Units and Measurements

A. Measurement

  • Measurement is the process of assigning a numerical value to a physical quantity based on a defined standard.
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B. Physical Quantities

Properties & Key Points:

  • Physical quantities are those that can be measured and expressed numerically. They are divided into two categories:
  • Fundamental Quantities: Mass, length, time, electric current, temperature, amount of substance, luminous intensity.
  • Derived Quantities: Speed, force, energy, pressure, etc., which are derived from fundamental quantities.
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C. SI Units

\[ \text{SI Units:} \]
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)
  • The International System of Units (SI) is the standard system for measurement used worldwide.
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D. Prefixes in SI

\[ \text{Prefix} = \text{Base Unit} \times 10^n \]
Kilo (k): \( 10^3 \)
Mega (M): \( 10^6 \)
Giga (G): \( 10^9 \)
Milli (m): \( 10^{-3} \)
Micro (μ): \( 10^{-6} \)
Nano (n): \( 10^{-9} \)
  • Prefixes are used to express large or small quantities by multiplying the base unit by a power of 10.
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E. Measurement of Length

\[ L = \text{measured length} \]
  • The length is measured using instruments like a ruler, vernier caliper, micrometer screw gauge, or optical methods.
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F. Measurement of Mass

\[ m = \text{measured mass} \]
  • Mass is measured using balances (e.g., a beam balance, electronic balance) or a spring scale.
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G. Measurement of Time

\[ t = \text{measured time} \]
  • Time is measured using clocks, stopwatches, or atomic clocks.
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H. Dimensional Analysis

\[ [M^a L^b T^c] \]

Properties & Key Points:

  • Dimensional analysis is a method used to check the consistency of equations and to derive relations between physical quantities.
  • \( M \): Mass dimension.
  • \( L \): Length dimension.
  • \( T \): Time dimension.
  • \( a, b, c \): Exponents representing the power of each dimension.
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I. Significant Figures

Properties & Key Points:

  • Significant figures are the digits in a number that carry meaningful information about its precision.
  • Rules for significant figures:
  • Non-zero digits are always significant.
  • Zeros between non-zero digits are significant.
  • Leading zeros are not significant.
  • Trailing zeros in a decimal number are significant.
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J. Accuracy and Precision

\[ \text{Accuracy} = \left| \text{Measured value} - \text{True value} \right| \]
\[ \text{Precision} = \frac{1}{N} \sum_{i=1}^{N} \left( \text{Measured value}_i - \bar{x} \right)^2 \]

Properties & Key Points:

  • Accuracy refers to how close a measured value is to the true value.
  • Precision refers to how close multiple measurements are to each other.
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K. Error in Measurement

Properties & Key Points:

  • Systematic Error: Consistent and predictable error due to faulty instruments or methods.
  • Random Error: Variations in measurement that occur due to uncontrollable factors.
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L. Conversion of Units

\[ \text{Value in new unit} = \text{Value in old unit} \times \text{Conversion factor} \]
  • Units are converted using conversion factors, which are ratios between equivalent units.
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M. Scalar and Vector Quantities

Properties & Key Points:

  • Scalar Quantities: Quantities that have only magnitude (e.g., mass, time, temperature).
  • Vector Quantities: Quantities that have both magnitude and direction (e.g., displacement, velocity, force).
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N. Physical Constants

Properties & Key Points:

  • Gravitational Constant: \( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)
  • Speed of Light: \( c = 3 \times 10^8 \, \text{m/s} \)
  • Planck’s Constant: \( h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s} \)
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