Study Guide on Measurements and Experimentation

MEASUREMENTS AND EXPERIMENTATION

Syllabus Overview

• International System of Units (S.I.)
  • Understand required S.I. units with correct symbols.
  • Overview of other commonly used unit systems: FPS (Foot-Pound-Second) and CGS (Centimeter-Gram-Second).
• Measurement Tools
  • Common instruments:
  • Vernier callipers for measuring length.
  • Micrometer screw gauge for precise length measurement.
  • Simple pendulum for time measurement.
• Scope
  • Measurement techniques, least count and accuracy.
  • Zero error concept for Vernier callipers and screw gauge.
  • Understanding time period and frequency of a simple pendulum.
  • Graphical representation of length versus T², with slope interpretation related to pendulum studies.

A. SYSTEMS OF UNIT AND UNITS IN S.I. SYSTEM

1.1 NEED OF UNIT FOR MEASUREMENT

• Definition of Measurement:
  • Measurement is comparing a physical quantity to a known standard quantity.
  • Unit is defined as a quantity used to express magnitudes of other quantities of the same kind.
  • Formula: Physical quantity = (numerical value) × (unit).
  • Examples:
  • Length of cloth: 10 metres means length is expressed in metres and is contained 10 times in that length.
  • Mass of sugar: 5 kilograms indicates mass is expressed in kilograms and is contained 5 times in that mass.

1.2 CHOICE OF UNIT

• Properties of a Good Unit:
  1. Convenient size.
  2. Clear definition without ambiguity.
  3. Reproducibility.
  4. Consistency in value across time and space.

1.3 KINDS OF UNIT

1. Fundamental (Basic) Units:
   • Independent units that cannot be derived from other measurements.
   • Examples include mass (kg), length (m), time (s), temperature (K), electric current (A), amount of substance (mol).
2. Derived Units:
   • Units derived from fundamental units which can be expressed in terms of them.
   • Examples of derived units:
     • Area: length × breadth = m².
     • Volume: length × breadth × height = m³.
     • Speed: length/time = m/s.

1.4 SYSTEMS OF UNITS

• In mechanics, three fundamental quantities: length, mass, and time.

1. C.G.S. System:
   • Length: centimeter (cm).
   • Mass: gram (g).
   • Time: second (s).
2. F.P.S. System:
   • Length: foot (ft).
   • Mass: pound (lb).
   • Time: second (s).
3. M.K.S. System:
   • Length: meter (m).
   • Mass: kilogram (kg).
   • Time: second (s).
4. S.I. System:
   • Seven fundamental quantities and two complementary fundamental quantities.
   • Units and symbols:
     • Length: meter (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).

B. MEASUREMENTS OF LENGTH

1.5 S.I. UNIT OF LENGTH

• Definition of Metre:
  • Original definition: distance between two specified marks on a platinum-iridium rod kept at 0°C.
  • Redefined to be 1,650,763.73 times the wavelength of a specified spectral line in krypton-86 (1960).
  • In 1983, it was further defined in relation to the speed of light as the distance light travels in vacuum in $\frac{1}{299792458}$ seconds.

1.6 SUB-UNITS OF METRE

• Common sub-units:
  • Centimetre (cm) = 10⁻² m.
  • Millimetre (mm) = 10⁻³ m.
  • Micron (μ) = 10⁻⁶ m.
  • Nanometre (nm) = 10⁻⁹ m.
• Multiple Units of Metre:
  • Kilometre (km) = 1,000 m = 10³ m.
• Non-Metric Units of Length:
  • Astronomical Unit (A.U.) = $1.496 imes 10^{11} m$ (distance from Earth to the Sun).
  • Light Year (ly) = distance light travels in one year ≈ $9.46 imes 10^{15} m$.
  • Parsec = distance at which 1 A.U. subtends an angle of one arcsecond ≈ $3.086 imes 10^{16} m$.

1.7 UNITS OF MASS

• S.I. UNIT OF MASS:
  • Defined as kilogram (kg).
  • Original definition: mass of a cylindrical platinum-iridium piece; also 1 litre of water at 4°C.

1.8 SUB-UNITS OF KILOGRAM

• Common sub-units:
  • Gram (g) = 10⁻³ kg.
  • Milligram (mg) = 10⁻⁶ kg.
• Larger units of mass:
  • Quintal = 100 kg.
  • Metric tonne = 1,000 kg = 10 quintals.

1.9 NON-METRIC UNIT OF MASS

• Expressed in Atomic Mass Unit (a.m.u or u).
  • Defined as $\frac{1}{12}$ the mass of a carbon-12 atom.
  • $1 a.m.u = 1.66 imes 10^{-27} kg$.
• Solar Mass:
  • Defined as the mass of the Sun: $1 ext{ solar mass} = 2 imes 10^{30} kg$.

1.10 UNITS OF TIME

• S.I. UNIT OF TIME:
  • Defined as second (s) based on mean solar day with more modern definitions based on the cesium atom oscillation frequency.
• Sub Units of Time:
  • Millisecond (ms): 10⁻³ s.
  • Microsecond (μs): 10⁻⁶ s.
  • Nanosecond (ns): 10⁻⁹ s.

C. MEASUREMENT OF TIME AND SIMPLE PENDULUM

1.14 MEASUREMENT OF TIME

• Definition of Simple Pendulum:
  • A simple pendulum consists of a point mass (bob) suspended from a rigid support by a massless string.
  • The bob's mean position is crucial for time measurement during oscillation.
• Oscillation Definition:
  • One complete to-and-fro motion of the bob (from mean position to extremes and back).
• Effective Length:
  • The distance between point of suspension and center of gravity of the bob.
• Measurement of Time Period (T):
  • Time taken to complete one oscillation. Formula: $T = 2 ext{π} imes \frac{l}{g}$, with l as length and g as gravity acceleration.
  • Frequent measurement is typically for multiple oscillations to improve precision.

EXAMPLES OF SIMPLE PENDULUM MEASUREMENTS

1. Calculation of Length:
   • To find length of seconds' pendulum using acceleration due to gravity as a constant value.
2. Variation in Time Period:
   • Reflects relationship of time period with length (T is proportional to $ext{squared root of length}$).
3. Factors Influencing Period:
   • Length (increases time period), gravity (decreases time period at higher altitudes), and mass (independent).