Overview
This lecture covers essential ultrasound physics formulas, how to manipulate and interpret them, relationships between variables, metric units, unit conversions, scientific notation, exponents, reciprocals, and axis concepts.
Manipulating Formulas
- Formulas express a variable (answer) in terms of others (e.g., a = b/c).
- To solve for a variable, perform the same operation on both sides to isolate it.
- Example: To solve a = b/c for b, multiply both sides by c: b = a × c.
- To solve for c: c = b/a.
Describing Relationships in Formulas
- Five relationship types: unrelated, related, directly related/proportional, inversely related/proportional, and reciprocal.
- Direct relationship: when one variable increases, the other increases.
- Inverse relationship: when one variable increases, the other decreases.
- Reciprocal: two variables multiplied together equal 1 (e.g., frequency × period = 1).
Understanding Variables and Formula Components
- Variables in formulas are "factors"; their products or quotients are answers.
- Rule 1: Factors are directly related to their product (if a or b increases, so does c in a × b = c).
- Rule 2: In a quotient, the numerator is directly related; the denominator is inversely related to the result.
Units and Dimensional Analysis
- Always specify units: e.g., time (seconds), length (meters), area (m²), volume (m³), velocity (m/s), frequency (Hz).
- Common measurement systems: metric (preferred, based on powers of ten) and empirical (U.S. customary).
Metric System & Unit Conversion
- Metric prefixes: giga (G), mega (M), kilo (k), hecto (h), deca (da), base, deci (d), centi (c), milli (m), micro (μ), nano (n).
- Conversion: move decimal right when converting to smaller units, left to larger units, counting 'stairs' between prefixes.
- Example: 0.1509 meters = 150,900,000 nanometers (move decimal 9 places right).
Scientific Notation & Exponents
- Scientific notation: express numbers as a × 10ⁿ; n positive for numbers >10, negative for <1.
- To convert to scientific notation: shift decimal to make a number between 1 and 10, count the moves for exponent.
- Positive exponents increase number size; negative exponents decrease it (small numbers).
- Any number to the 0th power is 1.
Working with Exponents and Reciprocals
- Raising numbers to exponents means multiplying base number repeatedly.
- Negative exponents: a⁻ⁿ = 1/(aⁿ).
- Reciprocals: two numbers whose product is 1 (e.g., 4 × ¼ = 1).
Fractions, Decimals, and Graph Axes
- Convert fractions to decimals via long division.
- Recognize repeating decimals in conversions if the remainder repeats.
- X, Y, Z axes represent dimensions; common in ultrasound data display (time, velocity, depth).
Key Terms & Definitions
- Numerator — The top part of a fraction/division bar; direct relation to quotient.
- Denominator — The bottom of a fraction/division bar; inverse relation to quotient.
- Product — Result of multiplying two or more factors.
- Quotient — Result of dividing one number by another.
- Reciprocal — A number which, when multiplied by a given number, equals 1.
Action Items / Next Steps
- Create a formula sheet with definitions, variables, and units.
- Practice manipulating formulas and understanding relationships.
- Memorize metric system prefixes, values, symbols, and exponents.
- Complete workbook practice problems and metric conversions.
- Review and answer nerd check questions to prepare for the quiz.