Physical World, Units, and Measurements Revision Notes BTER Polytechnic Classes

Physical World, Units, and Measurements

This topic forms the foundation of classical and modern physics. Understanding physical quantities, units of measurement, dimensional analysis, and errors in measurements is crucial for both theoretical understanding and practical application in engineering and science.

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1.1 Physical Quantities

1.1.1 Fundamental and Derived Quantities

1. Fundamental Quantities: These are basic quantities that are not defined in terms of other quantities. They form the foundation of measurement and have their own units. In the International System (SI), there are 7 fundamental quantities:

   • Length ($L$) – Unit: meter (m)
   • Mass ($M$) – Unit: kilogram (kg)
   • Time ($T$) – Unit: second (s)
   • Electric current ($I$) – Unit: ampere (A)
   • Temperature ($\theta$) – Unit: kelvin (K)
   • Amount of substance ($N$) – Unit: mole (mol)
   • Luminous intensity ($J$) – Unit: candela (cd)

2. Derived Quantities: These are quantities derived from the fundamental quantities through multiplication or division. Examples include:

   • Velocity = $\frac{\text{Length}}{\text{Time}}$, Unit: m/s
   • Force = $\text{Mass} \times \text{Acceleration}$, Unit: newton (N) = $\text{kg} \cdot \text{m/s}^2$
   • Energy = $\text{Force} \times \text{Distance}$, Unit: joule (J) = $\text{kg} \cdot \text{m}^2/\text{s}^2$

1.1.2 Dimensions and Dimensional Formulae of Physical Quantities

1. Dimensions: Dimensions are the powers to which the fundamental quantities must be raised to express a physical quantity. Every physical quantity has a dimensional formula.

2. Dimensional Formula: The dimensional formula represents the dependence of a physical quantity on the fundamental quantities.

   • Example: The dimensional formula for force (F) is $[M^1 L^1 T^{-2}]$.
     • Explanation: Force depends on mass, length, and time, so its dimensional formula is $M^1 L^1 T^{-2}$.

3. Dimensional Formula for Common Physical Quantities:

   • Velocity: $[L T^{-1}]$
   • Acceleration: $[L T^{-2}]$
   • Work (Energy): $[M L^2 T^{-2}]$
   • Pressure: $[M L^{-1} T^{-2}]$
   • Density: $[M L^{-3}]$

1.1.3 Principle of Homogeneity of Dimensions

The principle of homogeneity states that the dimensions of each term in a physical equation must be the same. In other words, the dimensions on both sides of the equation must be balanced.

• Example: In Newton's second law, $F = ma$, the dimensional formula of force ($F$) should be equal to the dimensional formula of mass ($m$) times the dimensional formula of acceleration ($a$).
  • $[F] = [M L T^{-2}]$ and $[ma] = [M] \times [L T^{-2}] = [M L T^{-2}]$, which confirms the homogeneity.

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1.2 Measurements

1.2.1 Measuring Instruments and Least Count

1. Measuring Instruments: These are tools used to measure physical quantities. Common measuring instruments include:

   • Ruler/Measuring tape for length
   • Vernier caliper for precise measurements of length, inner/outer diameters, and depth
   • Micrometer screw gauge for measuring small dimensions with high precision
   • Stopwatch for measuring time
   • Thermometers for temperature

2. Least Count: The least count of an instrument is the smallest quantity that can be measured with it. It is the smallest division on the scale of the instrument.

   • Example: In a Vernier caliper, the least count is the difference between one main scale reading and one Vernier scale reading, typically 0.01 cm.

1.2.2 Types of Measurement (Direct and Indirect)

1. Direct Measurement: This is the measurement of a physical quantity using a single instrument. Examples include measuring length with a ruler or time with a stopwatch.

2. Indirect Measurement: This involves calculating the value of a physical quantity by using a formula, based on other quantities that are measured directly. For example:

   • Speed can be calculated as $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$, where distance and time are measured directly.

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1.3 Errors in Measurements

Errors are inevitable in all measurements. These errors arise from the limitations of measuring instruments, environmental conditions, and human factors.

1.3.1 Absolute Error

The absolute error is the difference between the measured value and the true value of a quantity. It gives the magnitude of the error without considering its direction.

• Formula:

  $$\text{Absolute Error} = |\text{Measured Value} - \text{True Value}|$$

• Example: If the measured value of length is 5.2 cm and the true value is 5.0 cm, then the absolute error is $|5.2 - 5.0| = 0.2$ cm.

1.3.2 Relative Error

The relative error is the ratio of the absolute error to the true value of the quantity. It indicates the accuracy of the measurement relative to the true value.

• Formula:

  $$\text{Relative Error} = \frac{\text{Absolute Error}}{\text{True Value}}$$

• Example: If the absolute error in measuring length is 0.2 cm and the true value is 5.0 cm, then the relative error is:

  $$\text{Relative Error} = \frac{0.2}{5.0} = 0.04$$

1.3.3 Significant Figures

Significant figures are the digits in a measurement that are known with certainty, plus one digit that is estimated. They represent the precision of the measurement.

1. Rules for Significant Figures:

   • Non-zero digits are always significant (e.g., 123 has 3 significant figures).
   • Zeros between non-zero digits are significant (e.g., 1003 has 4 significant figures).
   • Leading zeros (zeros before non-zero digits) are not significant (e.g., 0.0045 has 2 significant figures).
   • Trailing zeros in a decimal number are significant (e.g., 45.00 has 4 significant figures).
   • In whole numbers without a decimal point, trailing zeros are ambiguous and may or may not be significant (e.g., 1500 can have 2 or 4 significant figures depending on context).

2. Using Significant Figures in Calculations:

   • When adding or subtracting, the result should be rounded to the least number of decimal places in the original measurements.
   • When multiplying or dividing, the result should have the same number of significant figures as the measurement with the least number of significant figures.

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Summary of Key Concepts

1. Physical Quantities: Fundamental quantities (e.g., length, mass) and derived quantities (e.g., velocity, force).
2. Dimensional Formulae: Dimensions are expressed using fundamental quantities (e.g., [M^1 L^1 T^-2] for force).
3. Homogeneity of Dimensions: Ensures consistency in physical equations.
4. Measurement Types: Direct (e.g., using a ruler) and indirect (e.g., calculating speed).
5. Errors: Absolute error (direct difference between measured and true value), relative error (compared to true value), and significant figures (indicating precision).

Understanding these concepts is essential for accurate measurements and error analysis in physics, engineering, and other scientific fields.