Physical Quantities and Measurements

Physics: The branch of science that deals with matter, energy, and their relationship. It involves the study of physical universe and the laws governing it.

All measurable quantities are called Physical Quantities. Examples include length, mass, time, and temperature.

A physical quantity possesses at least two characteristics in common: numerical magnitude and a unit.

• Base Quantities: The quantities on the basis of which other quantities are expressed. There are 7 base quantities: Length, Mass, Time, Electric Current, Temperature, Intensity of Light, and Amount of Substance.
• Derived Quantities: The quantities that are expressed in terms of base quantities. Examples: Area, Volume, Speed, Force, Work, Energy, Power, Electric Charge.

In 1960, an international committee agreed on a set of definitions and standards to describe physical quantities. This system is known as the International System of Units (SI).

Prefixes are words or letters added before SI units relative to the power of ten. They are useful to express very large or very small quantities.

Examples: kilo ($10^3$), milli ($10^{-3}$), micro ($10^{-6}$).

Scientific Notation is a way of writing numbers as a product of a number between 1 and 10 and an integer power of 10.

General Form: $M \times 10^n$, where $1 \le M < 10$ and $n$ is an integer.

• Distance of Moon from Earth: $384,000,000 m = 3.84 \times 10^8 m$
• Diameter of Hydrogen Atom: $0.0000000001 m = 1 \times 10^{-10} m$

Metre Rule: A length measuring instrument used in laboratories. It is 1 metre long.

• Least Count: 1 mm (or 0.1 cm). It cannot measure accurately less than 1 mm.
• Parallax Error: Error caused by incorrect positioning of the eye. The eye must be vertically above the reading mark.

Used to measure length with greater accuracy (up to 0.1 mm). It consists of a Main Scale and a Vernier Scale.

• Least Count (LC): Difference between one Main Scale Division (MSD) and one Vernier Scale Division (VSD).
  Formula: $LC = 1 MSD - 1 VSD = 1 mm - 0.9 mm = 0.1 mm$ (or $0.01 cm$).
  Also: $LC = \frac{\text{Smallest reading on Main Scale}}{\text{Total divisions on Vernier Scale}}$.
• Zero Error: Occurs if the zero of Vernier scale does not coincide with zero of Main scale when jaws are closed.
  - Positive Zero Error: Zero of Vernier is to the right of Main zero. Correction: Subtract error.
  - Negative Zero Error: Zero of Vernier is to the left of Main zero. Correction: Add error.

Used to measure small lengths (e.g., diameter of wire) with accuracy up to 0.01 mm.

• Pitch: Distance moved by the spindle for one complete rotation (usually 0.5 mm or 1 mm).
• Least Count: $LC = \frac{\text{Pitch}}{\text{Total divisions on Circular Scale}}$.
  Example: If Pitch=1mm and Div=100, $LC = 0.01 mm$. If Pitch=0.5mm and Div=50, $LC = 0.01 mm$ ($0.001 cm$).
• Zero Error: Similar to Vernier, check if circular scale zero coincides with index line.

• Physical Balance: Used to measure mass. Works on the principle of moments (Beam Balance).
• Stopwatch: Used to measure time intervals.
  - Mechanical Stopwatch: Accuracy 0.1 s.
  - Digital Stopwatch: Accuracy 0.01 s.

Used to measure the volume of a liquid or an irregular solid (by displacement method).

• Meniscus: The curved surface of liquid. Read lower meniscus for water/colorless liquids, upper meniscus for mercury.
• Volume of Irregular Solid: $V_{solid} = V_{final} - V_{initial}$.

Significant Figures: All the accurately known digits and the first doubtful digit in an expression are called significant figures. They reflect the precision of a measurement.

1. Non-zero digits are always significant. (e.g., 27 has 2 sig figs)
2. Zeros between two significant figures are significant. (e.g., 2705 has 4 sig figs)
3. Final or ending zeros on the right in a decimal fraction are significant. (e.g., 275.00 has 5 sig figs)
4. Zeros written on the left side of the decimal point for the purpose of spacing the decimal point are NOT significant. (e.g., 0.00207 has 3 sig figs: 2, 0, 7)

• If the drop digit > 5, increase last retained digit by 1. (1.47 -> 1.5)
• If the drop digit < 5, leave last retained digit unchanged. (1.43 -> 1.4)
• If the drop digit = 5, round to nearest even number. (1.35 -> 1.4, 1.45 -> 1.4). (Note: Some textbooks say round up, but even rule is standard in physics).

• Precision: Refers to how close a set of measurements are to each other. It depends on the Least Count of the instrument.
  Example: A measurement with Vernier Callipers (LC 0.01cm) is more precise than a Metre Rule (LC 0.1cm).
• Accuracy: Refers to how close a measured value is to the actual or true value. It depends on the Fractional Uncertainty.
  Example: A measurement can be precise but not accurate (if there is a zero error).