Measurement and Motion Lecture Review

Fundamentals of Measurement

Measurement is defined as the process of determining or estimating the magnitude of a quantity. This process inherently involves a comparison with a standard. There are two primary methods used to obtain measurements:

Direct Method: This involves taking a measurement through the direct use of a measuring device. This method produces Fundamental or Basic Physical Quantities.
Indirect Method: This process involves taking measurements using a formula or a combination of direct methods. This approach results in Derived Physical Quantities.

Fundamental SI Quantities and Units

The International System of Units (SI) identifies specific basic quantities and their corresponding units:

Length / Distance: Meter ( $m$ )
Mass: Kilogram ( $kg$ )
Time: Second ( $s$ )
Electric Current: Ampere ( $A$ )
Thermodynamic Temperature: Kelvin ( $K$ )
Amount of Substance: Mole ( $mol$ )
Luminous Intensity: Candela ( $cd$ )

Basic Measuring Tools and Their Functions

Thermometer: Used to measure the average kinetic energy of a substance, which is defined as temperature.
Tape Measure / Meter Tape: Utilized for measuring the length of a material.
Ammeter: Used specifically to measure electric current.
Stop Watch: Used to measure the duration or amount of time.
Digital Balance: Designed to measure the amount of a substance.
Weighing Scale: Used to measure the mass of matter.
Light Meter: Used to measure the intensity of a light source.

Historical and Scientific Context of Units

Kelvin: This unit was named in honor of Lord Kelvin, a Scottish mathematician and physicist who is credited with developing the Kelvin temperature scale.
Ampere: This unit for electric current was named after the physicist André-Marie Ampère, who performed foundational studies on the behavior of electrical charges.

Conversion Rates for English and Metric Systems

Measurements can be converted between systems using techniques such as the Factor Label Method or Dimensional Analysis Method.

Length Conversion

$1\text{ inch} = 2.54\text{ centimeters}$
$0.39\text{ inch} = 1\text{ centimeter}$
$1\text{ foot} = 30.48\text{ centimeters}$
$3.28\text{ feet} = 1\text{ meter}$
$1\text{ yard} = 0.91\text{ meters}$
$1.09\text{ yards} = 1\text{ meter}$
$1\text{ mile} = 1.61\text{ kilometers}$
$0.62\text{ miles} = 1\text{ kilometer}$

Mass Conversion

$1\text{ ounce} = 28.35\text{ grams}$
$0.035\text{ ounces} = 1\text{ gram}$
$1\text{ pound} = 0.45\text{ kilograms}$
$2.21\text{ pounds} = 1\text{ kilogram}$
$1\text{ ton} = 0.91\text{ metric tons}$
$1.10\text{ tons} = 1000\text{ kilograms}$

Volume Conversion

$1\text{ teaspoon} = 4.92\text{ milliliters}$
$0.20\text{ teaspoons} = 1\text{ milliliter}$
$1\text{ tablespoon} = 14.79\text{ milliliters}$
$0.68\text{ tablespoons} = 10\text{ milliliters}$
$1\text{ fluid ounce} = 29.57\text{ milliliters}$
$3.38\text{ fluid ounces} = 100\text{ milliliters}$
$1\text{ pint} = 0.47\text{ liters}$
$2.11\text{ pints} = 1\text{ liter}$
$1\text{ quart} = 0.95\text{ liters}$
$1.06\text{ quarts} = 1\text{ liter}$
$1\text{ gallon} = 3.79\text{ liters}$
$0.26\text{ gallons} = 1\text{ liter}$

Analysis of Mass vs. Weight

Mass

Definition: The quantity of matter contained within a body.
Property: It is a scalar quantity.
Consistency: Mass remains constant everywhere in the universe.
Formula: $m = \frac{F}{a}$
SI Unit: Kilogram ( $kg$ )

Weight

Definition: The gravitational force acting upon the mass of a body.
Property: It is a vector quantity.
Consistency: Weight changes depending on the geographical or celestial location.
Formula: $W = m \times g$
SI Unit: Newton ( $N$ )

Historical Standards: The King's Foot

In ancient times, the measurement of a "foot" was inconsistent, recorded as $11\frac{1}{42}\text{ inches}$ at one point. In modern standards, it is defined as $12\text{ inches}$ , roughly the length of an average man's foot. The Greeks developed the "foot" as a fundamental unit of length, legendary based on the actual foot measurement of Hercules. In England, the King's foot became the regional standard. Specifically, during the reign of King Henry 1, the foot was standardized at $12\text{ inches}$ , initiating a priority for the standardization of measurement.

Radio Station Frequencies and SI Prefixes

Radio stations transmit signals at specific frequencies measured in Hertz ( $Hz$ ).

FM (Frequency Modulation) Radio

Units: Megahertz ( $MHz$ ). $1\text{ MHz} = 10^6\text{ Hz}$ .
Radio Love: $90.7\text{ MHz}$ or $90.7 \times 10^6\text{ Hz}$ .
Easy Rock: $96.3\text{ MHz}$ or $96.3 \times 10^6\text{ Hz}$ .

AM (Amplitude Modulation) Radio

Units: Kilohertz ( $kHz$ ). $1\text{ kHz} = 10^3\text{ Hz}$ .
Veritas: $846\text{ kHz}$ or $846 \times 10^3\text{ Hz}$ .
Super Radyo: $594\text{ kHz}$ or $594 \times 10^3\text{ Hz}$ .

Detailed Definitions of SI Base Units

Ampere (A): Represents electric current. It is equivalent to the flow of approximately $6 \text{ quintillion}$ ( $6 \times 10^{18}$ ) electrons passing a single point every second.
Mole (mol): Used for measuring the amount of a substance. One mole of any item (atoms, molecules, etc.) is equal to approximately $600 \text{ sextillion}$ items, or $6 \times 10^{23}$ .
Kilogram (kg): Measures mass (matter in an object). If a hammer has a mass of $1\,kg$ , its weight can be calculated relative to the gravity of Earth or the Moon.
Candela (cd): Measures luminous intensity. One candela is approximately the amount of light emitted by a single candle.
Meter (m): Defines length as the distance light travels through a vacuum in exactly $\frac{1}{299,792,458}$ of a second.
Kelvin (K): Measures temperature. Water freezes at $273\,K$ and boils at $373\,K$ . Absolute zero ( $0\,K$ ) is the coldest possible temperature where atoms are nearly motionless.
Uncertainty: Although not a base unit, it identifies the potential error in a measurement and how much a value may deviate from the true value.

Significant Figures (Sig Figs)

Significant figures represent the digits in a number that carry meaning regarding the resolution of a measurement.

Rules for Identifying Significant Figures

Non-zero digits: Always significant (e.g., $123$ has 3 sig figs).
Captive zeros: Zeros between non-zero digits are always significant (e.g., $105$ has 3 sig figs).
Leading zeros: Zeros at the start are placeholders and never significant (e.g., $0.003$ has 1 sig fig).
Trailing zeros: Significant only if a decimal point is explicitly present (e.g., $500$ has 1 sig fig; $500.$ has 3 sig figs).
Exact numbers: Defined quantities (like $12\text{ inches} = 1\text{ foot}$ ) or counted objects have an infinite number of significant figures.

Calculation Rules

Multiplication/Division: Round the final answer to the same number of significant figures as the number in the calculation with the fewest significant figures.
- Example: $4.6\, (2\text{ sig figs}) \times 3.52\, (3\text{ sig figs}) = 16.192 \rightarrow 16$
- Example: $5.64\, (3\text{ sig figs}) \times 12.458\, (5\text{ sig figs}) = 70.26312 \rightarrow 70.3$
- Example: $96.752\, (5\text{ sig figs}) \div 3.541\, (4\text{ sig figs}) = 27.32335 \rightarrow 27.32$
Addition/Subtraction: Round the final answer to the same number of decimal places as the value with the fewest decimal places.
- Example: $2.36\, (2\text{ places}) + 12.1\, (1\text{ place}) = 14.46 \rightarrow 14.5$
- Example: $4.328\, (3\text{ places}) + 13\, (0\text{ places}) + 5.45\, (2\text{ places}) = 22.778 \rightarrow 23$

Scientific Notation

Scientific notation allows for the handling of very large or very small numbers using powers of 10.

Large Numbers (Positive Exponents)

Shift the decimal point to the left. The number of places moved is the exponent.

$45,000 \rightarrow 4.5 \times 10^4$
$3,750 \rightarrow 3.75 \times 10^3$
$580,000 \rightarrow 5.8 \times 10^5$
$72,000,000 \rightarrow 7.2 \times 10^7$
$9,300,000,000 \rightarrow 9.3 \times 10^9$

Small Numbers (Negative Exponents)

Shift the decimal point to the right. The number of places moved is the negative exponent.

$0.0023 \rightarrow 2.3 \times 10^{-3}$
$0.00076 \rightarrow 7.6 \times 10^{-4}$
$0.049 \rightarrow 4.9 \times 10^{-2}$
$0.00000541 \rightarrow 5.41 \times 10^{-6}$
$0.000000000835 \rightarrow 8.35 \times 10^{-10}$

Converting to Standard Form

Positive Exponents: Shift the decimal to the right (e.g., $2.4 \times 10^2 \rightarrow 240$ ).
Negative Exponents: Shift the decimal to the left (e.g., $3.7 \times 10^{-3} \rightarrow 0.0037$ ).

Accuracy, Precision, and Statistical Analysis

Definitions

Accuracy: The closeness of a measurement to the true or accepted value. It can be determined by a single measurement and may be affected by systematic error.
Precision: The reproducibility of measurements (getting the same answer every time). It requires multiple measurements to determine and may be affected by random error.

Statistical Metrics

Average (Mean): Calculated by adding all measurements and dividing by the total count.
Variance: The average of the squared differences from the mean (measures how scattered answers are).
Standard Deviation: The square root of the variance. A smaller standard deviation indicates data is near the average; a large one means data is spread out.
Standard Error: Indicates whether the calculated average is trustworthy.
Percent Difference: Used to compare two separate measurements.
Relative Error: Indicates the size of the error relative to the measurement.
Range: The spread from the lowest to the highest measurement.

Acceptable Error Thresholds

Analytical Chemistry: May require errors below $1\%-2\%$ .
Pendulum Experiments: Reasonably allow up to $5\%$ error.
Simple Field Investigations: May allow $5\%-10\%$ error.

Scalar vs. Vector Quantities

Scalar Quantities

Fully described by magnitude (numerical value) alone.
Examples: Distance ( $25\,m$ ), Time, Speed, Temperature, Pressure, Volume, Current, Energy, Mass, Specific Heat Capacity, Voltage, Charge.

Vector Quantities

Fully described by both magnitude and direction.
Representation: Symbol or letter of the physical quantity with an arrow above it.
Components: Found using trigonometric functions ( $\sin, \cos, \tan$ via SOH & CAH).
Examples: Displacement ( $25\,m\text{ North}$ ), Velocity, Acceleration, Force, Weight, Momentum, Gravitational Field Strength, Friction, Buoyancy Force.

Vector Addition Methods

Head-Tail Method: Draw the first vector, place the tail of the second at the head of the first, and draw the resultant vector from the first tail to the final head.
Parallelogram Method: Place both vectors starting at the same point. Draw parallel lines to form a parallelogram. The resultant vector is the diagonal from the starting point to the opposite corner.

Motion and Mechanics

Motion is the change in the position of an object over time. Mechanics is the branch of physics focused on motion and the forces affecting objects.

Types of Motion

Rectilinear Motion: Movement in a straight path (e.g., a car on a straight road).
Curvilinear Motion: Movement in a curved path (e.g., a ball thrown through the air).
Rotational Motion: Movement around a fixed point or axis (e.g., a spinning wheel).
Vibratory Motion: Repetitive back-and-forth motion around a central position (e.g., a swinging pendulum).
Translatory Motion: All points of the object move in the same direction and at the same speed (e.g., a car moving straight).

Acceleration and Deceleration

Acceleration: Occurs when an object speeds up or changes direction (Vector quantity).
Deceleration: Occurs when an object slows down (also called negative acceleration).
Uniform Motion: Movement at a steady, constant speed in a straight line without change in direction.

Interpreting Motion Graphs

Straight Upward Line: Steady acceleration; speed increases at a constant rate.
Straight Horizontal Line: Steady/constant speed.
Straight Downward Line: Steady deceleration; speed decreases at a constant rate.
Curved Upward Line: Increasing acceleration; speed increases faster and faster.
Curved Downward Line: Decreasing acceleration; speed increases more slowly.
Line on the Time Axis: Stationary; zero speed.
Steepness: The steeper the line, the greater the acceleration or deceleration.

Laboratory Procedures for Precision Tools

Using a Micrometer Caliper (Measuring Ballpen Diameter)

Place the ballpen body between the anvil and spindle.
Lock the position using the thimble.
Use a phone to photograph the main scale and circular scale for a clear view.
Perform three measurements total.
Calculate the mean of the measurements and tabulate findings.
Use the Percent Error Formula to check accuracy.
Determine precision by finding the difference between the highest/lowest measurement and the mean.
Ensure all answers are rounded to 3 significant figures.

Using a Vernier Caliper (Measuring Marker Cap Inner Diameter)

Handle with care due to sharp parts.
Place the inner cap on the upper measuring jaws by moving the slider.
Lock the position using the screw.
Perform the measurement three times total.
Calculate the mean and determine accuracy using the Percent Error Formula.

Formulas and Percent Error Steps

Percent Error Calculation

Find the absolute difference between the experimental (measured) and true (accepted) value: $|\text{Experimental} - \text{True}|$
Divide the difference by the true/accepted value.
Multiply by $100\%$ .

Smaller percent error indicates higher accuracy.

Calculating Average (Mean)

Sum all measurements.
Count the number of measurements.
Divide the total sum by the count of measurements.
Round the final result according to the precision of the initial measurements.