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Units, Significant Digits, and Scientific Notation
Significant Digits
Significant digits (or significant figures) are the digits in a measurement or calculation that are known with certainty, plus one estimated digit. They reflect the precision of the measuring instrument or the calculation process.
Definition: Significant digits are all the digits in a number that contribute to its accuracy, starting with the first nonzero digit.
Rules for Counting Significant Digits:
All nonzero digits are significant.
Zeros between nonzero digits are significant.
Leading zeros are not significant.
Trailing zeros are significant only if there is a decimal point.
Example: In the number 0.34500, there are 5 significant digits. In 7000, the number of significant digits is ambiguous unless written as 7.000 × 103 (4 significant digits) or 7 × 103 (1 significant digit).
Significant Digits in Calculations
When performing calculations, the number of significant digits in the result should reflect the least precise measurement used in the calculation.
Multiplication/Division: The result should have as many significant digits as the measurement with the fewest significant digits.
Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places.
Example: Calculating speed: (rounded to 3 significant digits, the least precise value).
Beware: Subtracting similar numbers can lead to a loss of significant digits. Keep extra digits in intermediate steps and round only the final result.
Scientific Notation
Scientific notation expresses numbers as a product of a coefficient and a power of ten, making it easier to handle very large or small numbers and to indicate significant digits clearly.
Format: , where and is an integer.
All digits in the coefficient are significant.
Example: has 1 significant digit; has 4 significant digits.
Calculation Example:
Calculate:
Write with a common power of 10:
Round to (4 significant digits).
Units and SI System
Physics uses the International System of Units (SI) for consistency and clarity. The base units for mechanics are kilograms (kg), meters (m), and seconds (s). All other units are derived from these.
Common Derived Units: Velocity (m/s), Acceleration (m/s2), Force (kg·m/s2 or Newton).
Other Units: Outside SI, you may encounter miles, yards, hours, pounds, etc.
Unit Conversion
Unit conversion involves multiplying by conversion factors (ratios equal to 1) to change from one unit to another, ensuring unwanted units cancel out.
Example: Convert 25 m/s to mph:
Key Point: Always check that units cancel appropriately, leaving only the desired units.
Units of Velocity
Speeds are often given in km/h rather than m/s. The conversion is:
Example: (also about 56 mph).
SI Prefixes
Prefixes are used to indicate powers of ten for units, making it easier to express very large or small quantities.
Prefix | Symbol | Power of Ten |
|---|---|---|
kilo | k | |
mega | M | |
giga | G | |
tera | T | |
deci | d | |
centi | c | |
milli | m | |
micro | μ | |
nano | n | |
pico | p |
Powers in Unit Conversions
When converting squared or cubed units, remember to apply the power to the conversion factor as well.
Example:
Example:
Units of Volume
Volume is often measured in cubic centimeters (cm3) or liters (L). These units are commonly used in laboratory and fluid measurements.
Conversions:
Common Mistakes with Units
It is incorrect to add or subtract quantities with different units or physical meanings. For example, adding a time, a distance, and a number of people is not meaningful in physics.
Key Point: Only quantities with the same units and physical meaning can be added or subtracted.
