Units, Scientific Notation, and Significant Figures

Scientific Notation

Scientific notation is a way to express very large or very small numbers using powers of ten. This is especially useful in physics for clarity and precision.

• Definition: A number is written as a × 10n, where 1 ≤ a < 10 and n is an integer.

• Example: 0.0012 can be written as .

Significant Figures

Significant figures (sig figs) indicate the precision of a measured or calculated quantity.

• Definition: The digits in a number that are known with certainty plus one estimated digit.

• Rules:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros after a decimal point are significant.

• Example: The number 0.003010 has 4 significant figures (3, 0, 1, 0).

Rounding and Calculations with Significant Figures

• When multiplying or dividing, the result should have as many significant figures as the value with the fewest significant figures.

• When adding or subtracting, the result should have the same number of decimal places as the value with the fewest decimal places.

• Example: Multiplying 1.125 m × 0.606 m = 0.68175 m2 should be rounded to 0.682 m2 (3 significant figures).

Unit Conversions

Unit conversions are essential for solving physics problems and ensuring consistency in calculations.

• Example: 1.71 × 102 kg = 1.71 × 106 g = 1.71 × 108 mg

• To convert between units, multiply by the appropriate conversion factor.

Vectors and Vector Operations

Vector Magnitude and Direction

Vectors are quantities that have both magnitude and direction, such as displacement, velocity, and force.

• Magnitude: The length or size of the vector.

• Direction: The orientation of the vector in space, often given as an angle or in component form.

• Example: If and , then .

• Magnitude of :

• Direction:

Kinematics and Motion

Basic Kinematic Quantities

• Displacement: The change in position of an object.

• Velocity: The rate of change of displacement with respect to time.

• Acceleration: The rate of change of velocity with respect to time.

Free Fall and Acceleration Due to Gravity

• When an object is in free fall, the only force acting on it is gravity.

• Acceleration due to gravity: (downward)

• Equation: , where

• If upward is positive, then for free fall.

Interpreting Position-Time Graphs

• The slope of a position vs. time graph gives the velocity of the object.

• The acceleration at a point is given by the slope of the velocity vs. time graph at that point.

Unit Conversion Examples

• Speed Conversion: To convert 4.50 km/h to ft/min:

  • 1 km = 1000 m, 1 m = 3.28 ft, 1 h = 60 min

• Mass Loss Conversion: If a person loses 1.6 kg in a week, to find micrograms lost:

  • 1 kg = mg, 1 mg = μg

Sample Table: Significant Figures in Calculations

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Where original questions referenced specific numbers or calculations, full academic context and step-by-step methods have been provided.