Measurement of Basic Quantities

Measurement Of Length

Measurement Of Large Distances

Measuring large distances often requires indirect methods, as direct measurement with a ruler or tape measure is impractical. Common techniques include:

• Parallax Method: This is used for measuring distances to nearby stars. By observing a star from two different points in Earth's orbit (e.g., six months apart), the apparent shift in the star's position against a distant background can be measured. This angular shift (stellar parallax) is used to calculate the distance. If $ p $ is the parallax angle in arcseconds, and the distance $ d $ is in parsecs, then $ d = 1/p $. (1 parsec is approximately 3.26 light-years).
• Radar Method: For distances within the solar system (e.g., to planets), radio waves (radar) are transmitted from Earth, reflected off the planet's surface, and detected back on Earth. Knowing the speed of light ($ c $) and the time taken for the round trip ($ t $), the distance ($ D $) can be calculated using $ D = \frac{1}{2} c t $.
• Laser Ranging: Similar to radar, lasers can be used to measure distances to the Moon by timing the reflection of laser pulses.
• Triangulation: For measuring distances on Earth (e.g., to inaccessible points), a baseline is established, and angles are measured to the target from the ends of the baseline. Using trigonometry, the distance can be calculated.

Estimation Of Very Small Distances: Size Of A Molecule

Estimating the size of molecules requires clever indirect methods, often involving macroscopic quantities that depend on the molecular size.

• Monolayer Method (Oleic Acid Experiment): This classic experiment estimates the size of a molecule like oleic acid. A known small quantity of oleic acid is dissolved in alcohol and spread as a thin film (monolayer) on a large water surface. The volume of oleic acid ($ V $) and the area it covers ($ A $) on water are known. Assuming the film is one molecule thick, the volume of a single molecule can be approximated as $ V_{\text{molecule}} \approx V/N $, where $ N $ is the number of molecules. If the molecule is approximated as a cube or sphere with side/radius $ a $, then $ V_{\text{molecule}} \approx a^3 $. From $ V_{\text{molecule}} = A \times \text{thickness} $, the thickness (molecular size) can be estimated as $ \frac{V}{A} $.
• X-ray Diffraction: This is a more precise method used to determine the precise atomic and molecular structure of materials. When X-rays pass through a crystalline solid, they are diffracted by the regular arrangement of atoms, creating a diffraction pattern. Analyzing this pattern allows for the determination of interatomic distances and molecular dimensions.

Range Of Lengths

Lengths in the universe span an incredible range, from the subatomic to the cosmological:

Here's a glimpse of the scale:

• Size of a Nucleus: $ \sim 10^{-15} $ m (femtometre or fermi)
• Size of an Atom: $ \sim 10^{-10} $ m (Angstrom)
• Size of a Molecule: $ \sim 10^{-9} $ m (nanometre)
• Wavelength of Light: $ \sim 10^{-7} $ m
• Thickness of a Paper: $ \sim 10^{-4} $ m
• Size of a Person: $ \sim 10^0 $ m (1 metre)
• Size of the Earth: $ \sim 10^7 $ m
• Distance to the Sun: $ \sim 10^{11} $ m (1 Astronomical Unit, AU)
• Size of the Solar System: $ \sim 10^{13} $ m
• Distance to the Nearest Star (Proxima Centauri): $ \sim 10^{16} $ m (4.24 light-years)
• Size of the Milky Way Galaxy: $ \sim 10^{21} $ m
• Size of the Observable Universe: $ \sim 10^{26} $ m

Measurement Of Mass

Mass is a fundamental property of matter that quantifies its inertia and its interaction via gravity. Measuring mass can be done directly using a balance or indirectly using its effects.

• Using a Physical Balance: A physical balance compares the unknown mass with known standard masses. The principle is to balance the torques produced by the masses on the two arms of the balance. When balanced, the unknown mass is equal to the sum of the standard masses used.
• Using a Spring Balance: A spring balance measures weight ($ W = mg $), which is the force of gravity on an object. If the gravitational acceleration ($ g $) is known, the mass ($ m $) can be determined by measuring the weight.
• Inertial Balance: This method relies on the definition of mass as a measure of inertia. An object is attached to a spring, and its oscillation period ($ T $) is measured. The period depends on the mass ($ m $) and the spring constant ($ k $) according to $ T = 2\pi\sqrt{m/k} $. By comparing the period of the unknown mass with the period of a known mass, the unknown mass can be determined.
• Mass Spectrometry: This is a powerful analytical technique used to determine the mass-to-charge ratio of ions. It is used to identify elements and isotopes and to determine the molecular mass of compounds.

Range Of Masses

Similar to lengths, masses also span an enormous range in the universe:

• Mass of an Electron: $ \sim 9.11 \times 10^{-31} $ kg
• Mass of a Proton/Neutron: $ \sim 1.67 \times 10^{-27} $ kg
• Mass of a DNA Molecule: $ \sim 10^{-21} $ kg
• Mass of a Bacterium: $ \sim 10^{-15} $ kg
• Mass of a Human: $ \sim 10^2 $ kg
• Mass of the Earth: $ \sim 6 \times 10^{24} $ kg
• Mass of the Sun: $ \sim 2 \times 10^{30} $ kg
• Mass of the Milky Way Galaxy: $ \sim 10^{42} $ kg
• Mass of the Observable Universe: $ \sim 10^{53} $ kg

Measurement Of Time

Time is a fundamental dimension that orders the sequence of events, measures their duration, and quantifies the intervals between them. Measuring time is crucial in physics for understanding motion, periodicity, and the evolution of systems.

• Atomic Clocks: The most accurate timekeeping devices are atomic clocks. The SI definition of the second is based on the frequency of radiation emitted or absorbed by caesium-133 atoms during a transition between two hyperfine levels of the ground state. An atomic clock measures time by counting the oscillations of this radiation, providing extremely precise and stable time measurements. The frequency is $ \nu_{Cs} = 9,192,631,770 \, \text{Hz} $, and one second is defined as the time taken for $ 9,192,631,770 $ periods of this radiation.
• Pendulum Clocks: Historically, pendulum clocks were used for accurate timekeeping. The period of a simple pendulum is given by $ T = 2\pi\sqrt{L/g} $, where $ L $ is the length of the pendulum and $ g $ is the acceleration due to gravity. However, their accuracy is affected by variations in $ g $ and air resistance.
• Quartz Clocks: These clocks use the piezoelectric property of a quartz crystal. When an electric voltage is applied, the crystal oscillates at a very precise frequency. These oscillations are used to regulate the timekeeping.
• Astronomical Observations: Historically, timekeeping was based on the Earth's rotation (solar day) and its orbit around the Sun (year). While less precise than atomic clocks, astronomical events still provide a framework for larger time scales.

The SI unit of time is the second (s).