Measurement and Experimentation

Measurements and Experimentation Syllabus

• The syllabus includes the International System of Units (SI) and other common systems like FPS and CGS
• Measurements using instruments like Vernier callipers, micrometre screw gauge, and simple pendulum are considered
• Vernier callipers and micrometre screw gauge are used for measuring length
• Decreasing the least count increases accuracy
• The syllabus includes the least count (L.C.) of Vernier callipers and screw gauge as well as zero error
• Time period, frequency, and graphs of length vs. T² in simple pendulums are referenced, as well as the formula T = 2π√(l/g)

Need of Unit for Measurement

• Physics involves experimental studies and measurements
• Physical quantities are measured by comparing them to a standard quantity of the same nature
• Measurement compares a given physical quantity to a known standard of the same nature
• A unit measures a given physical quantity
• Quantities of different nature use different units
• A unit is a constant magnitude used to measure the magnitudes of other quantities of the same nature
• The measurement of a physical quantity involves a unit and a numerical value
• Physical quantity = (numerical value) × (unit)
• 10 metre means the length is measured in the unit metre and this unit is contained 10 times in the length
• 5 kilogram, meaning the mass is measured in the unit kilogram and the unit is contained 5 times in the quantity

Choice of Unit

• A unit should be of convenient size, definable without ambiguity, reproducible, and unchanging with space and time
• Units are of two kinds: fundamental or basic and derived

Fundamental or Basic Units

• A fundamental unit is independent of any other unit, unchangeable, and unrelated to any other fundamental unit
• Mass, length, time, temperature, current, and amount of substance use fundamental units

Derived Units

• Units of quantities other than those measured with fundamental units are derived units and can be expressed in terms of fundamental units
• Area is expressed as length × length or (length)²
• Volume is expressed as length x length x length or (length)³
• Speed is expressed as length / time
• Area, volume, and speed use derived units

Systems of Units

• Length, mass, and time are the three fundamental quantities in mechanics
• The C.G.S. system uses centimetre (cm) for length, gram (g) for mass, and second (s) for time
• The F.P.S. system uses foot (ft) for length, pound (lb) for mass, and second (s) for time
• The M.K.S. system uses metre (m) for length, kilogram (kg) for mass, and second (s) for time
• The SI system of units is an enlarged and modified version of the metric system

Système Internationale d'Unites (or S.I. system)

• In 1960, it was recommended that temperature, luminous intensity, current, amount of substance, angle, and solid angle also use fundamental units
• There are seven fundamental and two complementary fundamental units

Fundamental quantities, units and symbols in S.I. system

• Length (metre, m)
• Mass (kilogram, kg)
• Time (second, s)
• Temperature (kelvin, K)
• Luminous intensity (candela, cd)
• Electric current (ampere, A)
• Amount of substance (mole, mol)
• Angle (radian, rd)
• Solid angle (steradian, st-rd) Nowadays 1 mol means 1 kg mol equal to 6-02 x 1026 entities (i.e., atoms or molecules or ioms).

Use of Prefix with a Unit

• Prefixes like deca, hecto, and kilo are used with units to express large measurements

Prefixes used for big measurements

• deca (da) - 10¹
• hecto (h) - 10²
• kilo (k) - 10³
• mega (M) - 10⁶
• giga (G) - 10⁹
• tera (T) - 10¹²
• peta (P) - 10¹⁵
• exa (E) - 10¹⁸
• zetta (Z) - 10²¹
• yotta (Y) - 10²⁴
• Small measurements use prefixes like deci, centi, milli, and micro

Prefixes used for small measurements

• deci (d) - 10⁻¹
• centi (c) - 10⁻²
• milli (m) - 10⁻³
• micro (µ) - 10⁻⁶
• nano (n) - 10⁻⁹
• pico (p) - 10⁻¹²
• femto (f) - 10⁻¹⁵
• atto (a) - 10⁻¹⁸
• zepto (z) - 10⁻²¹
• yocto (y) - 10⁻²⁴
• 2.5 GHz = 2.5 × 10⁹ Hz
• 5.0 pF = 5.0 × 10⁻¹² F
• 5.0 MΩ = 5.0 × 10⁶ Ω
• 2.0 ms = 2.0 × 10⁻³ s

Units of Length

• The SI unit of length is the metre (m)
• The metre is equal to 1,650,763.73 wavelengths of orange-red spectral line in Krypton-86
• 1 metre is 1,553,164.1 times the wavelength of the red line in the emission spectrum of cadmium
• 1 metre is the distance travelled by light in 1/299,792,458 of a second in air (or vacuum)

Subunits of Metre

• Centimetre (cm): 1 cm = 1/100 m = 10⁻² m
• Millimetre (mm): 1 mm = 1/1000 m = 10⁻³ m = 1/10 cm
• Micrometre or micron (μ): 1 micron (μ) = 10⁻⁶ metre = 10⁻⁴ cm = 10⁻³ mm
• Nanometer (nm): 1 nm = 10⁻⁹ m

Multiple Units of Metre

• Kilometer (km): 1 km = 1000 m (or 10³ m)

Non-Metric Units of Length

• Astronomical unit (A.U.): 1 A.U. = 1.496 x 10¹¹ metre, equals the mean distance between the earth and the sun
• Light year (ly): 1 light year = 9.46 x 10¹⁵ m = 9.46 × 10¹² km, the distance light travels in a vacuum in one year
• 1 light minute = 3 x 10⁸ m s⁻¹×60 s = 1.8 × 10¹⁰ m
• 1 light second = 3 x 10⁸ m s⁻¹ x 1 s = 3 x 10⁸ m
• Parsec: 1 Parsec = 3.08 x 10¹⁶ m, the distance from where the semi major axis of orbit of earth (1 A.U.) subtends an angle of one second

Smaller Units of length

• Angstrom (Å): 1 Å = 10⁻¹⁰ metre = 10⁻⁸ cm = 10⁻¹ nm
• 1 micron = 10,000 Å
• 1 nm = 10 Å
• Accepted today, wavelength, inter-atomic or inter-molecular separation, etc. are now commonly expressed in nm
• Fermi (f): 1 fermi (f) = 10⁻¹⁵ m

Summary of Length Units

• Smaller units:
  • cm = 10⁻² m
  • mm = 10⁻³ m
  • μ (or µm) = 10⁻⁶ m
  • nm = 10⁻⁹ m
  • Å = 10⁻¹⁰ m
  • f = 10⁻¹⁵ m
• Bigger units:
  • km = 10³ m
  • A.U. = 1.496 x 10¹¹ m
  • ly = 9.46 x 10¹⁵ m
  • parsec = 3.08 x 10¹⁶ m

Units of Mass

• The SI unit of mass is the kilogram (kg)
• Kilogram equals the mass of a platinum-iridium alloy cylinder
• 1 litre (= 1000 ml) of water at 4°C is also 1 kilogram

Subunits of Kilogram

• Gram (g): 1 g = 1/1000 kg = 10⁻³ kg, OR 1 kg = 1000 g
• Milligram (mg): 1 mg = 10⁻⁶ kg OR 1 mg = 10⁻³ g

Multiple Units of Kilogram

• Quintal: 1 quintal = 100 kg
• Metric tonne: 1 metric tonne = 1000 kg = 10 quintal

Non-Metric Unit of Mass

• Atomic mass unit (a.m.u) or unified atomic mass unit (u): 1 a.m.u. (or u) is 1/12th the mass of one carbon-12 atom
• The mass of 6-02 x 10²⁶ atoms of carbon -12 is 12 kg

Conversions

• 1 a.m.u (or u) = 1/12× 6.02×1026 kg = 1.66 × 10⁻²⁷ kg
• Solar mass: One solar mass equals the mass of the sun (2 x 10³⁰ kg)

Summary of Mass Units

• Smaller units:
  • g = 10⁻³ kg
  • mg = 10⁻⁶ kg
  • u (or a.m.u.) = 1.66 x 10⁻²⁷ kg
• Bigger units:
  • quintal = 100 kg
  • metric tonne = 1000 kg
  • solar mass = 2x10³⁰ kg

Units of Time

• The SI unit of time is the second (s)
• A second is defined as 1/86400 th part of a mean solar day
• One solar day is the time taken by the earth to complete one rotation on its own axis
• A second is 1/31,556,925.9747th part of the year 1900
• One second is the time interval of 9,192,631,770 vibrations of radiation

Smaller Units of Time

• millisecond (ms) = 10⁻³ s
• microsecond (µs) = 10⁻⁶ s
• shake = 10⁻⁸ s
• nanosecond (ns) = 10⁻⁹ s

Bigger Units of Time

• minute (min): 1 min = 60 s
• hour (h): 1 h = 60 min = 3600 s
• day: 1 day = 24 h = 86400 s, is the time for the earth to rotate once
• month: about 30 days
• lunar month: about 29.5 days
• year (yr): 1 yr = 365 days, is the time for the earth to revolve around the sun

Other time periods

• Leap year: the month of February is of 29 days and has 366 days
• Decade = 10 years = 3·1536 × 10⁸ s
• Century = 100 years = 3.16 × 10⁹ s, has 24 years of 366 days
• Millennium: 1 Millennium = 1000 years = 3.16 × 10¹⁰ s

Values of bigger units of time

• min = 60 s
• h = 3600 s
• day = 86400 s
• month = 2.592x10⁶ s
• year = 3.1536 x 10⁷ s
• Decade = 3.1536 x 10⁸ s
• Century= 3·16 × 10⁹ s
• Millennium = 3.16 × 10¹⁰ s

Examples of Derived Units

• Derived units are obtained by multiplying or dividing fundamental units:
  • Area (length x breadth): metre x metre = m²
  • Volume (length x breadth × height): metre x metre x metre = m³
  • Density (mass / volume): kilogram / (metre)³ = kg m⁻³
  • Speed or velocity (distance / time): metre / second = m s⁻¹
  • Acceleration (velocity / time): (metre/second) / second = m s⁻²
  • Force (mass x acceleration): kilogram x (metre / second²) = kg m s⁻² or newton (N)
  • Work or energy (force x displacement): kilogram x (metre / second²) x metre = kg m² s⁻² or joule (J)
  • Momentum (mass x velocity): kilogram x (metre / second) = kg m s⁻¹ or N s
  • Moment of force or torque (force x distance): kilogram x (metre / second²) x metre = kg m² s⁻² or newton-metre (N m)
  • Power (work / time): kilogram (metre)²/ (second)³ = kg m² s⁻³ or joule/second or watt (W)
  • Pressure (force / area): kilogram x metre / (second)²/(metre)² = kg m⁻¹ s⁻² or newton/(metre)² or pascal (Pa)
  • Frequency (1 / time period): 1 / second = s⁻¹ or hertz (Hz)
  • Electric charge (current x time): ampere x second = As or C
  • Electric potential or electromotive force (e.m.f.) (work/charge): kilogram × metre² / second² / ampere x second = kg m² A⁻¹ s⁻³ or joule/coulomb or volt
  • Electrical resistance (potential / current): kilogram × metre² / ampere × second³ / ampere = kg m² A⁻² s⁻³ or volt / ampere or ohm
  • Electrical power (potential x current): volt x ampere or watt = VA or W
• Some derived units have special names related to scientists who worked in that field like newton for force, joule for work, watt for power, pascal for pressure, hertz for frequency, coulomb for electric charge, volt for electric potential and ohm for electrical resistance

Guidelines for writing the Units

• Unit symbols not named after a scientist are written in lowercase (e.g., m for metre, s for second, kg for kilogram)
• Unit symbols named after a scientist are written with the first letter capitalized (e.g., N for newton, J for joule, W for watt, Pa for pascal, Hz for hertz, C for coulomb, V for volt)
• Full names of units are written in lowercase, regardless of origin (e.g., kilogram, metre, newton, joule, watt)
• Compound units have multiplication expressed with a dot, cross, or space (e.g., N.m, N x m, Nm)
• Negative powers indicate division of one unit by the other

Examples of Expressing Units

• Velocity in metre / second = m s⁻¹
• Power in joule / second = J s⁻¹
• Short-form units are never pluralized, for example, 10 metres, not 10 ms (ms means millisecond)
• Prefixes can be used to avoid powers of ten, for example, use GW instead of kMW
• When using a prefix with a unit symbol, the combination is a single symbol

Least Count of a Measuring Instrument

• Instruments are needed to measure physical quantities
• Every instrument has a definite limit for accuracy in terms of its least count
• The least count of an instrument is the smallest measurement that can be taken accurately with it
• A measuring instrument has a graduated scale for measurement
• The least count is the value of one smallest scale division

Measurement of Length

• Metre rules are often used to measure objects
• Metre rules have 10 subdivisions in each centimetre, so the value of one small division is 1 mm
• Metre rules can measure to 1 mm accuracy
• Vernier callipers and screw gauges are more accurate, with least counts smaller than 0.1 cm.

Principle of Vernier

• The vernier enables length measurement up to the 2nd decimal place of a cm
• Two scales are used, the main scale that is fixed and the vernier scale that slides along the main scale
• the main scale usually has divisions of 1 mm
• vernier scale graduations equal n divisions of the vernier scale to (n - 1) divisions of the main scale

Least count of vernier

• The least count of vernier is equal to the difference between values of one main scale division and one vernier scale division
• L.C. = value of 1 main scale division - value of 1 vernier scale division
• L.C. = Value of one main scale division / Total number of divisions on vernier
• Example: If the main scale graduated reads to 1mm, and the Vernier scale on which the length of 10 divisions is equal to the length of 9 divisions on the main scale L.C.=1mm/10 = 0.1mm = 0.01cm
• To illustrate the use of a vernier scale main scale and vernier scale are so made that when the movable vernier scale touches the fixed end, its zero mark coincides with the zero mark of the main scale

Vernier callipers

• A vernier callipers is a slide calliper
• It can measure: the diameter of a sphere, hollow cylinder, and depth of a small beaker
• Vernier scale can have 20, 25, or 50 divisions
• The least count = Value of one main scale division / Total number of divisions on vernier

Zero Error on Vernier Calliper

• When the movable jaw 2 in contact with the fixed jaw 1 the zero mark of vernier scale should be coincide the zero mark of main scale
• If it's not it's said to be have zero error . this zero error is equal to the length of between zero mark of the main scale and the zero mark of the vernier
• The two types of error on vernier callier are as follows: a. positive zero error b. Negative zero error
• to get correct/True measurement of a vernier calliper : True measurement = Observed reading - Zero error (with sign)

Principle of a Screw

• An ordinary screw has threads on it at an equal distance along its length
• the pitch of the screw: can be moved along its axis by the screw in one complete rotation of its head

Least count of Screw

• Pitch of screw: if a screw is 1mm and if you have 100 divisions on its head on rotation of 100 divisions of its circular scale , the pointed end of the screw moves by distance equal to 1mm. hence distance by the screw along is axis that the rotation is divided per a circular scale will approximately be 1/100 = 0.01mm it can also say L.C

Screw Gauge

• It works on principle of Screw. it use to measure diameter. It measures wire,paper L.C = pitch of screw gauge/ Total N.O on circular scale