Honors Physics: Unit 1

1. Scientific Notation

Scientific notation is a way to express very large or very small numbers compactly. It follows the format M \times 10^n, where M (the significand) is a number greater than or equal to 1 and less than 10 (1 \le |M| < 10), and n (the exponent) is an integer.

1.1 Converting to Scientific Notation

  • Move the decimal point until there is only one non-zero digit to its left.

  • The number of places the decimal point was moved determines the exponent n.

    • If moved to the left, n is positive.

    • If moved to the right, n is negative.

    • Example: 123,000 = 1.23 \times 10^5 (decimal moved 5 places left).

    • Example: 0.000045 = 4.5 \times 10^{-5} (decimal moved 5 places right).

1.2 Converting from Scientific Notation

  • If n is positive, move the decimal point n places to the right.

  • If n is negative, move the decimal point n places to the left.

    • Example: 6.02 \times 10^{23} = 602,000,000,000,000,000,000,000

    • Example: 1.6 \times 10^{-19} = 0.00000000000000000016

2. Metric Prefixes

Metric prefixes are used to denote multiples or submultiples of a base unit by powers of 10.

Prefix

Symbol

Multiplier (10^n)

Giga

G

10^9

Mega

M

10^6

Kilo

k

10^3

Hecto

h

10^2

Deka

da

10^1

(Base)

10^0 (1)

Deci

d

10^{-1}

Centi

c

10^{-2}

Milli

m

10^{-3}

Micro

\mu

10^{-6}

Nano

n

10^{-9}

Pico

p

10^{-12}

3. Factor Label Method (Dimensional Analysis)

The factor label method is a systematic approach to convert units using conversion factors. A conversion factor is a ratio equal to one that expresses the same quantity in different units (e.g., 1 \text{ m}/100 \text{ cm}).

  • Steps:

    1. Start with the given quantity and its units.

    2. Multiply by conversion factors to cancel out unwanted units and introduce desired units.

    3. Ensure units cancel diagonally (one in the numerator, one in the denominator).

    4. Perform the calculation.

  • Example: Convert 2.5 meters to centimeters.
    2.5 \text{ m} \times \frac{100 \text{ cm}}{1 \text{ m}} = 250 \text{ cm}

4. Significant Figures

Significant figures (sig figs) indicate the precision of a measurement. They include all known digits plus one estimated digit.

4.1 Rules for Identifying Significant Figures

  1. Non-zero digits: All are significant. (e.g., 234.5 has 4 sig figs)

  2. Zeros between non-zero digits: Always significant. (e.g., 2007 has 4 sig figs)

  3. Leading zeros: Not significant (placeholders). (e.g., 0.0025 has 2 sig figs)

  4. Trailing zeros with a decimal point: Always significant. (e.g., 25.00 has 4 sig figs, 250. has 3 sig figs)

  5. Trailing zeros without a decimal point: Ambiguous; assume not significant unless otherwise indicated. Scientific notation removes ambiguity. (e.g., 2500 likely has 2 sig figs, but 2.50 \times 10^3 has 3 sig figs).

  6. Exact numbers: Have an infinite number of significant figures (e.g., counted items, definitions like 1 m = 100 cm).

5. Operations with Significant Figures
5.1 Addition and Subtraction

  • The result must have the same number of decimal places as the measurement with the fewest decimal places.

  • Example: 2.345 \text{ (3 decimal places)} + 1.2 \text{ (1 decimal place)} = 3.545 \Rightarrow \text{round to 1 decimal place} \Rightarrow 3.5

5.2 Multiplication and Division

  • The result must have the same number of significant figures as the measurement with the fewest significant figures.

  • Example: 2.5 \text{ (2 sig figs)} \times 3.45 \text{ (3 sig figs)} = 8.625 \Rightarrow \text{round to 2 sig figs} \Rightarrow 8.6