × !× ." × #
. " = 1.45768982 × 10
(ii) 72.1 (iii) 2.7543
3.42 4.10
0.003 1.2373
75.523 8.1273
Ans. (i) The final result up to appropriate precision is 1.46 × 10 . It is because of the reason that the
factor 3.64 × 10 , is the least accurate measurement which has three significant figures. Therefore
the answer should be written to the three significant figures.
(ii) The final result up to appropriate precision is 75.5. It is because of the reason that the factor 72.1
has smallest number of decimal places. Thus, the answer should be rounded off to one decimal place.
(iii) The final result up to appropriate precision is 8.13. It is because of the reason that the factor 4.10
has smallest number of decimal places. Thus, the answer should be rounded off to two decimal places.
Q # 19. Differentiate among precision and accuracy.
Precision Accuracy
(i) The precise measurement is one which has
least absolute uncertainty.
(ii) The precision of measurement depends on
the instrument or device being used.
(i) An accurate measurement is one which has less
fractional or percentage uncertainty.
(ii) The accuracy in any measurement not only
depends on instrument being used, but also on
the total measurement taken.
Q # 20. Which of the following measurement is more precise and which of them is mare
accurate.
(i) Length of object is recorded as 25.5 cm using meter rod.
(ii) The length of object is measured as 0.45cm using vernier calipers.
Solution.
(i) Length of object is recorded as 25.5 cm using meter rod.
%&'()*+, -./,0+12.+3 = 4,1'+ /(*.+ (5 6,+,0 0(7 = 0.1 /6
8,0/,.+19, -./,0+12.+3 =
:;<=>?@A BCDAE@FGCG@H
I=@F> JAF<?EAKAC@
× 100 =
.
.
× 100 = 0.4 %
(ii) The length of object is measured as 0.45 cm using vernier calipers.
%&'()*+, -./,0+12.+3 = 4,1'+ /(*.+ (5 M,0.2,0 /1))2N,0' = 0.01 /6
8,0/,.+19, -./,0+12.+3 =
:;<=>?@A BCDAE@FGCG@H
I=@F> JAF<?EAKAC@
× 100 =
.
.
× 100 = 2 %
Result: The measurement (ii) is more precise because it has less absolute uncertainty. The
measurement (i) is more accurate as it has less percentage uncertainty.
Q # 21. Assess the total uncertainty in the final result for following cases
(i) Find out displacement between points e = 10.5 ± 0.1 and e = 26.8 ± 0.1
(ii) If the potential difference of V = 5.2 ± 0.1 volt applied across the ends of conductor, and as
the result the current i = 0.84 ± 0.05 pass through conductor. Determine the resistance of
conductor.
(iii) Find out volume of sphere whose radius 0 = 2.25 ± 0.01 /6.
(iv) The six measurements were taken of the diameter of wire using screw gauge which are 1.20,
1.22, 1.23, 1.19, 1.22, 1.21. Determine the uncertainty in final result.
(v) The simple pendulum completes 30 vibrations 50.6 s. the least count of the stop watch is 0.01
s. Find out uncertainty in the time period of simple pendulum.
Ans. (i) Given points are
e = 10.5 ± 0.1, e = 26.8 ± 0.1
Displacement e =?
e = e − e
= (26.8 ± 0.1) − (10.5 ± 0.1)
= 16.3 ± 0.2 /6
(ii) Resistance l = ?
Given quantities are
Potential Difference V = 5.2 ± 0.1 volt
Current i = 0.84 ± 0.05
By Ohm’s law,
l =m
n = .
. = 6.2 o ---------------------- (1)
Uncertainty = ?
% -./,0+12.+3 2. p = 0.1
5.2 × 100 = 2 %
% -./,0+12.+3 2. i = 0.05
0.84 × 100 = 6 %
Therefore,
l = 6.2 o with 8 % uncertainty
or
l = 6.2 ± 0.5 o
(iii) Volume of sphere p = ?
Given
Radius 0 = 2.25 ± 0.01 /6
As Volume of sphere p =
q0
=
q(2.25) = 47.7 /6 ---------------------- (1)
Uncertainty = ?
% -./,0+12.+3 2. 0 = 0.01
2.25 × 100 = 0.4 %
r(+1) -./,0+12.+3 2. 0 = 3 × 0.4 % = 1.2 %
Hence
Volume of sphere p = 47.7 /6 with 1.2 % uncertainty
or p = 47.7 ± 0.6/6
(iv) The measurements of diameter of wire are 1.20, 1.22, 1.23, 1.19, 1.22, 1.21
Average diameter of wire = . t . t . t . ut . t .
" = 1.21
Deviation of each measurement from average value are 0.01, 0.01, 0.02, 0.02, 0.1,0
Mean Deviation = . t . t . t . t . t
" = 0.01
Thus uncertainty in mean value of diameter = 0.01
Hence Diameter of wire =1.21 ± 0.01 66
(v) Given that
Time for 30 Vibrations = 54.6 s
Time Period = Time for 1 Vibration = ."
= 1.82 s
Uncertainty = ?
-./,0+12.+3 = vAF<@ w=?C@
I=@F> x?K;AE =y mG;EF@G=< = .
= 0.003 '
Thus time period is expressed as
r = 1.82 ± 0.003 s
Q # 22. What do you know about the dimension analysis?
Ans. To express any physical quantity in by scientific symbols of corresponding base quantities,
written within the square brackets, called the dimensions. The scientific symbols used to express the
dimensions of different physical quantities are as follows
Dimension of Mass = [ M ]
Dimension of Length = [ L ]
Dimension of Time = [ T ]
Q # 23. Write down the dimensions of velocity, acceleration and force?
Dimension of Velocity = Dimensions of Displacement
Dimensions of Time ⟹[ v ] = [ L ]
[ T ] = [ LT
]
Dimension of acceleration = Dimensions of Velocity
Dimensions of Time ⟹[ a ] = [ LT
]
[ T ] = [ LT
]
Dimension of Force = (Dimensions of Mass)(Dimensions of acceleration)
⟹[ F ] = [ m ][ a ] = [ M ][ LT
] = [ MLT
]
Q # 24. What are the advantages of dimension analysis?
Ans. The dimension analysis may be used for
(i) Checking the correctness of a physical equation
(ii) Deriving a possible formula of a physical quantity
Q # 25. What is homogeneity principle?
Ans. According to homogeneity principle “If the dimensions of a physical quantity on both sides of
equation are the same, then the equation will be dimensionally correct”.
Q # 26. Write down any two drawbacks of dimension analysis?
Ans. The major drawbacks of dimension analysis are
(i) The dimension analysis is unable to find the values of various constant in physical equations.
(ii) The dimension analysis cannot be applied to the physical quantities involving trigonometric
and logarithmic functions.
Q # 27. Determine the dimension of following physical quantities?
(i) Nuclear Energy (ii) Angle (θ)
Ans. (i)
‚26,.'2(.' (5 ƒ.,093 = ‚26,.'2(. (5 „(0…
[ƒ.,093] = [„] = [†. 7]
[ƒ.,093] = [†]. [7] = [‡4r
][4] = [‡4 r
]
(ii) We know that
ˆ = 0‰
⟹‰ =Š
E ⟹[‰] = [Š]
[E] = [v]
[v] = 1
Therefore, angle is a dimensionless quantity.
Chapter 1(Physics 1st Year) Measurements
8
Written and composed by: Prof. Muhammad Ali Malik, Govt. Degree College, Naushera, 03016775811
EXERCISE SHORT QUESTIONS
Q # 1. Name several repetitive phenomenon occurring in nature which can serve as reasonable
time standards.
Ans. Any natural phenomenon that repeats itself after exactly same time interval can be used as time
standard. The following natural phenomenon can be used as time standard.
The rotation of earth around the sun and about its own axis
The rotation of moon around earth
Atomic vibrations in solids
Q # 2. Give the drawbacks to use the time period of a pendulum as a time standard.
Ans. The time period of the simple pendulum depends upon its length and value of ‘g’ (gravitational
acceleration) at any place. Therefore, the drawbacks to use the time period of a pendulum as a time
standard are
The value of ‘g’ changes at different places
The variation in the length of simple pendulum due to change in temperature in different
seasons
Air resistance may affect the time period of simple pendulum
Q # 3. Why we use it useful to have two units for the amount of substance, the kilogram and the
mole?
Ans. The kilogram and mole are the units to determine the amount of a substance. Both units are
useful in different cases describe below
The unit kilogram is useful when we want to describe the macroscopic behavior of an object
without considering the interaction between the atoms present in it
The unit mole is useful when we want consider a fix number of atoms of a system. It is used
to determine the microscopic behavior of any object.
Q # 4. Three students measured the length of a needle with a scale on which minimum division
is 1 mm and recorded as (i) 0.2145 (ii) 0.21 (iii) 0.214. Which record is correct and why?
Ans. The record (iii) is correct.
Reason: As the scale used for measurement has the least count of 1 mm = 0.001 m. So the reading
must be taken up to three decimal places. Therefore, the reading 0.214 m is correct.
Q # 5. An old saying is that “A chain is only as strong as its weakest link”. What analogous
statement can you make regarding experimental data used in computation?
Ans. The analogous statement for experimental data used in computation will be
“A result obtained by mathematical computation of experimental data is only as much
accurate as its least accurate reading in measurements”.
Q # 6. The time period of the simple pendulum is measured by a stop watch. What type of
errors are possible in the time period?
Ans. The possible errors that might occur are the personal error and systematic error. The personal
error occurs due to negligence or inexperience of a person, while the systematic error may be due to
the poor calibration of equipment or incorrect marking etc.
Q # 7. Does the dimensional analysis gives any information on constant of proportionality that
may appear in an algebraic expression. Explain?
Ans. Dimension analysis does not give any information about constant of proportionality in any
expression. This constant can be determined experimentally of theoretically.
Example: In the expression of time period of simple pendulum, the constant of proportionality cannot
be determined from dimension analysis.
Q # 8. What are the dimensions of (i) Pressure (ii) Density
(i) By definition, Pressure = ŒŽ
‘Ž’
Dimension of Pressure = Dimensions of Force
Dimensions of Area ⟹[ P ] = [ F ]
[ A ]
∵ [ F ] = [ m ][ a ] = [ M ][ LT
] = [ MLT
]
⟹[ P ] = [ F ]
[ A ] = [ MLT
]
[ L ]
⟹[ P ] = [ ML
T
]
(ii) By definition, Density = •’––
—˜™š
Dimension of Density = Dimensions of Mass
Dimensions of Volume ⟹[ › ] = [ mass ]
[ volume ]
∵ [ mass ] = [ M ]
∵ [ volume ] = [ L ]
⟹[ › ] = [ M ]
[ L ] = [ ML
]
Q # 9. The wavelength œ of a wave depends on the speed of the wave and its frequency ž. Decide which
of the following is correct, ž = œ or (ii) ž =
œ
(i) 5 = M Ÿ
Dimension of LHS = [ 5] = [ T
]
Dimension of RHS = [ MŸ] = [M] [Ÿ]
∵ [M] = [ LT
]
∵ [Ÿ] = [L]
Dimension of RHS = [ MŸ] = [ LT
] [L]
= [L T
]
As Dimension of LHS ≠ Dimension of RHS
Hence, the equation 5 = M Ÿ is not
dimensionally correct.
(ii) 5 = £
¤
Dimension of LHS = [ 5] = [ T
]
Dimension of RHS = ¥ M
Ÿ¦ = [M]
[Ÿ] = [ LT
]
[L]
= [ T
]
As
Dimension of LHS = Dimension of RHS
Hence, the equation 5 = £
¤
is
dimensionally correct.
Chapter 2 (1st Year Physics) Vectors and Equilibrium
IMPORTANT QUESTIONS WITH ANSWERS
Q # 1. Differentiate among scalars and vectors.
Scalars Vectors
(i) The physical quantities that are completely
described by magnitude with proper unit are
called scalars.
(ii) Mass, length, time and speed are examples of
scalars.
(i) The physical quantities that are completely
described by magnitude with proper unit and
direction are called vectors.
(ii) Displacement, velocity, acceleration, force and
momentum are examples of vectors.
Q # 2. What do you know about rectangular coordinate system? Describe its significance.
Ans. The lines which are drawn perpendicular to each other are called coordinate axis and a system of
coordinate axis is called the rectangular or Cartesian coordinate system. A coordinate system is used to describe
the location of a body with respect to a reference point, called origin.
Q # 3. Describe the Head to Tail rule.
Ans. The vectors can be added graphically by head to tail rule. According to this rule, the addition of two
vectors A and B consists of following steps:
(i) Place the tail of vector B on the head of vector A.
(ii) Draw a vector from the tail of vector A to the head of vector
B, called the resultant vector.
Q # 4. What do you know about the Resultant Vector?
Ans. The vector which has the same effect as that of all
component vectors is called resultant vector. Consider four vectors A, B, C and D are added by head to tail rule
and R is their resultant vector, as shown in the figure.
The vector R has the same effect as the combined effect of vectors
A, B, C and D.
Q # 5. Define following
(i) Negative of a Vector
The vector which has the same magnitude as that of vector A, but
opposite in direction is called negative of vector A.
(ii) Vector Subtraction
Subtraction of a vector is equivalent to the addition of one vector
into negative of second vector. Consider two vectors A and B. In
order to subtract B from A, the negative of vector B is added to
vector A by head to tail rule.
The resultant C is given by
–
(iii) Equal Vector
Two vectors are said to be equal if they have same magnitude and direction.
(iv) Null Vector
A vector of zero magnitude and arbitrary direction is called null vector.
(v) Component of a Vector
A component of a vector is its effective value in a specific direction.
(vi) Rectangular Component
The components of a vector which are perpendicular to each other are called rectangular components.
(vii) Position Vector
The position vector describes the location of a point with respect to origin. In two dimension, the
position vector ‘ ’ of point
,
is describe as
ı̂ ȷ̂
The magnitude of this position vector will be
In three dimensional Cartesian coordinate system, the position vector ‘ ’ of point
, ,
is describe as
ı̂ ȷ̂ k
The magnitude of this position vector will be
Q # 6. Discuss the different cases of multiplication of a vector by a scalar (number).
Case -1
If any scalar > 0 is multiplied by a vector ‘A’, then the magnitude of the resultant ‘n ’ will become
n times (|nA|) but the direction remains same as that of A.
Case-2
If any scalar < 0 is multiplied by vector, then the magnitude of the resultant vector will become n
times and the direction will reverse.
Q # 7. What do you about Unit Vector? Describe its significance.
Ans. A vector having the unit magnitude is called the unit vector. It is used to indicate the direction of any
vector. The unit vector in the direction of vector A is expressed as
A
|A|
where A is the unit vector in the direction of vector A and |A| is its magnitude. In space, the direction of x, y
and z-axis are represented by unit vectors ı̂, ȷ̂ and k , respectively.
Q # 8. Find out the rectangular component of a vector.
Ans. Consider a vector A, represented by a line "!"" " which makes an angle # with the x-axis.
We want to find out rectangular components of vector A. For
this, we draw a perpendicular from point ‘ ’ on x-axis. Projection "!"$"
being along x-direction is represented by %&ı ̂ and projection "$""" " along
y -direction represented by %'ȷ̂ . By head to tail rule:
%&ı̂ %'ȷ̂ --------------- (1)
For x component
()#
"!""$""
"!"" "
%&
%
⟹%& % ()#
For y component
)+ #
"""$""
"!"" "
%'
%
⟹%' %)+ #
Putting values of %& and %' in eq. (1), we get
% ()# ı̂ %)+ # ȷ̂
Q # 9. Determine a vector from its rectangular component.
Ans. Let %& and %' are the rectangular components of vector A which is represented by a line "!"" " as shown in
the figure below
We want to determine the magnitude and direction of vector A with
x-axis.
Magnitude
The magnitude of vector A can be find using Pythagorean
Theorem. In triangle !$
"!"" "
!"""$""
"$""" "
% %&
%'
% ,%&
%'
This expression gives the magnitude of resultant
Direction
In right angle triangle !$
-
#
$"""" "
"!""$""
-
#
%'
%&
Chapter 2 (1st Year Physics) Vectors and Equilibrium
4
Written and composed by: Prof. Muhammad Ali Malik, Govt. Degree College, Naushera, 03016775811
# tan01 2
%'
%&
3
This expression gives the direction of the vector A with respect to x-axis.
Q # 10. Describe the vector addition in terms of rectangular components.
Ans. Consider two vectors A and B represented by lines !"""$"" and "$""" ", respectively . By head to tail rule, the
resultant vector is given by
4
Let 5& and 5' are the rectangular components of resultant
vector R along x and y-axis respectively, then we can write
4 5& ı̂ 5' ȷ̂ ----------- (1)
From figure,
"!""5" "!""6"" "6""5"
5& %& 7& ---------- (2)
Also,
5""" " 5""8"" "8"" "
5' %' 7' ---------- (3)
Putting values of 5& and 5' in eq. (1), we get
4 5& ı̂ 5' ȷ̂
4 %& 7&
ı̂ 9%' 7': ȷ̂
Which is the expression of resultant in terms of rectangular components.
Magnitude of Resultant
The magnitude of resultant can be expressed as
R ,5&
5'
Putting the values of 5&and 5',
R , %& 7&
9%' 7':
Direction
The direction of resultant can be find out using expression,
# tan01 2
5'
5&
3
# tan01 2
%' 7'
%& 7&
3
Generalization
If 4 is the resultant vector of a large number of coplanar vectors represented by %, 7, <,……, then the
expression for the magnitude of resultant will become
R , %& 7& <& ⋯
9%' 7' <' ⋯:
The direction of resultant vector 4 with x-axis can be find out using expression
# tan01 2
%' 7' <' ⋯
%& 7& <& ⋯
3
Q # 11. Differentiate among scalar and vector product.
Scalar Product Vectors Product
(i) When two vectors are multiplied to give a scalar quantity,
then the product of vectors is called the scalar or dot
product. The scalar product of two vectors and is
written as . and is defined as
. %7 ()#
where % and 7 are the magnitudes of vector and and
# is the angle between them.
(ii) The work done @ is the dot product of force A and
displacement B is an example of scalar product.
Mathematically, it is written as
@ A.B CD ()#
(i) When two vectors are multiplied to give a vector quantity,
then the product of vectors is called the vector or cross
product. The vector product of two vectors and is
written as E and is defined as
E %7)+ # FG
where % and 7 are the magnitudes of vector and and
# is the angle between them and FG is the unit vector
perpendicular to the plane containing and .
(ii) The turning effect of force is called the torque and is
determined from the vector product of force A and position
vector . Mathematically, it is written as
Torque H E A
Q # 12. Show that the scalar product is commutative.
Consider two vectors and . Place the both vector tail to tail as shown in Fig. (a)
Then, from Fig. (b)
. $
I +-JDK (L
(MK -+( (L (
OR
. $
I +-JDK (L
<(NO( K- (L + -PK D+ K -+( (L
. %
7 ()#
%7 ()# ---------------------- (1)
Similarly, from Fig. (c)
. 7
% ()#
7% ()# %7 ()# ---------------------- (2)
Thus, from eq. (1) and (2)
. .
Hence, the scalar product is commutative.
Q # 13. Show that the vector product is non-commutative.
Consider two vectors and . Place the both vector and tail to tail to define the plane of and .
The direction of the vector product will be perpendicular to the plane of and and can be determined using
right hand rule.
By applying the right hand rule on the vector products of E and E [Fig.(a) and Fig. (b)], it is
clear that product vectors E and E are anti-parallel to each other i.e.,
Q E R S R E Q
Hence, the vector product is not commutative.
Q # 14. Compare the main characteristics of scalar and vector product.
Scalar Product Vector Product
(i) Scalar product is commutative.
That is, for vectors and , . .
(ii) Scalar product of two mutually perpendicular vectors is
zero.
If the two vectors are and mutually perpendicular
to each other, then
. %7 () 90˚ 0
(iii) The scalar product of two parallel vectors is equal to the
product of their magnitudes.
If the two vectors are and parallel to each other,
then . %7 () 0˚ %7
If the two vectors are and anti-parallel to each
other, then . %7 () 180˚ %7
(iv) The self scalar product of vector is equal to the square
of its magnitudes. . %% () 0˚ %
(v) Scalar product of vectors and in terms of their
rectangular components will be
. 9%& ı̂ %' ȷ̂ %X k :. 7& ı̂ 7' ȷ̂ 7X k
. %& 7& %'7' %X7X
(vi) The angle between these vector can be find out by
putting the value of . in above equation
. %7 ()# %& 7& %'7' %X7X
()#
%& 7& %'7' %X7X
%7
(i) Vector product is non-commutative.
That is, for vectors and , Q E R S R E Q
(ii) Vector product of two mutually perpendicular vectors
has maximum magnitude.
If the two vectors are and mutually perpendicular
to each other, then
E %7)+ 90˚ FG %7 FG
(iii) The vector product of two parallel and anti-parallel
vectors is the null vector.
If the two vectors are and parallel to each other,
then E %7)+ 0˚ FG Y
If the two vectors are and anti-parallel to each
other, then E %7)+ 180˚ FG Y
(iv) The self vector product of vector is the null vector.
E %%)+ 0˚ FG Y
(v) Vector product of vectors and in terms of their
rectangular components will be
E 9%& Ẑ %' [̂ %X \] : E 7& Ẑ 7' [̂ 7X \]
E 9%' 7X %X7': Ẑ %X 7& %&7X
[̂
9%& 7' %'7&: \]
E ^
Ẑ [̂ \]
%& %' %X
7& 7' 7X
^
(vi) The magnitude of E is equal to the area of
parallelogram formed with and as two adjacent sides
Q # 15. Describe the right hand rule.
According to right hand rule, the right hand is placed on the first vector and fingers
are curled towards the second vector, keeping the thumb erect. The erected thumb will give
the direction of product vector.
Q # 16. Define the term ‘torque’.
Ans. The turning effect of a force is called torque. The torque ‘H’ acting on a body under the
action of force ‘A’ is described as
H E A
Where is the position vector of point of application of force with respect to pivot point ‘O’.
Anticlockwise torque is taken as positive, while
the clockwise torque is considered as negative.
Q # 17. Derive the expression for torque
produce in a rigid body under action of any
force.
Ans. Let the force ‘A’ acts on rigid body at
point whose position vector relative to pivot ‘O’
is r.
We want to find out the expression torque about point ‘O’ acting on the rigid body due to
force ‘A’.
The force ‘A’ makes an angle ‘_’ with horizontal, therefore, it can be resolved in two
rectangular components i.e., ‘C ()_’ and ‘C)+ _’. The torque due to ‘C ()_’ about point
‘O’ is zero as its line of action posses through this point. Therefore, the ‘C)+ _’ component
of forces is responsible for producing torque in the body about point ‘O’.
Now the torque,
` Force
Moment Arm
` C)+ _
` C)+ _
H E A
This is the required expression of torque.
Q # 18. Define the term “equilibrium”. Write down different types of equilibrium.
Ans. A body is said to be in state of equilibrium if it is at rest or moving with uniform
velocity. There are two types of equilibrium.
(i) Static Equilibrium
If a body is at rest, then it is said to be in static equilibrium.
(ii) Dynamic Equilibrium
If the body is moving with uniform velocity, then it is said to be in dynamic
equilibrium.
Q # 19. Write down different conditions of equilibrium.
Ans. There are two conditions of equilibrium.
First Condition of Equilibrium
The vector sum of all forces acting on any object must be zero. Mathematically,
hA 0
In case of coplanar forces, this conditions is expressed usually in terms of x and y
components of forces. Hence, the 1st condition of equilibrium for coplanar forces will be
ΣAj 0, ΣAk 0
When the first condition of equilibrium is satisfied, there will be no linear acceleration
and body will be in translational equilibrium.
Second Condition of Equilibrium
The vector sum of all torque acting on any object must be zero. Mathematically,
hH 0
When the second condition of equilibrium is satisfied, there is no angular acceleration
and body will be in rotational equilibrium.
Q # 20. State the complete requirement for a body to be in equilibrium?
Ans. A body will be in the state of complete equilibrium, when the sum of all the forces and
torques acting on the body will be equal to zero. Mathematically, it is described as
(i) ΣA 0 i.e. ΣAj 0, ΣAk 0
(ii) ΣH 0
EXERCISE (SHORT QUESTIONS)
Q # 1. Define the terms (i) Unit Vector (ii) Position Vector (iii) Component of a Vector.
(viii)Unit Vector
A vector having the unit magnitude is called the unit vector. It is used to indicate the direction of any
vector. The unit vector in the direction of vector A is expressed as
A
|A|
where A is the unit vector in the direction of vector A and |A| is its magnitude.
(ix) Position Vector
The position vector describes the location of a point with respect to origin. In two dimension, the
position vector ‘ ’ of point
,
is describe as
ı̂ ȷ̂
The magnitude of this position vector will be
In three dimensional Cartesian coordinate system, the position vector ‘ ’ of point
, ,
is describe as
ı̂ ȷ̂ k
The magnitude of this position vector will be
(x) Component of a Vector
A component of a vector is its effective value in a specific direction.
Q # 2. The vector sum of three vectors gives a zero resultant. What can be the orientation of the vectors?
Ans. If the three vectors are such that they can be represented by the sides of a triangle taken in cyclic order,
then the vector sum of three vectors will be zero.
Let three vectors , and are the three vectors acting along the sides of
triangle ! 6 as shown in the figure.
As the head of coincides with the tail of , so by head to tail rule,
the resultant of these three vectors will be zero.
Q # 3. Vector lies in xy plane. For what orientation will both of its
rectangular components be negative? For what components will its
components have opposite signs?
Ans. i) When the vector lies in 3rd quadrant, then both of its
rectangular components of vector will negative.
ii) The components of a vector have opposite sign when
the vector lies in 2nd or 4th quadrant.
If %& and %' are the rectangular components of vector ,
then rectangular components of vectors in different quadrants will
be:
Q # 4. If one of the rectangular components of a vector is not zero, can its magnitude be zero? Explain.
Ans. If one of the components is not zero, then the magnitude of vector can’t be zero. If %& and %' are the
rectangular components of vector , then its magnitude will be:
Magnitude of % ,%& %'
If %& 0, then % ,0 %' %'
If %& 0, then % ,%& 0 %&
Q # 5. Can a vector have a component greater than the vector’s magnitude?
Ans. The magnitude of the component of a vector can never be greater than the vector’s magnitude because the
component of a vector is its effective value in a specific direction. The maximum value of magnitude of
component can be equal to the magnitude of the vector.
Q # 6. Can the magnitude of a vector have a negative value?
Ans. No, the magnitude of a vector cannot be negative, because the magnitude of vector can be described by
the formula:
Magnitude of = % ,%& %'
Where %& and %' are the rectangular components of . As the squares of real quantities always gives the
positive values. Therefore, the magnitude of a vector will always be positive.
Q # 7. If l, what can you say about the components of the two vectors.
Ans. Given that: 0
⟹
These vectors can be expressed in terms of rectangular components,
%&ı̂ %'ȷ̂ 97&ı̂ 7'ȷ̂:
%&ı̂ %'ȷ̂ 7&ı̂ 7'ȷ̂
Comparing the coefficients of unit vectors ı̂ and ȷ̂, we get:
%& 7& and %' 7'
Hence the components of both vectors are equal in magnitude but opposite in direction.
Q # 8. Under what circumstances would a vector have components that are equal in magnitude?
Ans. The components of a vector will have equal magnitude when it makes an angle of 45° with x-axis. If a
vector makes an angle of 45°, then its rectangular components will be:
%& % cos 45˚ = 0.707 %
%' = % sin 45˚ = 0.707 %
Q # 9. Is it possible to add a vector quantity to a vector quantity to a scalar quantity?
Ans. No it is not possible to add a vector quantity to a scalar quantity because the physical quantities of same
nature can be added. Vectors and scalars are different physical quantities. It means that vectors can be added to
vectors and scalars are added in scalars, but the vectors can’t be added to scalar.
Q # 10. Can you add zero to a null vector?
Ans. No, zero can’t be added to a null vector because zero is a scalar and scalars can’t be added to vectors. Only
the physical quantities of same nature can be added.
Q # 11. Two vectors have unequal magnitudes. Can their sum be zero? Explain.
Ans. No, the sum of two vectors having unequal magnitudes can’t be zero. The sum of two vectors will be zero
only when their magnitudes are equal and they act in opposite direction.
Q # 12. Show that the sum and the difference of two perpendicular vectors of equal lengths are also
perpendicular and of same length.
Ans. Consider two vectors and of equal magnitude which are perpendicular to
each other. The sum and the difference of both vectors gives the resultant 4 and 4′,
respectively, and are described below:
4 % ı̂ 7ȷ̂
4s % ı̂ 7ȷ̂
Magnitude of 4 5 √% 7 -------------- (1)
Magnitude of 4′ 5′ √% 7 -------------- (2)
From (1) and (2), it is clear that the sum and the difference of two perpendicular vectors of equal magnitude
have the same lengths. Now taking dot product of 4 and 4′, we get:
4 .4s % ı̂ 7ȷ̂
. % ı̂ 7ȷ̂
% 7 ∵ | | | | ⟹% 7
4 .4s % % 0
As 4 .4s 0, therefore, the sum and the difference of two perpendicular vectors of equal magnitude are
perpendicular to each other.
Q # 13. How would the two vector same magnitude have to be oriented, if they were to be combined to
give a resultant equal to a vector of same magnitude?
Ans. The two vectors of equal magnitudes are combined to give a resultant vector of same magnitude when they
act at an angle of 120˚ with each other.
Consider two vectors and of equal magnitude which makes an
angle of 120˚ with each other. Both vectors are added by head to tail rule to give
resultant 4 as shown in the figure below:
From figure it is clear that
4 and
|4| | | | |
Q # 14. The two vectors to be combined have magnitude 60 N and 35 N. Pick the correct answer from
those given below and tell why is it the only one of the three that is correct.
(i) 100 N (ii) 70 N (iii) 20 N
Ans. The correct answer is 70 N.
1. The resultant of two vectors has maximum magnitude when they act in same direction. Thus if both
vectors are parallel, then the magnitude of resultant will be: 60 x 35 x 95 x.
2. The resultant of two vectors has minimum magnitude when they act in opposite direction. Thus if both
vectors are anti-parallel, then the magnitude of resultant is 60 x 35 x 25 x.
Hence the sum can’t be less than 25 x and more than 95 x. Therefore, the only possible value for
correct answer is 70 x.
Q # 15. Suppose the sides of a closed polygon represent vector arranged head to tail. What is the sum of
these vectors?
Ans. If there are five vectors A, B, C, D and E which are acting along the
sides of close polygon as shown in the figure:
As the tail of the first vector meets with the head of last vector, so
by head to tail rule:
z { 0
Hence the sum of vectors arranged along the sides of polygon will be zero.
Q # 16. Identify the correct answer.
(i) Two ships X and Y are travelling in different direction at equal speeds. The actual direction of X is
due to north but to an observer on Y, the apparent direction of motion X is north-east. The actual
direction of motion of Y as observed from the shore will be
(A) East (B) West (C) South-east (D) South-West
Ans. The correct answer is (B) West
(ii) The horizontal force F is applied to a small object P of mass m at rest on a smooth plane inclined at
an angle | to the horizontal as shown in the figure
below. The magnitude of the resultant force acting up
and along the surface of the plane, on the object is
(a) A }~ € ‚ ƒF €
(b) A ƒF € ‚ }~ €
(c) A }~ € ‚ ƒF €
(d) A ƒF € ‚ }~ €
(e) ‚ „…F€
Ans. The forces acting up and along the surface of plane is
A }~ € ‚ ƒF €, therefore the correct option is (a)
Q # 17. If all the components of the vectors, † and ‡ were reversed, how would this alter † E ‡.
Ans. If all the components of the vectors † and ‡ are reversed, then both vectors will be represented as †
and ‡, respectively. Therefore,
† E ‡ † E ‡
Hence the vector product of two vectors will remain unchanged even when the components of the vectors are
reversed.
Q # 18. Name the three different conditions that could makes † E ‡ O
Ans. The conditions that could make the † E ‡ O are as follows:
If † is the null vector
If ‡ is the null vector
If the vectors † and ‡ are parallel or anti-parallel with each other.
Q # 19. Identify true or false statements and explain the reason.
(a) A body in equilibrium implies that it is neither moving nor rotating.
(b) If the coplanar forces acting on a body form a close polygon, then the body is said to be in
equilibrium.
Ans. i) Statement (a) is false. Because a body may be in equilibrium if it is moving or rotating with uniform
velocity.
ii) Statement (b) is correct. Since the vector sum of all the forces acting on the body along close polygon is
zero, then the first condition of equilibrium will be satisfied and the body will be in state of equilibrium.
Q # 20. A picture is suspended from a wall by two strings. Show by diagram the configuration of the
strings for which the tension in the string is minimum.
Ans. Consider a picture of weight w is suspended by two strings as shown in the figure.
From figure,
2‰ sin # @
⟹‰
Š
‹Œ Ž
Case 1: For # 0˚
‰
@
2 sin 0˚
@
0
∞
Case 2: For # 45˚
‰
@
2 sin 45˚
0.7 @
Case 3: For # 90˚
‰
@
2 sin 90˚
0.5 @
Hence it is clear that the tension will be minimum for # 90˚
Q # 21. Can a body rotate about its center of gravity under the action of its weight?
Ans. No a body can’t rotate about the center of gravity under the action of its weight. The whole weight of the
body acts on the center of gravity. The torque due to weight will be zero because the moment arm is zero in this
case. Hence, a body cannot rotate about center of gravity under the action of its weight.
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