Physics Fundamentals and Measurement Units

Fundamental Physical Quantity: Independent quantities such as length, mass, time, temperature, electric current, luminous intensity.
Derived Physical Quantity: Dependent quantities derived from fundamental quantities; examples include area, volume, density, speed, force, acceleration, power, energy, work, potential difference, magnetic moment.

Fundamental Unit: Unit for fundamental physical quantities.
Derived Unit: Unit for physical quantities derived from fundamental units.
System of Units: Consists of fundamental and derived units; commonly used units include:
- Length: foot, metre
- Mass: pound, kilogram
- Time: second
- Temperature: Fahrenheit, Celsius, Kelvin
- Electric Current: ampere
- Luminous Intensity: candela

Full unit names should not be written in plural form; use singular forms (e.g., meter, kilogram).
Symbols should be in singular without plural forms (e.g., 20 m, not 20 ms).
Only symbols named after individuals should capitalize (e.g., N for Newton, J for Joule).
Do not add punctuation after unit symbols.

Recognized system globally; basic units include:
- Length: 1 metre (m) defined based on the platinum-iridium bar.
- Mass: 1 kilogram (kg) defined by a prototype cylinder.
- Time: 1 second (s) defined as 1/86400 of a mean solar day.
- Electric Current: 1 ampere (A) defined by the force between conductors in a vacuum.

Scalar Quantity: Requires only magnitude (e.g., mass, time, speed).
Vector Quantity: Requires both magnitude and direction (e.g., velocity, force).

Dimensions: Expressed in terms of fundamental units; length [L], mass [M], time [T].
Dimensional Formula: Shows the dependence of a physical quantity on fundamental quantities (e.g., area [M^0 L^2 T^0]).

Used to check equations' correctness and to determine conversion factors for different units.

Instrumental Errors: Result from faulty instruments; maintain constant errors over time.
Random Errors: Occur due to unpredictable variations in experiment conditions.

Use tools like vernier calipers and micrometers to measure length and diameter with specified precision and significant figures.

Increase the last retained figure by 1 if dropping figure is 5 or greater; retain if less than 5.
Express measurements with the same number of decimal places as the least count of the measuring instrument.### Ancient Astronomical Instruments
Bhaskaracharya and Lalla were significant mathematicians who crafted and refined tools for examining celestial phenomena.

Accurate measurements employ vernier calipers and micrometer screw gauges, which have specific least counts that determine their precision.

Example of Width Measurement:
- Averaged width: 4.12 cm from readings of 4.13 cm, 4.11 cm, 4.10 cm, and 4.14 cm.
- Average absolute error calculated as 0.015 cm.
- Percentage error computed to be 0.364%.
Example of Cylinder Diameter Measurement:
- Diameter recorded as 9.32 ± 0.16 cm.
- Relative error formulated as 0.00322.
- Percentage error found to be 0.322%.
Example of Wire Diameter Measurement:
- Mean diameter determined as 0.2215 cm from readings: 0.221 cm, 0.223 cm, 0.220 cm, and 0.222 cm.
- Average absolute error calculated at 0.001 cm.
- Percentage error approximated as 0.0045%.
Comparative Accuracy Assessment:
- Measurements compared for accuracy:
  - (i) 3.75 ± 0.008 results in a percentage error of 0.2133%.
  - (ii) 2.44 ± 0.005 yields a percentage error of 0.2049%.
  - (iii) 4.18 ± 0.006 gives a percentage error of 0.1439%.
- Smaller percentage errors indicate greater accuracy.

Least Count Defined:
- The least count of an instrument is the smallest dimension that can be measured, determined through specific calculations based on the instrument's scale divisions.
Vernier Caliper Usage:
- Measures dimensions by aligning the object with jaws; the measurement is read directly from the calibrated scale.
Micrometer Screw Gauge:
- The least count is derived from the screw's pitch divided by the number of divisions on the circular scale.
- Enables high precision measurements, often down to thousandths of a centimeter.

Zero error occurs when the instrument's measurement doesn’t correspond to the expected zero reference point.
- Types of Zero Error:
  - Negative zero error: Circular scale zero is below the reference line, leading to an overestimation of measured values.
  - Positive zero error: Circular scale zero is above the reference line, leading to an underestimation.
Correct readings are obtained by adjusting observed values with identified zero errors.