Chapter
1: S.I. Units and Measurements
A physical quantity is a physical property of a substance,
phenomenon or a body that can be measured by a measuring instrument. Examples
of some physical quantities are mass, time, length, speed, force etc.
The standard value of a physical quantity is called
“unit”.
|
Quantity |
Unit |
|
Mass |
kilogram |
|
Length |
meter |
Fundamental
quantities and units
The physical quantities that are independent to one another
are called fundamental or base quantities and their units are called
fundamental or base units.
|
Quantity |
Unit |
|
Mass |
kilogram |
|
Length |
meter |
|
Time |
second |
The physical quantities that are derived from fundamental
quantities are called derived quantities and their units are called derived
units.
|
Derived
quantity |
Fundamental
quantity |
Unit |
|
Velocity |
Displacement / Time |
meter/second |
|
Acceleration |
Velocity / Time |
meter / second2 |
|
Force |
Mass X Acceleration |
Newton or |
The physical quantities, which are neither fundamental
nor derived, are called supplementary quantities and their units are called
supplementary units.
|
Supplementary
quantity |
Unit |
Symbol |
|
Plane angle |
Radian |
rad |
|
Solid angle |
Steradian |
Sr |
|
Radio activity |
Curie |
Ci |
Characteristics of units
- They should not vary with place and time.
- They should be easily reproducible.
- They should be well defined.
Measurement of units with different system.
|
Measurement system |
Measurement standard |
|||
|
Length |
Mass |
Time |
Current |
|
|
FPS |
Foot |
Pound |
Second |
- |
|
CGS |
Centimeter |
Gram |
Second |
- |
|
MKS |
Meter |
Kilogram |
Second |
- |
|
MKSA |
Meter |
Kilogram |
Second |
Ampere |
Note: 1 kilogram = 2.22
pound & 1 meter = 3.28 foot
SI is an abbreviation of Standard International.
Most of the countries use this measurement system.
|
Fundamental quantity |
Unit |
Symbol |
|
Length |
Metre |
m |
|
Mass |
Kilogram |
kg |
|
Time |
Second |
s |
|
Electric current |
Ampere |
A |
|
Temperature |
Kelvin |
K |
|
Luminous intensity |
Candela |
cd |
|
Quantity of matter |
Mole |
mol |
|
Derived quantity |
Formula |
SI unit |
|
|
Area |
length X breadth |
m2 |
|
|
Volume |
length X breadth X height |
m3 |
|
|
Density |
|
kg/m3 |
|
|
Speed |
|
m/s |
|
|
Velocity |
|
m/s |
|
|
Acceleration |
|
m/s2 |
|
|
Force |
|
kg m/s2 OR N |
N = newton |
|
Momentum |
|
kg m/s OR N
s |
|
|
Work (Energy) |
|
kg m2/s2 OR N
m OR J |
J = joule |
|
Pressure |
|
N/m2 OR Pa |
Pa = pascal |
|
Frequency |
|
s-1 OR Hz |
Hz = hertz |
The smallest possible measurement of a physical quantity
by a given instrument is called least count of the instrument. For e.g. least
count of ruler scale is 1 mm or 0.1 cm.
The biggest possible measurement of a physical quantity
by a given instrument is called the range of the instrument. For e.g. range of
a given scale is 150 mm or 15 cm. (1 cm = 10 mm)
Principle of Vernier: The marking on the Vernier
scale are such that the
length of 'N' divisions on the Vernier
scale is equal to (N-1)
divisions of the main scale.
Working: Vernier
caliper is more precise than ruler scale. Its least count is less than ruler
scale. This instrument measures internal diameter (by internal jaws), external
diameter (by external jaws) and depth (by depth measuring blade) too.
On Vernier scale N
divisions are marked, they equal (N-1) divisions of main scale.
  |
|
When jaws of a vernier caliper are closed, ZERO of main
scale must coincides with the ZERO of vernier scale, if they do not coincide
then it is said that a zero error in the vernier caliper.
When jaws of a vernier caliper are closed and the ZERO of
the vernier scale appears after the ZERO of the main scale, then it is said
that a positive zero error in the vernier scale.
For calculation
of positive error, find coinciding division mark on vernier scale and multiply
this mark with least count. For e.g. here in above figure, 3rd
division mark on vernier scale coincides
with one of the main scale mark, and when this mark is multiplied with least
count of vernier caliper, the positive error is found to be +0.3.
Here, coinciding
mark, n = 3
So, positive
error = n X L.C.
= 3 X 0.1 mm
= +0.3 mm
So, correction C
= - 0.3 mm
When jaws of a vernier caliper are closed and the ZERO of
the vernier scale appears before the ZERO of the main scale, then it is said
that a negative zero error in the vernier scale.
For calculation
of positive error, find coinciding division mark on vernier scale and then
multiply this mark with least count. For e.g., here 8th division
mark on vernier scale coincides with one of the main scale mark, and then this
mark is multiplied with least count.
Here, coinciding mark = (Total
mark on vernier – coinciding mark on vernier)
=
(N – n)
So, negative
error = (N – n) X L.C.
= (10 – 8) X 0.1 mm
= - 0.2 mm
So, correction C
=+ 0.2 mm
Note: if error
is positive then correction should be negative and vice-versa.
Numerical 1: Main
scale of a vernier caliper is calibrated in millimeter. If 20 divisions of
vernier scale equal to 19 divisions of main scale, calculate least count of the
vernier caliper.
Solution:
 |
OR Here,
20 VSD = 19 MSD So,
1 VSD = LC
= 1 MSD - LC
= (1 - LC
= LC
= LC
= 0.05 MSD LC
= 0.05 mm |
Numerical 2: Main
scale of a vernier caliper is calibrated in millimeter. If 10 divisions of
vernier scale equal to 9 divisions of main scale, calculate least count of the
vernier caliper.
Solution:
 |
OR Here,
10 VSD = 9 MSD So,
1 VSD = LC
= 1 MSD - LC
= (1 - LC
= LC
= LC
= 0.1 MSD LC
= 0.1 mm |
Numerical 3: In
a vernier caliper 50 divisions of vernier scale is equivalent to 49 divisions
of main scale, calculate least count of the vernier caliper.
Solution:
 |
OR Here,
50 VSD = 49 MSD So
1 VSD = LC
= 1 MSD - LC
= (1 - LC
= LC
= LC
= 0.02 MSD LC
= 0.02 UNIT |
The micrometer screw gauge is an instrument used
for measuring accurately the diameter of a thin wire or the thickness of a
sheet of metal. For that insert object between anvil and spindle by rotating
thimble outside. Read main scale reading and add rotating scale reading in it.
The pitch of the micrometer screw gauge is the distance moved by the
spindle per complete revolution of thimble.
|
Note: here, pitch distance is equal to value of single division on main scale. |
The lack in accuracy in
the measurement is called an error.
If the
zero mark on the circular scale does not coincide with the base line of the
main scale then it is called zero error.
If the zero mark on the circular scale is below the base line of the main scale, the error is said to be positive.
The number of circular scale division, which coincides with the base line is multiplied with the least count of the screw gauge to get the positive zero error.
|
|
 Positive Error = Coinciding mark on rotating scale x L.C |
|
|
|
= + 3 x 0.01 mm = + 0.03 mm So, correction C = - 0.03 mm |
|
If the zero mark on the
circular scale is above the base line of the main scale, the error is said to
be negative.
The number of circular scale division coinciding with the
base line is subtracted from the total number of divisions on the circular
scale and is multiplied with the least count of the screw gauge. This gives the
negative zero error.
|
|
Negative
Error = (Total mark on rotating scale - Coinciding mark on rotating scale ) x
L.C. |
|
|
|
=
(50 – 48 ) x 0.01 mm =
2 x 0.01 mm= -0.02 mm So,
correction C = +0.02 mm Note:
Coinciding mark is counted from zero always. |
|
Numerical 4: Calculate
the least count of a micrometer screw having pitch 1 mm and total number of
circular scale divisions are 100.
Solution:
Numerical 5: A
micrometer screw has least count 1 x 10-3 cm, if there are 50
divisions on its circular scale, find pitch distance of micrometer screw gauge.
Solution:
= 50 x 10-3 cm
= 0.05 cm
p = 0.5 mm
So, pitch of micrometer screw gauge is
0.5 mm.
Accuracy
|
Precision
|
|
Ø Accuracy is how close a measured value is to the actual value. |
Ø Precision is how close the measured values are to each other. It might be
far from actual value. |
 |
 |
|
|
|
|
|
Error is the difference between the actual
value of a quantity and the measured value.
1. Systematic
Error
2. Random
Error
The systematic
errors are those errors that tend to be in one direction, either positive
or negative and are displacement in measured value from true value by a fixed
magnitude.
Systematic errors cannot
be eliminated by averaging or by statistical means. That means such errors
cannot be dealt with simple repeated measurements. Repetition of data
collection should be done by using different techniques and different tools or
equipments and then the results should be compared to minimize the systematic
errors.
Some
sources of this error are as under:
a)
Instrumental
error
b)
Experimental
error
c)
Personal
error
d)
Errors
due to external environmental conditions
This error is due to defect in the
instrument or in the calibration of scale. For example, broken or defective
calibrated scale gives error, which may be positive or negative.
This error originates due to improper procedure of
experiment. For example, measurement of the volume of wooden block using
Archimedes principle by graduated cylinder.
This error originates due to incorrect
measurement or may use poor technique in taking a measurement by person.
These
errors are due to environmental factors like change in temperature, humidity
and variation in pressure. This kind of error can be avoided by providing
proactive cover or shield to the instrument.
Random (or indeterminate) errors occur due to
uncontrollable fluctuations in variables that affect experimental results. For
example, air fluctuations occurring as student open and close lab doors cause
changes in pressure readings.
The
difference between the true value and the measured value is called ‘absolute
error. If X1, X2, X3……………XN are the
measured values of any quantity X in an experiment performed N times, then the
average of these values is called the true value X of the quantity.
The absolute error in
measured values is given by
The
average of the magnitude of absolute errors in all the measurement is called
mean absolute error.
The
ratio of average absolute error to the true value is called relative error.
Percentage
value of relative error is called percentage error.
Percentage error =
Relative error x 100%
Numerical 6: In
ohm’s law experiment, observations of resistance are 5.25, 5.27, 5.28, 5.26 and
5.29 Ω. Find out average absolute error, relative error and percentage error in
above experiment.
Solution:
(I)
Values of resistance are:
R1=5.25 Ω, R2=5.27 Ω, R3= 5.28 Ω, R4=5.26 Ω and R5=5.29 Ω.
Therefore, Average Resistance, R:
Now, absolute error in each observation:
ΔR1 =
ΔR2 = R
ΔR3 =
ΔR4 =
ΔR5 =
(II) Average absolute
error:
(III) Relative error
(IV) Percentage error
In
the measured value of a physical quantity, the reliable digits plus
the first uncertain digit altogether are known as significant digits or significant
figures.
Significant figure means each of the digits of a number that are
used to express it to the required degree of accuracy, starting from the first
non-zero digit.
If we say the diameter of wire is 1.63 mm, the digits 1 and 6
are reliable and certain, while the digit 3 is uncertain. Thus, the measured
value has three significant figures.
1)
All non-zeros digits are significant
digits, e.g. 4325 kg has 4 significant digits 4,3,2,5.
2)
All zeros between non-zero digits are
significant digits, e.g. 20003 has 5 significant digits 2, 0,0,0,3.
3)
Trailing zeros in a number containing a
decimal point are significant, e.g. 52.100 kg has only five significant digits
5, 2,1,0,0.
4)
Leading zeros are not significant. e.g.
0.0003500 has 4 significant digits 3,5,0,0.
5)
In a number without a decimal point, trailing
zeros may not be significant.(In special case it may be significant.) e.g.
12020 has four significant digits 1,2,0,2.
Rules of rounding off significant
figures:
1)
If the last digit to be dropped is less
than 5, the new last digit remain unchanged
e.g. 56.34 rounding off to the three significant digits is 56.3
2)
If the last digit to be dropped is more
than 5, the new last digit increased by 1.
e.g. 56.36 rounding off to the three significant digits is 56.4
3)
If the last digit to be dropped is 5,
the new last digit remains unchanged if it is an even digit e.g. 56.65 rounding
off to the three significant digits is 56.6.
4)
If the last digit to be dropped is 5,
the new last digit increased by 1 if it is an odd digit e.g. 56.75 rounding off
to the three significant digits is 56.8.
Numerical 7: Find
out significant figures.
(1)
2205 (2) 0.0025 (3) 0.0100 (4) 3.531 (5)
8.300 (6) 05.532 (7) 8.1 x 10-5(8) 0.012 x 106 (9) 150 x
10-5
Solution:
(1) 2205 = 2,2,0,5 total significant
digits are 4
(2) 0.0025 = 2,5 total significant
digits are 2
(3) 0.0100 = 1,0,0 total significant
digits are 3
(4) 3.531 = 3,5,3,1 total significant
digits are 4
(5) 8.300 = 8,3,0,0 total significant
digits are 4
(6) 05.532 = 5,5,3,2 total
significant digits are 4
(7) 8.1 x 10-5 =8,1 total
significant digits are 2
(8) 0.012 x 106 = 1,2
total significant digits are 2
(9) 150 x 10-5= 1,5 total
significant digits are 2
(1)
1 Newton = 105 dyne (Newton is
SI unit and dyne is CGS unit of force)
(2)
1 Joule = 107 erg (Joule is SI
unit and erg is CGS unit of work or energy)
(3)
1 A0 = 10-10 m
1. Define: -Physical quantity, Fundamental physical quantity, Derived physical quantity, unit, Fundamental unit, Derived unit, supplementary quantity, supplementary unit
2. What
is unit? Write its properties.
3. Write
fundamental quantities and their units with symbol for FPS, CGS, MKS and SI
system.
4. Write
a short note on SI system.
5. What
is derived quantity? Sate any five derived quantities, their formula and units.
1. Draw
the neat diagram of vernier caliper and label its parts.
2.
Explain principle of vernier caliper and obtain its formula for least count
(L.C.).
3.
Explain how to measure dimensions of an object with vernier caliper.
4. What
is zero error? State types of zero error for vernier caliper.
5.
Explain positive error of vernier caliper with proper figure.
6.
Explain negative error of vernier caliper with proper figure.
7.
Define: least count.
1. Draw
the neat diagram of micrometer screw gauge and label its parts.
2.
Explain principle of micrometer screw gauge and obtain its formula for least
count (L.C.).
3.
Explain how to measure dimensions of an object with micrometer screw gauge.
4. What
is zero error? State types of zero error for micrometer screw gauge.
5.
Explain positive error of micrometer screw gauge with proper figure.
6.
Explain negative error of micrometer screw gauge with proper figure.
7. What
is pitch?
1.
Define error. Explain systematic error and random error with examples.
2.
Explain (a) absolute error, (b) average absolute error, (c) relative error and
(d) percentage error.
ACCURACY,
PRECISION AND SIGNIFICANT NUMBERS
1.
Define/Explain accuracy and precision.
2. What
is significant figure? State rules for significant figures.
(1)Joule is unit of ______
(A) Work (B) Force
(C) Time (D) Pressure
(2)Least count of Vernier Calipers is proportional to______
(3)Yotta means ____
(A)1024 (B)1023
(C)106 (D)107
(4) A vernier calipers has calibrated in mm and 19 of main scale division is equal to 20 division of vernier scale then L.C is____
(A)0.05mm (B)0.01mm (C)0.03 mm (D)0.01 cm
(5) How many significant figures in 9.4 X 10-10
(A) 1 (B)2 (C)3 (D)4
(6) Which is fundamental quantity?
(A) length (B) velocity (C)voltage (D)joule
(7) 1 dyne =_______
(A) 10-3 (B)10-4 (C)10-5 (D)10-7
(8) Force is
(A) Scalar quantity (B) Vector quantity (C) Fundamental quantity (D) None of these
(9) Unit of momentum is ________.
(A) kg.m/s (B) kg.m.s (C) kg. m (D) m.s
(10)Give the unit of pressure.
(A)N.m2 (B) N.m (C) N/m2 (D) N2 /m
(11) 1 Å = ________ cm.
(A) 10-8 (B)
10-10 (C) 10-12 (D) 10-6
(12) Give number of significant digits in 0.004
(A) two (B) three (C) one (D) four
(13) Pitch of Micrometer is 0.5 mm and there are 50 divisions on its circular scale. Least count of Micrometer is ________.
(A) 0.01mm (B) 0.01cm (C) 0.001 mm (D) 0.0001 mm
(14) Unit of temperature in SI is ________.
(A) Celsius (B) Kelvin (C) Fahrenheit (D) none of these
(15) 50 divisions of vernier scale are equal to 49 divisions on main scale. Main scale is in mm least count ________.
(A) 0.02 mm (B) 0.01 mm (C) 0.002mm (D) 0.001 mm
(16) 100 newton = ________ dyne.
(A) 10-3 (B) 10-5 (C) 103 (D) 10-7
(A)1010 (B)10-10 (C)109 (D) 1000
(18) Which of the following is a scalar quantity?
(A)Velocity (B) Acceleration (C) Speed (D) Momentum
(19) S.I .Unit of temperature is --------
(A) Joule (B) Kelvin (C) Pascal (D) Candela
(20) 1 Newton = ---------dyne.
(A) 104 (B) 106 (C) 105 (D) 1010
(21) Number of significant figure in 317000
(A) 4 (B) 6 (C) 3 (D) 5
(22) Precision of measurement is depend on----------
(A) Pressure (B) Least count of instrument (C) Error (D) None of above
(22) S.I. unit of power is ---------
(A) Volt (B) Watt (C) Ampere (D) Newton
(23)Which of the following is derived physical quantity?
(A)Length (B) Mass (C) Force (D) None of above
(24)1 microampere =---------Ampere.
(A) 104 (B) 10-4 (C) 106 (D) 10-6
(25) What is the number of significant figures in 0.780×103?
(A) 2 (B) 4 (C) 3 (D) 6
(26)The SI unit for _______ is ampere.
(A)
Luminous intensity (B) Electric current (C) Power (D) Electric voltage
(27)
Accuracy is defined as:
(A)
A measure of how often an experimental value can be repeated
(B)
The closeness of a measured value to the real value.
(C) The number of significant figures
used in a measurement.
(D) None of these
(28) Pa is the symbol for pascal, which is the SI unit for
(A)
Pressure (B) Capacitance (C) Electric charge (D) Force
(29)
Every measurement consists of a number and a __________
(A)Decimal (B)Standard (C)Exponent (D) Unit
(30) Light year is a unit of
(A) Time (B) Sunlight intensity (C) Distance
(D) Mass
(31) Which of the following instrument is more appropriate to measure thickness of Aluminum sheet?
(A)Scale (B)Meter tape (C)Vernier Callipers (D)Micrometer screw
(32)
The fraction 1/273.16 of the thermodynamic temperature of the triple point of
water is known as _____________
(A)Volt (B)Mole (C)Kelvin (D)Candela
(33)
Watt is a unit of
(A)Work (B)Power (C)Energy (D)Momentum
(34) The
significant numbers of the digit 1.054 is _______.
(A) 2 (B)3 (C)1 (D)4
(35) Internal
diameter of the hollow cylinder can be measured accurately by_______.
(A)
Vernier calipers (B) Measuring tape (C)
Meter rule (D) Micrometer screw
(36) Precision
depends on ______
(A) Light
(B) Pressure (C) Care taken by person (D) Least count of
instrument
(37) If
the zero of the vernier scale remains right side of main scale, then the error
is called-----.
(A)Negative
error (B) Positive error (C) Zero
error (D) Constant error
(38)
Which of the following unit is a fundamental Physical unit?
(A)m/s (B)m (C)N (D)m2
(39) No.
of Significant figures in 1.60 X 10 -19 is __________
(A)1 (B)2 (C)3 (D)4
(40) For
a vernier calliper 1 division of main scale is 1mm and total 20 divisions in
vernier scale. It has ……… least count
(A)0.05mm (B)0.02mm (C) 0.5mm (D)0.2mm
(41) S.I.
unit of work is __________
(A)Dyne (B)joule (C)Newton (D)Watt
(42) Energy/time
= __________
(A)Power (B)Speed (C)Acceleration (D)Efficiency
(43)Which
of the following instrument is used to measure thickness of wire?
(A)Vernier
callipers (B)Micrometer screw gauge (C)
Scale (D)Protector
“Two things are infinite: the universe and human
stupidity; and I'm not sure about the universe.” - Albert Einstein
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