Units And Measurement Notes

Introduction

In physics, measurements and units are fundamental concepts that form the basis for understanding the physical world. The NEET syllabus emphasizes the importance of mastering these concepts as they are foundational for solving problems in various fields of physics. This study note will cover the key aspects of units and measurement, breaking down complex ideas into simpler sections for better comprehension.

Physical Quantities

Fundamental and Derived Quantities

Fundamental Quantities: These are the basic quantities that are independent of other quantities. There are seven fundamental quantities in physics:

• Length (meter, $m$)
• Mass (kilogram, $kg$)
• Time (second, $s$)
• Electric current (ampere, $A$)
• Thermodynamic temperature (kelvin, $K$)
• Amount of substance (mole, $mol$)
• Luminous intensity (candela, $cd$)

Derived Quantities: These quantities are derived from the fundamental quantities through mathematical relationships. Examples include:

• Velocity ($m/s$)
• Acceleration ($m/s^2$)
• Force ($kg \cdot m/s^2$ or $N$)
• Energy ($kg \cdot m^2/s^2$ or $J$)

Systems of Units

SI Units

The International System of Units (SI) is the most widely used system for measurement. It is based on the seven fundamental quantities.

Examples of SI units:

• Length: meter ($m$)
• Mass: kilogram ($kg$)
• Time: second ($s$)
• Electric current: ampere ($A$)
• Temperature: kelvin ($K$)
• Amount of substance: mole ($mol$)
• Luminous intensity: candela ($cd$)

CGS, FPS, and MKS Systems

CGS System:

• Length: centimeter ($cm$)
• Mass: gram ($g$)
• Time: second ($s$)

FPS System:

• Length: foot ($ft$)
• Mass: pound ($lb$)
• Time: second ($s$)

MKS System:

• Length: meter ($m$)
• Mass: kilogram ($kg$)
• Time: second ($s$)

Measurement of Physical Quantities

Accuracy, Precision, and Errors

Accuracy: Refers to how close a measured value is to the true value.

Precision: Refers to how close the measured values are to each other.

Errors: Errors are the deviations of the measured value from the true value. They can be categorized as:

• Systematic Errors: Consistent and repeatable errors due to faulty equipment or bias.
• Random Errors: Errors that vary unpredictably.

Absolute Error: The difference between the measured value and the true value. $$ \Delta A = A_{\text{measured}} - A_{\text{true}} $$

Relative Error: The ratio of the absolute error to the true value. $$ \text{Relative Error} = \frac{\Delta A}{A_{\text{true}}} $$

Percentage Error: The relative error expressed as a percentage. $$ \text{Percentage Error} = \left( \frac{\Delta A}{A_{\text{true}}} \right) \times 100 $$

Significant Figures

Significant figures are the digits in a measurement that contribute to its precision. This includes all known digits plus one estimated digit.

Rules for Determining Significant Figures

1. All non-zero digits are significant.
2. Any zeros between significant digits are significant.
3. Leading zeros are not significant.
4. Trailing zeros in a decimal number are significant.

Operations with Significant Figures

Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.

Multiplication/Division: The result should have the same number of significant figures as the measurement with the fewest significant figures.

Dimensional Analysis

Dimensional analysis is a method to check the consistency of equations and convert units from one system to another.

Dimensional Formula

The dimensional formula expresses a physical quantity in terms of the fundamental quantities. For example, the dimensional formula for force is: $$ [F] = [M^1 L^1 T^{-2}] $$

Applications of Dimensional Analysis

1. Checking Dimensional Consistency: Ensure both sides of an equation have the same dimensions.
2. Deriving Relations: Use known dimensions to derive formulas.
3. Converting Units: Convert units from one system to another using conversion factors.

Conclusion

Understanding units and measurement is crucial for solving physics problems accurately. Mastery of these concepts will provide a strong foundation for tackling more complex topics in physics. Always remember to use the correct units, pay attention to significant figures, and apply dimensional analysis to check the validity of your equations.