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Units and Measurements
Need for measurement: Units of measurement; systems of
units; SI units, fundamental and derived units. Length,
mass and time measurements; accuracy and precision of
measuring instruments; errors in measurement; significant
figures. Dimensions of physical quantities, dimensional
analysis and its applications.
Revised syllabus -CORONA
PANDEMIC
2020
Units and Measurements
Physical Quantity
• Physical Quantity: A quantity which can
be measured and expressed in form of
laws is called a physical quantity.
• Physical quantity (Q) = n × u , Where n
represents the numerical value and u
represents the unit.
• As the unit(u) changes, the magnitude (n)
will also change but product nu will remain
same. i.e. n u = constant, or n1u1 = n2u2
= constant;
Fundamental and Derived Units
• Fundamental and Derived Units:
• Any unit of mass, length and time in
mechanics is called a fundamental,
absolute or base unit.
• Other units which can be expressed in
terms of fundamental units, are called
derived units
System of units
• System of units : A complete set of units, both
fundamental and derived for all kinds of
physical quantities is called system of units. (1)
CGS system, (2) MKS system, (3) FPS system.
(4) S.I. system : It is known as International
system of units.
• There are seven fundamental quantities in
this system. These quantities and their units
are given in the following table.
SI UNITS
• Quantity Name of Units Symbol
• Length Metre m
• Mass Kilogram kg
• Time Second s
• Electric Current Ampere A
• Temperature Kelvin K
• Amount of Substance Mole Mol
• Luminous Intensity Candela Cd
• Besides the above seven fundamental units two
supplementary units are also defined - Radian (rad) for
plane angle and Steradian (sr) for solid angle
In a practical unit if the unit of mass becomes double and that of unit
of time becomes half, then 8 joule will be equal to ______________
8 joules of work done can be expressed,
As work= 8 joules = 8kg m²/ s²
where normal units
mass = kg
length= m
Time = second=s
let the new units be KG,M and S.
According to question ,
KG = 2kg
M= no change mentioned = so assume same = m
S= s/2
work =8 kg m/ s=8 (KG/2)*M²/(2S)²=8/8(KGM²/S²)
=1 Joule of work.
LARGEST AND SMALLEST PRACTICAL
UNITS
Q LARGEST PRACTICAL UNIT SMALLEST PRACTICAL
UNIT
LENGTH 1A.U 1 FERMI
MASS 1 C.S.L 1a.m.u
TIME 1 CENTURY 1 SHAKE
DIMENSIONAL ANALYSIS
Dimensions of a Physical Quantity
• Dimensions of a Physical Quantity: When a
derived quantity is expressed in terms of
fundamental quantities, it is written as a
product of different powers of the
fundamental quantities.
• The powers to which fundamental quantities
must be raised in order to express the given
physical quantity are called its dimensions.
Following examples show how dimensions of
the physical quantities are combinations of the
powers of M, L and T :
• (i) Volume requires 3 measurements in length.
So it has 3 dimensions in length (L3).
• (ii) Density is mass divided by volume. Its
dimensional formula is ML–3.
• (iii) Speed is distance travelled in unit time or
length divided by time. Its dimensional
formula is LT–1.
• (iv) Acceleration is change in velocity per unit
time, i.e., length per unit time per unit time.
Its dimensional formula is LT–2.
Dimensions of acceleration are 1 in length, -2 in
time.
• (v) Force is mass multiplied by acceleration. Its
dimensions are given by the formula MLT–2.
TRY DIMENSIONS OF OTHER PHYSICAL
QUANTITIES GIVEN BELOW:
Important Dimensions
• S.NO Quantity Dimension
1. Velocity or speed (v) LT-1
2. Acceleration LT-2
3. Distance L
4. Force MLT-2
5. Pressure ML-1T-2
6. Stress(F/A) ML-1T-2
7. Strain DIMENSIONLESS
8. Angle DIMENSIONLESS
9. Gravitational constant M-1L3T-2
10. Acceleration due to gravity LT-2
Application of Dimensional Analysis.
• (1) To find the unit of a physical quantity in a given
system of units.
• (2) To find dimensions of physical constant or
coefficients.
• (3) To convert a physical quantity from one system to
the other.
• (4) To check the dimensional correctness of a given
physical relation: This is based on the ‘principle of
homogeneity’. According to this principle the
dimensions of each term on both sides of an equation
must be dimensionally same.
• (5) To derive relation between various physical
quantities..
EXAMPLES 1: To find the unit of a physical quantity in a given
system of units
The dimensions of mv2 /2 are M.(LT–1)2,
or ML2T–2. (Remember that the numerical
factors have no dimensions.)
So S.I. unit is kgm2s-2 which is the Unit of
energy
EXAMPLE2 : To find the dimensions of
epsilon
• Electric permittivity:
• Electrostatic force F= 1/4π εO (q1 q2/r2)
εO= (1/4π F) q1 q2/r2
Unit of εO= coulomb2/m2N
Dimension of εO = A2T2/L2MLT-2
= M–1L–3T4A2
EXAMPLE 3: Consistency of a Dimensional Equation
The similarity of dimensions of physical quantities is called consistency of a
dimensional equation
• Check whether the given equation is dimensionally correct.
• W = 1/2 mv2 – mgh
where W stands for work done, m means mass, g stands for gravity, v
for velocity and h for height.
To check the above equation as dimensionally correct, we first write
dimensions of all the physical quantities mentioned in the equation.
• W = Work done = Force × Displacement = [MLT-2] × [L] = [ML2T-2]
• 1/2 mv2 = Kinetic Energy = [M] × [L2T-2] = [ML2T-2]
• mgh = Potential Energy = [M] × [LT-2] × [L] = [ML2T-2]
Since all the dimensions on left and right sides are equal it is a
dimensionally correct equation.
EXAMPLE 4
• Check the consistency of the equation
S-ut-1/2at2=0
L.H.S.
• s= distance = [L]
• ut = velocity × time = [LT-1] × [T] = [L]
• at2 = acceleration × time2 = [LT-2] × [T2] = [L]
R.H.S
“ 0” takes the dimensions of any physical quantity with which it is
associated.
0= dimension of length= [L]
Since dimensions of left hand side equals to dimension on right hand side,
equation is said to be consistent and dimensionally correct.
EXAMPLE 5: Derive relations between physical
quantities involved in physical phenomena
• The distance covered by a car, say x, starting from rest and having uniform acceleration
depends on time t and acceleration a. Use dimensional analysis to find expression for the
distance covered.
Suppose x depends on the mth power of t and nth power of a.
Then we may write
x ∝ tm. an
Expressing the two sides now in terms of dimensions,
we get
L1 ∝ Tm (LT–2)n,
or,
L1 ∝ Tm–2n Ln.
Comparing the powers of L and T on both sides, you will get
m-2n=0
n=1
By solving or n and m we get
n = 1, and m = 2.
Hence, we have
x ∝ t2a1, or x ∝ at2 or x =k at2
Here k is experimental constatnt.
This is as far as we can go with dimensional
analysis. It does not help us in getting the
numerical factors, since they have no dimensions.
To get the numerical factors, we have to get input
from experiment or theory.(here experimental
value of k is ½). In this particular case, of course, we
know that the complete relation is x = (1/2)at2
EXAMPLE 6:
• Convert 1 Newton into dyne.
• Let N1 and U1 be numerical value and unit of force in S.I system
(Newton) and N2 and U2 be numerical value of force in CGS system
(Dyne)
• N1U1= N2U2
• N1[M1L1T1
-2] = N2[M2L2T2
-2]
• N2 = N1[M1L1T1
-2]/[M2L2T2
-2]
• N2 = 1 [Mass in Kg/Mass in gm × Length in m/Length in cm × (Time
in seconds/Time in seconds)-2]
• N2 = [ 1000gm/ gm × 100cm/cm × (1 second/ 1 second)-2]
• N2 = [ 105]
• Which means 1 Newton = 105 Dyne
QUESTIONS FOR PRACTICE.
Q1. Experiments with a simple pendulum show that its
time period depends on its length (l) and the acceleration
due to gravity (g). Use dimensional analysis to obtain the
dependence of the time period on l and g .
Q2. Consider a particle moving in a circular orbit of radius
r with velocity v and acceleration a towards the centre of
the orbit. Using dimensional analysis, show that a ∝ v2/r .
Q3. You are given an equation: mv = Ft, where m is mass,
v is speed, F is force and t is time. Check the equation for
dimensional correctness
Limitations of Dimensional Analysis
• (1) For same dimensions, physical quantity
may not be unique. e.g. work and torque
• (2) Numerical constant(different from physical
constants) don’t have any dimensions.
• (3) The method of dimensions can not be
used to derive relations other than algebraic
functions.
• (4) The method of dimensions cannot be
applied to derive formula consist of more than
3 physical quantities.
Units , Measurement and Dimensional Analysis
Significant Figures
• Significant Figures in the measured value of a
physical quantity tell the number of digits in
which we have confidence.
• Larger the number of significant figures
obtained in a measurement, greater is the
accuracy of the measurement. The reverse is
also true.
Significant Figures-RULES
• (1) All non-zero digits are significant.
• (2) A zero becomes significant figure if it
appears between two non-zero digits.
• (3) Leading zeros or the zeros placed to the
left of the number are never significant.
Example : 0.543 has three significant figures.
0.006 has one significant figures.
Significant Figures-RULES
• (4) Trailing zeros or the zeros placed to the
right of the number are significant, if they
come after a decimal point. Example : 4.330
has four significant figures. 343.000 has six
significant figures.
• (5) In exponential notation, the numerical
portion gives the number of significant
figures. Example : 1.32 x102 has three signifi
cant figures.
Units , Measurement and Dimensional Analysis
Units , Measurement and Dimensional Analysis
Rounding Off
• (1) If the digit to be dropped is less than 5, then
the preceding digit is left unchanged. Example : x
= 7.82 is rounded off to 7.8, again x = 3.94
rounded off to 3.9.
• (2) If the digit to be dropped is more than 5, then
the preceding digit is raised by one. Example : x =
6.87 is rounded off to 6.9, again x = 12.78 is
rounded off to 12.8.
• (3) If the digit to be dropped is 5 followed by
digits other than zero, then the preceding digit is
raised by one. Example : x = 16.351 is rounded off
to 16.4, again x = 6.758 is rounded off to 6.8
Rounding Off
• (4) If digit to be dropped is 5 or 5 followed by
zeros, then preceding digit is left unchanged, if it
is even. Example : x = 3.250 becomes 3.2 on
rounding off, again x = 12.650 becomes 12.6 on
rounding off.
• (5) If digit to be dropped is 5 or 5 followed by
zeros, then the preceding digit is raised by one, if
it is odd. Example : x = 3.750 is rounded off to
3.8, again x = 16.150 is rounded off to 16.2.
Significant Figures in Calculation
• The following two rules should be followed to
obtain the proper number of significant figures
in any calculation.
• (1) The result of an addition or subtraction in the
number having different precisions should be
reported to the same number of decimal places
as are present in the number having the least
number of decimal places.
• (2) The answer to a multiplication or division is
rounded off to the same number of significant
figures as is possessed by the least precise term
used in the calculation
• Accuracy
• Accuracy is defined as the closeness of measured value to a
standard value. Suppose you weigh a box and noted 3.1 kg but its
known value is 9 kg, then your measurement is not accurate.
•
Precision
• Precision is defined as the closeness between two or more
measured values to each other. Suppose you weigh the same box
five times and get close results like 3.1, 3.2, 3.22, 3.4, and 3.0 then
your measurements are precise.
• Remember: Accuracy and Precision are two independent terms.
You can be very accurate but non-precise, or vice-versa.
NOTE. More number of decimal places in a measured value means
reading is more precise.
EXAMPLE:7
If the true value of a certain length is 3.678 cm
and two instruments with different resolutions,
up to 1 (less precise) and 2 (more precise)
decimal places respectively, are used. If first
measures the length as 3.5 and the second as
3.38 then the first has more accuracy but less
precision while the second has less accuracy and
more precision.
Order of Magnitude
• Order of magnitude of quantity is the power of
10 required to represent the quantity.
• For determining this power, the value of the
quantity has to be rounded off. While rounding
off, we ignore the last digit which is less than 5. If
the last digit is 5 or more than five, the
preceding digit is increased by one. For example,
(1) Speed of light in vacuum = 3 X108 m/s
• = 108 m/s (ignoring 3 < 5)
• (2) Mass of electron = = 9.1X 10 -31 kg= 10-30 kg
(as 9.1 > 5).
Errors of Measurement.
• The measured value of a quantity is always somewhat
different from its actual value, or true value.
• This difference in the true value and measured value
of a quantity is called error of measurement.
• Error
The uncertainty in the measurement of a
physical quantity is called an error.
The errors in measurement can be classified as
(i) Systematic errors and (ii) Random errors
Systematic error
• A clock running consistently 5% late. Hard to detect. Errors
of this type affect all measurements in same way. They
may result from faulty calibration or bias on part of the
observer.
• These are the errors that tend to be either positive or
negative. Sources of systematic errors are
(i) Instrumental errors
(ii) Imperfection in experimental technique or procedure
(iii) Personal errors
Instrumental Errors: The errors which
occur due to lack of accuracy in an
instrument are called instrumental
errors.
• If the markings of a thermometer are
improperly calibrated, let’s say it’s 108°C
instead of 100°C, then it is called An
Instrumental Error
• If a meter scale is worn off at its end
Imperfection in Technique: If the
experiment is not performed under
proper guidelines or physical
conditions around are not constant,
• If we place thermometer under the armpit
instead of the tongue, the temperature will
always come out to be lower than that of
body, as the technique of using thermometer
is incorrect
Personal Errors: These errors occur
due to improper setting of apparatus,
lack of observation skills in an
experiment and are based on the
carelessness of individual only.
• For measuring height of an object, if the
student don’t place his head in a proper way, it
may lead to parallax and readings won’t be
correct
How to reduce systematic errors?
• Improving experimental techniques by performing
experiment as per the guidelines and precautions of
the experiment
• By using correct, rightly accurate instruments and
sending old worn out instruments for maintenance
• Concentrating more while performing an experiment in
order to avoid silly mistakes in taking the readings of
the measurement
• Removing personal mistakes as far as possible and
keeping instruments safely after the experiment
• Fluctuation in observations.
• Reducing them is essentially one of improving experimental methods
• These errors which occur irregularly .These errors arise due to
unpredictable fluctuations in experimental conditions . Here the cause is
not known so can not be rectified. These errors can be reduced only by
repeated measurements.
• If a person repeats an experiment he is more likely to get different
observations
• These errors can be calculated by the use of mathematical tools as
shown below.
RANDOM ERRORS
Least Count Error
• The smallest value that can be measured in an
instrument is called Least Count of the
Instrument.
Instrument Least count
Vernier Caliper 0.01 cm
Spherometer 0.001 cm
Micrometer 0.0001 cm
To reduce least count error, we perform the
experiment several times and take arithmetic mean
of all the observations. The mean value is always
almost close to the actual value of the measurement.
Absolute error
• The magnitude of the difference between the individual
measurement and the true value of the quantity is called the
absolute error of the measurement.
suppose we perform an experiment in which readings are a1, a2, a3, a4,
a5 …. up to an and total number of observations is ‘n’,
amean = a1+a2+a3+………….+an/n
• Absolute error is denoted by the notation |Δa| and errors in
individual measurements can be calculated as:
• Δa1 = amean – a1
• Δa2 = amean – a2
• Δa3 = amean – a3
• ……. ……….. …
• Δan = amean – an
Remember that Δa may be a positive or negative sign, but will always
focus on the magnitude of it
• Mean Absolute Error
• The arithmetic mean of all the absolute errors
is taken as the final or mean absolute error of
the value of the physical quantity a. It is
represented by Δa mean
• arithmetic mean of all absolute error is the
final mean of absolute error of experiment.
• Δamean = Δa1 + Δa2+ Δa3+……………..+ Δan/n
•
• Relative error - it is the ratio of the mean
absolute error to the true value.
• Relative error = Δa mean/ a mean
• Percentage Error : When the relative error is
expressed in per cent, it is called the
percentage error (δa).
• Percentage error =( Δa mean/ a mean) ×100
EXAMPLE 7
• If the actual value of a quantity is 50 and its
measured value is 49.8. Then calculate the
absolute error and relative error in it.
• We have amean = 50 ( amean and actual value are
same thing)
• Measured value = 49.8
• Absolute error = Actual Value – Measured Value
• = 50 – 49.8
• = 0.2
• Relative Error = 0.2/50 = 0.4
Combination of Errors
• Error in sum of the quantities :
• Suppose x = a + b
Let Δa = absolute error in measurement of a Δb
= absolute error in measurement of b
Δx = absolute error in calculation of x i.e. sum of
a and b.
• The maximum absolute error in x is
Δx = (Δa + Δb)
• Error in difference of the quantities—Suppose x = a – b
The maximum absolute error in x is
Δx = (Δa + Δb)
• Example
The length of two scales is given as l1 = 20 cm ± 0.5 cm and
l2 = 30 cm ± 0.5 cm, then
the final length by adding length of both scales will be
given as
50cm ± 1 cm
• Error in product of quantities—
• Suppose x = a × b
The maximum fractional error in x is :
Δx /x = Δ a/a + Δ b/ b
• Example
The mass of a substance is 100 ± 5 g and
volume is 200 ± 10 cm3,
then the relative error in density will be the sum of
percentage error in mass i.e. 5/100 × 100 = 5% and
percentage error in volume i.e. 10/200 ×100 = 5%,
Δx /x= 10%.
• Error in division of quantities—
• Suppose x = a/b
• The maximum fractional error in x is :
• Δx /x = Δ a/a + Δ b/ b
Error in case of a measured quantity
rose to some power
• Error in quantity raised to some power—
Suppose x = (a) m/ (b )n
The maximum fractional error in x is :
Δx /x = m (Δ a/a) + n (Δ b/ b)
• Example
The relative error in S = A3B4C2, will be written
as,
ΔS/S = 3ΔA/A + 4ΔB/B + 2 ΔC/C

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Units , Measurement and Dimensional Analysis

  • 1. Units and Measurements Need for measurement: Units of measurement; systems of units; SI units, fundamental and derived units. Length, mass and time measurements; accuracy and precision of measuring instruments; errors in measurement; significant figures. Dimensions of physical quantities, dimensional analysis and its applications. Revised syllabus -CORONA PANDEMIC 2020 Units and Measurements
  • 2. Physical Quantity • Physical Quantity: A quantity which can be measured and expressed in form of laws is called a physical quantity. • Physical quantity (Q) = n × u , Where n represents the numerical value and u represents the unit. • As the unit(u) changes, the magnitude (n) will also change but product nu will remain same. i.e. n u = constant, or n1u1 = n2u2 = constant;
  • 3. Fundamental and Derived Units • Fundamental and Derived Units: • Any unit of mass, length and time in mechanics is called a fundamental, absolute or base unit. • Other units which can be expressed in terms of fundamental units, are called derived units
  • 4. System of units • System of units : A complete set of units, both fundamental and derived for all kinds of physical quantities is called system of units. (1) CGS system, (2) MKS system, (3) FPS system. (4) S.I. system : It is known as International system of units. • There are seven fundamental quantities in this system. These quantities and their units are given in the following table.
  • 5. SI UNITS • Quantity Name of Units Symbol • Length Metre m • Mass Kilogram kg • Time Second s • Electric Current Ampere A • Temperature Kelvin K • Amount of Substance Mole Mol • Luminous Intensity Candela Cd • Besides the above seven fundamental units two supplementary units are also defined - Radian (rad) for plane angle and Steradian (sr) for solid angle
  • 6. In a practical unit if the unit of mass becomes double and that of unit of time becomes half, then 8 joule will be equal to ______________ 8 joules of work done can be expressed, As work= 8 joules = 8kg m²/ s² where normal units mass = kg length= m Time = second=s let the new units be KG,M and S. According to question , KG = 2kg M= no change mentioned = so assume same = m S= s/2 work =8 kg m/ s=8 (KG/2)*M²/(2S)²=8/8(KGM²/S²) =1 Joule of work.
  • 7. LARGEST AND SMALLEST PRACTICAL UNITS Q LARGEST PRACTICAL UNIT SMALLEST PRACTICAL UNIT LENGTH 1A.U 1 FERMI MASS 1 C.S.L 1a.m.u TIME 1 CENTURY 1 SHAKE
  • 9. Dimensions of a Physical Quantity • Dimensions of a Physical Quantity: When a derived quantity is expressed in terms of fundamental quantities, it is written as a product of different powers of the fundamental quantities. • The powers to which fundamental quantities must be raised in order to express the given physical quantity are called its dimensions.
  • 10. Following examples show how dimensions of the physical quantities are combinations of the powers of M, L and T : • (i) Volume requires 3 measurements in length. So it has 3 dimensions in length (L3). • (ii) Density is mass divided by volume. Its dimensional formula is ML–3. • (iii) Speed is distance travelled in unit time or length divided by time. Its dimensional formula is LT–1.
  • 11. • (iv) Acceleration is change in velocity per unit time, i.e., length per unit time per unit time. Its dimensional formula is LT–2. Dimensions of acceleration are 1 in length, -2 in time. • (v) Force is mass multiplied by acceleration. Its dimensions are given by the formula MLT–2. TRY DIMENSIONS OF OTHER PHYSICAL QUANTITIES GIVEN BELOW:
  • 12. Important Dimensions • S.NO Quantity Dimension 1. Velocity or speed (v) LT-1 2. Acceleration LT-2 3. Distance L 4. Force MLT-2 5. Pressure ML-1T-2 6. Stress(F/A) ML-1T-2 7. Strain DIMENSIONLESS 8. Angle DIMENSIONLESS 9. Gravitational constant M-1L3T-2 10. Acceleration due to gravity LT-2
  • 13. Application of Dimensional Analysis. • (1) To find the unit of a physical quantity in a given system of units. • (2) To find dimensions of physical constant or coefficients. • (3) To convert a physical quantity from one system to the other. • (4) To check the dimensional correctness of a given physical relation: This is based on the ‘principle of homogeneity’. According to this principle the dimensions of each term on both sides of an equation must be dimensionally same. • (5) To derive relation between various physical quantities..
  • 14. EXAMPLES 1: To find the unit of a physical quantity in a given system of units The dimensions of mv2 /2 are M.(LT–1)2, or ML2T–2. (Remember that the numerical factors have no dimensions.) So S.I. unit is kgm2s-2 which is the Unit of energy
  • 15. EXAMPLE2 : To find the dimensions of epsilon • Electric permittivity: • Electrostatic force F= 1/4π εO (q1 q2/r2) εO= (1/4π F) q1 q2/r2 Unit of εO= coulomb2/m2N Dimension of εO = A2T2/L2MLT-2 = M–1L–3T4A2
  • 16. EXAMPLE 3: Consistency of a Dimensional Equation The similarity of dimensions of physical quantities is called consistency of a dimensional equation • Check whether the given equation is dimensionally correct. • W = 1/2 mv2 – mgh where W stands for work done, m means mass, g stands for gravity, v for velocity and h for height. To check the above equation as dimensionally correct, we first write dimensions of all the physical quantities mentioned in the equation. • W = Work done = Force × Displacement = [MLT-2] × [L] = [ML2T-2] • 1/2 mv2 = Kinetic Energy = [M] × [L2T-2] = [ML2T-2] • mgh = Potential Energy = [M] × [LT-2] × [L] = [ML2T-2] Since all the dimensions on left and right sides are equal it is a dimensionally correct equation.
  • 17. EXAMPLE 4 • Check the consistency of the equation S-ut-1/2at2=0 L.H.S. • s= distance = [L] • ut = velocity × time = [LT-1] × [T] = [L] • at2 = acceleration × time2 = [LT-2] × [T2] = [L] R.H.S “ 0” takes the dimensions of any physical quantity with which it is associated. 0= dimension of length= [L] Since dimensions of left hand side equals to dimension on right hand side, equation is said to be consistent and dimensionally correct.
  • 18. EXAMPLE 5: Derive relations between physical quantities involved in physical phenomena • The distance covered by a car, say x, starting from rest and having uniform acceleration depends on time t and acceleration a. Use dimensional analysis to find expression for the distance covered. Suppose x depends on the mth power of t and nth power of a. Then we may write x ∝ tm. an Expressing the two sides now in terms of dimensions, we get L1 ∝ Tm (LT–2)n, or, L1 ∝ Tm–2n Ln. Comparing the powers of L and T on both sides, you will get m-2n=0 n=1 By solving or n and m we get n = 1, and m = 2.
  • 19. Hence, we have x ∝ t2a1, or x ∝ at2 or x =k at2 Here k is experimental constatnt. This is as far as we can go with dimensional analysis. It does not help us in getting the numerical factors, since they have no dimensions. To get the numerical factors, we have to get input from experiment or theory.(here experimental value of k is ½). In this particular case, of course, we know that the complete relation is x = (1/2)at2
  • 20. EXAMPLE 6: • Convert 1 Newton into dyne. • Let N1 and U1 be numerical value and unit of force in S.I system (Newton) and N2 and U2 be numerical value of force in CGS system (Dyne) • N1U1= N2U2 • N1[M1L1T1 -2] = N2[M2L2T2 -2] • N2 = N1[M1L1T1 -2]/[M2L2T2 -2] • N2 = 1 [Mass in Kg/Mass in gm × Length in m/Length in cm × (Time in seconds/Time in seconds)-2] • N2 = [ 1000gm/ gm × 100cm/cm × (1 second/ 1 second)-2] • N2 = [ 105] • Which means 1 Newton = 105 Dyne
  • 21. QUESTIONS FOR PRACTICE. Q1. Experiments with a simple pendulum show that its time period depends on its length (l) and the acceleration due to gravity (g). Use dimensional analysis to obtain the dependence of the time period on l and g . Q2. Consider a particle moving in a circular orbit of radius r with velocity v and acceleration a towards the centre of the orbit. Using dimensional analysis, show that a ∝ v2/r . Q3. You are given an equation: mv = Ft, where m is mass, v is speed, F is force and t is time. Check the equation for dimensional correctness
  • 22. Limitations of Dimensional Analysis • (1) For same dimensions, physical quantity may not be unique. e.g. work and torque • (2) Numerical constant(different from physical constants) don’t have any dimensions. • (3) The method of dimensions can not be used to derive relations other than algebraic functions. • (4) The method of dimensions cannot be applied to derive formula consist of more than 3 physical quantities.
  • 24. Significant Figures • Significant Figures in the measured value of a physical quantity tell the number of digits in which we have confidence. • Larger the number of significant figures obtained in a measurement, greater is the accuracy of the measurement. The reverse is also true.
  • 25. Significant Figures-RULES • (1) All non-zero digits are significant. • (2) A zero becomes significant figure if it appears between two non-zero digits. • (3) Leading zeros or the zeros placed to the left of the number are never significant. Example : 0.543 has three significant figures. 0.006 has one significant figures.
  • 26. Significant Figures-RULES • (4) Trailing zeros or the zeros placed to the right of the number are significant, if they come after a decimal point. Example : 4.330 has four significant figures. 343.000 has six significant figures. • (5) In exponential notation, the numerical portion gives the number of significant figures. Example : 1.32 x102 has three signifi cant figures.
  • 29. Rounding Off • (1) If the digit to be dropped is less than 5, then the preceding digit is left unchanged. Example : x = 7.82 is rounded off to 7.8, again x = 3.94 rounded off to 3.9. • (2) If the digit to be dropped is more than 5, then the preceding digit is raised by one. Example : x = 6.87 is rounded off to 6.9, again x = 12.78 is rounded off to 12.8. • (3) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one. Example : x = 16.351 is rounded off to 16.4, again x = 6.758 is rounded off to 6.8
  • 30. Rounding Off • (4) If digit to be dropped is 5 or 5 followed by zeros, then preceding digit is left unchanged, if it is even. Example : x = 3.250 becomes 3.2 on rounding off, again x = 12.650 becomes 12.6 on rounding off. • (5) If digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd. Example : x = 3.750 is rounded off to 3.8, again x = 16.150 is rounded off to 16.2.
  • 31. Significant Figures in Calculation • The following two rules should be followed to obtain the proper number of significant figures in any calculation. • (1) The result of an addition or subtraction in the number having different precisions should be reported to the same number of decimal places as are present in the number having the least number of decimal places. • (2) The answer to a multiplication or division is rounded off to the same number of significant figures as is possessed by the least precise term used in the calculation
  • 32. • Accuracy • Accuracy is defined as the closeness of measured value to a standard value. Suppose you weigh a box and noted 3.1 kg but its known value is 9 kg, then your measurement is not accurate. • Precision • Precision is defined as the closeness between two or more measured values to each other. Suppose you weigh the same box five times and get close results like 3.1, 3.2, 3.22, 3.4, and 3.0 then your measurements are precise. • Remember: Accuracy and Precision are two independent terms. You can be very accurate but non-precise, or vice-versa. NOTE. More number of decimal places in a measured value means reading is more precise.
  • 33. EXAMPLE:7 If the true value of a certain length is 3.678 cm and two instruments with different resolutions, up to 1 (less precise) and 2 (more precise) decimal places respectively, are used. If first measures the length as 3.5 and the second as 3.38 then the first has more accuracy but less precision while the second has less accuracy and more precision.
  • 34. Order of Magnitude • Order of magnitude of quantity is the power of 10 required to represent the quantity. • For determining this power, the value of the quantity has to be rounded off. While rounding off, we ignore the last digit which is less than 5. If the last digit is 5 or more than five, the preceding digit is increased by one. For example, (1) Speed of light in vacuum = 3 X108 m/s • = 108 m/s (ignoring 3 < 5) • (2) Mass of electron = = 9.1X 10 -31 kg= 10-30 kg (as 9.1 > 5).
  • 35. Errors of Measurement. • The measured value of a quantity is always somewhat different from its actual value, or true value. • This difference in the true value and measured value of a quantity is called error of measurement. • Error The uncertainty in the measurement of a physical quantity is called an error. The errors in measurement can be classified as (i) Systematic errors and (ii) Random errors
  • 36. Systematic error • A clock running consistently 5% late. Hard to detect. Errors of this type affect all measurements in same way. They may result from faulty calibration or bias on part of the observer. • These are the errors that tend to be either positive or negative. Sources of systematic errors are (i) Instrumental errors (ii) Imperfection in experimental technique or procedure (iii) Personal errors
  • 37. Instrumental Errors: The errors which occur due to lack of accuracy in an instrument are called instrumental errors. • If the markings of a thermometer are improperly calibrated, let’s say it’s 108°C instead of 100°C, then it is called An Instrumental Error • If a meter scale is worn off at its end
  • 38. Imperfection in Technique: If the experiment is not performed under proper guidelines or physical conditions around are not constant, • If we place thermometer under the armpit instead of the tongue, the temperature will always come out to be lower than that of body, as the technique of using thermometer is incorrect
  • 39. Personal Errors: These errors occur due to improper setting of apparatus, lack of observation skills in an experiment and are based on the carelessness of individual only. • For measuring height of an object, if the student don’t place his head in a proper way, it may lead to parallax and readings won’t be correct
  • 40. How to reduce systematic errors? • Improving experimental techniques by performing experiment as per the guidelines and precautions of the experiment • By using correct, rightly accurate instruments and sending old worn out instruments for maintenance • Concentrating more while performing an experiment in order to avoid silly mistakes in taking the readings of the measurement • Removing personal mistakes as far as possible and keeping instruments safely after the experiment
  • 41. • Fluctuation in observations. • Reducing them is essentially one of improving experimental methods • These errors which occur irregularly .These errors arise due to unpredictable fluctuations in experimental conditions . Here the cause is not known so can not be rectified. These errors can be reduced only by repeated measurements. • If a person repeats an experiment he is more likely to get different observations • These errors can be calculated by the use of mathematical tools as shown below. RANDOM ERRORS
  • 42. Least Count Error • The smallest value that can be measured in an instrument is called Least Count of the Instrument.
  • 43. Instrument Least count Vernier Caliper 0.01 cm Spherometer 0.001 cm Micrometer 0.0001 cm To reduce least count error, we perform the experiment several times and take arithmetic mean of all the observations. The mean value is always almost close to the actual value of the measurement.
  • 44. Absolute error • The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. suppose we perform an experiment in which readings are a1, a2, a3, a4, a5 …. up to an and total number of observations is ‘n’, amean = a1+a2+a3+………….+an/n • Absolute error is denoted by the notation |Δa| and errors in individual measurements can be calculated as: • Δa1 = amean – a1 • Δa2 = amean – a2 • Δa3 = amean – a3 • ……. ……….. … • Δan = amean – an Remember that Δa may be a positive or negative sign, but will always focus on the magnitude of it
  • 45. • Mean Absolute Error • The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by Δa mean • arithmetic mean of all absolute error is the final mean of absolute error of experiment. • Δamean = Δa1 + Δa2+ Δa3+……………..+ Δan/n •
  • 46. • Relative error - it is the ratio of the mean absolute error to the true value. • Relative error = Δa mean/ a mean • Percentage Error : When the relative error is expressed in per cent, it is called the percentage error (δa). • Percentage error =( Δa mean/ a mean) ×100
  • 47. EXAMPLE 7 • If the actual value of a quantity is 50 and its measured value is 49.8. Then calculate the absolute error and relative error in it. • We have amean = 50 ( amean and actual value are same thing) • Measured value = 49.8 • Absolute error = Actual Value – Measured Value • = 50 – 49.8 • = 0.2 • Relative Error = 0.2/50 = 0.4
  • 48. Combination of Errors • Error in sum of the quantities : • Suppose x = a + b Let Δa = absolute error in measurement of a Δb = absolute error in measurement of b Δx = absolute error in calculation of x i.e. sum of a and b. • The maximum absolute error in x is Δx = (Δa + Δb)
  • 49. • Error in difference of the quantities—Suppose x = a – b The maximum absolute error in x is Δx = (Δa + Δb) • Example The length of two scales is given as l1 = 20 cm ± 0.5 cm and l2 = 30 cm ± 0.5 cm, then the final length by adding length of both scales will be given as 50cm ± 1 cm
  • 50. • Error in product of quantities— • Suppose x = a × b The maximum fractional error in x is : Δx /x = Δ a/a + Δ b/ b • Example The mass of a substance is 100 ± 5 g and volume is 200 ± 10 cm3, then the relative error in density will be the sum of percentage error in mass i.e. 5/100 × 100 = 5% and percentage error in volume i.e. 10/200 ×100 = 5%, Δx /x= 10%.
  • 51. • Error in division of quantities— • Suppose x = a/b • The maximum fractional error in x is : • Δx /x = Δ a/a + Δ b/ b
  • 52. Error in case of a measured quantity rose to some power • Error in quantity raised to some power— Suppose x = (a) m/ (b )n The maximum fractional error in x is : Δx /x = m (Δ a/a) + n (Δ b/ b) • Example The relative error in S = A3B4C2, will be written as, ΔS/S = 3ΔA/A + 4ΔB/B + 2 ΔC/C