Measurement and Units in Physics

Measurement in Daily Life

• Measurement is a crucial part of daily life, used in many activities.
• It helps us understand and interact with our environment accurately.
• Measurements are used in cooking, construction, transportation, healthcare, and more.
• They allow us to quantify, compare, and make informed decisions.
• Examples of physical quantities measured include mass, volume, length, temperature, time, density, pressure, energy, power, area, velocity/speed, and electrical/angular quantities.

Need for Measurement in Physics

• Experiments are essential for understanding phenomena in physics.
• Experiments require measurements of physical properties like length, mass, time, and temperature.
• Measurement is vital for experimental verification of laws and theories.
• Measurement focuses on:
  • Quantity: A physical property with a specific value and unit, convertible between units.
  • Unit: A measurement unit defined by its name and symbol.
  • Measurement: A quantity paired with a timestamp.

Physical Quantity

• A physical quantity can be measured and has a name, dimension, and unit.
• It can be manipulated mathematically and assigned a numerical value.
• Quantities can be basic/fundamental or derived, and scalar or vector.

Classification of Physical Quantities

A. Fundamental and Derived Quantities

1. Fundamental Quantity: Cannot be broken down into other quantities (e.g., mass, length, time). These are directly measured and have a single unit.
2. Derived Quantities: Combination of two or more fundamental quantities (e.g., area, speed, force, energy).

B. Scalar and Vector Quantities

1. Scalar Quantity: Defined by magnitude (value and unit) alone, represented by a single number.
2. Vector Quantity: Defined by both magnitude and direction, represented by an arrow.

Systems of Measurement

• Measurement is comparing an unknown quantity to a known standard.
• Every measurement has a number and a unit.

Standard Sets of Measurements/Units:

1. International System (SI) or MKS System (Meter-Kilogram-Second):
   • Fundamental units for length, mass, and time are meter, kilogram, and second, respectively.
   • First introduced in France.
2. Gaussian or CGS system (Centimeter-Gram-Second):
   • Uses centimeter, gram, and second as basic units.
   • Introduced in France and named after Carl Friedrich Gauss.
3. English System or FPS (Foot-Pound System):
   • Uses foot for length, pound for mass, and second for time.
   • Introduced in Britain.

SI (Système International d'Unités)

• The modern form of the metric system, most widely used globally.
• Provides a coherent set of units for physical quantities.
• A General Conference on Weights and Measures decided this system in 1971.

Seven Base Units/Fundamental Quantities

1. Meter (m): Length unit, distance traveled by light in a vacuum in \frac{1}{299,792,458} of a second.
2. Kilogram (kg): Mass unit, defined by a platinum-iridium cylinder stored at BIPM in France.
3. Second (s): Time unit, duration of 9,192,631,770 periods of radiation from the cesium-133 atom.
4. Ampere (A): Electric current unit, the current that produces a force of 2 \times 10^{-7} newtons per meter between two parallel conductors.
5. Kelvin (K): Temperature unit, \frac{1}{273.16} of the thermodynamic temperature of water's triple point.
6. Mole (mol): Amount of substance, the amount containing as many entities as atoms in 0.012 kilograms of carbon-12.
7. Candela (cd): Luminous intensity, the intensity of a source emitting monochromatic radiation at 540 × 10^12 hertz with a radiant intensity of \frac{1}{683} watt per steradian.

• These base units combine to derive units for other quantities.

Rules for Writing SI Units

1. Full name of unit starts with a small letter, even if named after a person.
2. Symbol for unit named after a scientist is a capital letter.
3. Symbols for all units are written in small letters.
4. One space between the last digit of the numeral and the symbol.
5. Units do not have plural forms.
6. No full stop after the units.
7. No space between the symbols for units.

SI Prefixes

• Decimal-based prefixes to denote multiples or fractions of base units.
• Allow expression of measurements from very small to very large values.

Scientific Notation

• Shorthand for very large or very small numbers, commonly used in science.
• Form: a \times 10^n, where a is between 1 and 10, and n is an integer.
• Example: 2.48 \times 10^{23}

Mathematical Operations

1. Addition and Subtraction:
   • Exponents must be the same before adding or subtracting.
   • Example: 2.48 \times 10^3 + 4.25 \times 10^4 - 3.2 \times 10^2 = 0.248 \times 10^4 + 4.25 \times 10^4 - 0.032 \times 10^4 = 4.466 \times 10^4
2. Multiplication:
   • (a \times 10^n) (b \times 10^m) = (a \times b) \times 10^{n + m}
3. Division:
   • (a \times 10^n) / (b \times 10^m) = (a / b) \times 10^{n - m}
4. Power:
   • (a \times 10^n)^m = a^m \times 10^{mn}

Conversion of Units

• Process of changing a measurement from one unit to another.
• Rules:
  1. Units are treated as algebraic quantities and can be multiplied/divided.
  2. Multiplying or dividing by 1 does not change the value.

Some Conversion Factors:

Time:

• 1 hour = 60 min = 3600 s
• 1 day = 1440 min = 8.64 \times 10^4 s
• 1 year = 365.2 days = 3.156 \times 10^7 s

Length:

• 1 meter (m) = 100 cm = 1000 mm = 39.37 in. = 3.281 ft
• 1 cm = 10 mm = 0.3937 in.
• 1 km = 1000 m = 0.6214 mi
• 1 foot (ft) = 12 in. = 0.3048 m = 30.48 cm
• 1 inch (in.) = 2.54 cm
• 1 mile (mi) = 5280 ft = 1.609 km
• 1 nautical mile (nmi) = 6076 ft = 1.152 mi = 1.852 km
• 1 light year = 9.461 \times 10^{15} m

Mass:

• 1 kilogram (kg) = 1000 grams (g)
• 1 g = 6.58 \times 10^{-5} slug
• 1 slug = 14.59 kg
• 1 atomic mass unit (amu) = 1.660 \times 10^{-27} kg

Volume:

• 1 m^3 = 10^3 liters (l) = 1 \times 10^6 cm^3 = 35.32 ft^3
• 1 liter (l) = 1000 ml = 10^3 cm^3
• 1 ml = 1 cubic centimeter (cm^3)
• 1 ft^3= 28.32 liters = 7.481 gallons

Force:

• 1 newton (N) = 10^5 dyne = 0.2248 lb
• 1 lb = 4.448 N = 4.448 \times 10^5 dyne

Pressure:

• 1 pascal (Pa) = 1 N/m^2
• 1 bar = 10^5 Pa = 14.5 lb/in^2
• 1 lb/in^2 = 6.895 \times 10^3 Pa
• 1 atm = 1.013 \times 10^5 Pa = 1.013 bar = 14.70 lb/in^2

Energy:

• 1 joule (J) = 10^7 ergs = 0.239 cal = 2.39 \times 10^{-4} kcal = 0.7376 ft . lb
• 1 cal = 4.186 J
• 1 kcal = 4185 J = 3077 ft . lb
• 1 Btu = 1054 J = 252 cal = 778 ft . lb
• 1 kWh = 3.6 \times 10^6 J = 2.655 \times 10^6 ft . lb = 860.4 kcal

Angle:

• 1 radian (rad) = 57.30^\circ = \frac{180^\circ}{\pi}
• 1^\circ = 0.01745 \text{ rad} = \frac{\pi}{180^\circ}
• 1 revolution (rev) = 360^\circ

Examples:

1. Convert 350 slug to kg:
   • 350 \text{ slug} \times \frac{14.59 \text{ kg}}{1 \text{ slug}} = 5106.5 \text{ kg}
2. Convert 40 mi/hr to:
   • a) ft/s: 40 \frac{\text{mile}}{\text{hr}} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 58.67 \text{ ft/s}
   • b) m/s: 40 \frac{\text{mile}}{\text{hr}} \times \frac{1609 \text{ m}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 17.88 \text{ m/s}

Practical Units for Measuring Length

• 1 micron = 10^{-6} m
• 1 nanometer = 10^{-9} m
• 1 angstrom = 10^{-10} m
• 1 fermi = 10^{-15} m
• Astronomical unit:
  • Mean distance of Earth from the Sun.
  • 1 astronomical unit = 1.5 \times 10^{11} m
• Light-year:
  • Distance traveled by light in a vacuum in one year.
  • 1 light year = 9.5 \times 10^{15} m
• Parsec:
  • Distance at which an arc of 1 AU subtends an angle of 1".
  • Largest practical unit of distance in astronomy.
  • 1 parsec = 3.1 \times 10^{16} m

Mass

• 1 metric ton = 1000 kg
• 1 quintal = 100 kg
• 1 pound = 0.454 kg
• 1 slug = 14.59 kg
• 1 Chandrasekhar limit = 1.4 \times \text{mass of sun} = 2.785 \times 10^{30} \text{ kg}
  • Biggest practical unit for measuring mass.
• 1 atomic mass unit = \frac{1}{12} \times \text{mass of single } C \text{ atom}
  • 1 atomic mass unit = 1.66 \times 10^{-27} kg
  • Smallest practical unit for measuring mass.

Time

• 1 Solar day = 24 h
• 1 Sidereal day = 23 h & 56 min
• 1 Solar year = 365 solar day = 366 sidereal day
• 1 Lunar month = 27.3 Solar day
• 1 shake = 10^{-8} s

Accuracy vs Precision

• Accuracy: Closeness of a measured value to the true value.
• Precision: Consistency or reproducibility of measurements.

Classifications of Accuracy:

1. Point Accuracy: Accuracy at a particular point on the scale.
2. Accuracy as Percentage of Scale Range: Determined by the uniform scale range.
   • Example: A thermometer with a scale range up to 500°C and accuracy of ±0.5% of scale range (0.005 x 500 = ±2.5°C) will have a maximum error of ±2.5°C.
3. Accuracy as Percentage of True Value: Determined by measured value with respect to its true value within ±0.5%.

• Repeatability: Variation when conditions are identical and measurements are repeated in a short time.
• Reproducibility: Variation using the same process among different instruments and operators over longer times.

Errors in the Realm of Measurement

1. Gross Errors:
   • Due to human oversight (e.g., misreading).
   • Avoidable with care in reading, recording, and calculating data.
   • Reduced by increasing the number of experimenters and averaging readings.
2. Random Errors:
   • Irregular errors due to unpredictable fluctuations (e.g., temperature, voltage).
3. Systematic Errors:
   1. Environmental Errors:
      • Arise from external conditions (e.g., temperature, pressure).
   2. Interference Error:
      • Occur when external factors interfere with the measurement process.
   3. Observational Errors:
      • Due to bias or carelessness.
   4. Parallax error:
      • Affected by the position of the observer or viewing angle.
4. Human Error:
   • Mistakes or inaccuracies by individuals.
   • Minimizable through proper training and supervision.
5. Personal error
   • Wide range of mistakes due to human factors
6. Instrumental Errors:
   • Due to faulty construction and calibration.
   • Examples include zero error, worn-off scales, and loading effects.

Errors Calculation

• Absolute Error:
  • |VA - VE|, where VA is the actual value and VE is the measured value.
• Percent Error:
  • (\frac{VA - VE}{V_E}) \times 100
• Relative Error:
  • \frac{\text{Absolute error}}{\text{Actual value}}

How to Reduce Errors in Measurement?

1. Monitor the procedure.
2. Ensure correct formulas.
3. Cross-check measurements.
4. Use high-precision instruments.
5. Pilot test measuring instruments.
6. Use multiple measures.
7. Perform measurements under controlled conditions.

How to Read a Ruler?

1. Understand the Graduations: Standard rulers are divided into inches and fractions of an inch (1/2, 1/4, 1/8, 1/16).
2. Identify the Inch Marks: Look for longer, numbered lines representing whole inches.
3. Locate the Fractional Marks: Smaller lines between inch marks represent fractions of an inch.
4. Count the Lines: Count lines from the nearest whole-inch mark.
5. Interpret Fractional Measurements: Determine the fractional part of the measurement.
6. Consider the Object Length: Estimate additional length if the object extends beyond the ruler.
7. Practice: Familiarize yourself with markings and improve precision.

Measurements

• ruler graduations with inches
• converted to equivalent decimal and millimeter metric values included in the document.
• Examples of using a ruler in inches and centimeters included in the document.

Conclusion

• Understanding physical quantities, units, and measurement is crucial in biology and psychology.
• In biology: Quantifies phenomena like organism size, concentrations, and reaction rates.
• In psychology: Assesses behavior, cognition, and physiology using surveys, tests, neuroimaging, and monitoring.
• Accurate measurement is essential for reliable data, testing hypotheses, and improving health and well-being.
• Interdisciplinary research integrates measurement techniques for comprehensive insights.