Foundations of General and Thermal Physics: Measurements, Errors, and Vectors

Physical Quantities and the International System of Units (SI)

A physical quantity is defined as any quantity that can be measured. Every physical quantity consists of two essential components: a numerical value and a standard or unit.

Units are particular physical quantities defined and adopted by convention. They serve as the standard with which other quantities of the same kind are compared to express their value. The International System of Units (SI) is founded on seven SI base units for seven base quantities, which are assumed to be mutually independent.

SI Base Quantities and Units

A base quantity is a quantity that cannot be expressed in terms of other quantities. There are exactly seven base quantities:

• Mass: kilogram ($\text{kg}$)

• Length: meter ($\text{m}$)

• Time: second ($\text{s}$)

• Temperature: Kelvin ($\text{K}$)

• Electric current: Ampere ($\text{A}$)

• Light intensity: candela ($\text{cd}$)

• Amount of substance: mole ($\text{mol}$)

Derived Quantities and Units

Derived quantities are combinations of two or more base quantities achieved through multiplication, division, or both. These are obtained using equations involving the seven SI base units.

Examples of Derived Quantities and their Relations:

• Area: $\text{Length} \times \text{width}$

• Volume: $\text{Length} \times \text{width} \times \text{Height}$

• Density: $\frac{\text{Mass}}{\text{Volume}}$

• Speed: $\frac{\text{Distance}}{\text{Time}}$

• Acceleration: $\frac{\text{Distance}}{\text{Time} \times \text{Time}}$

• Force: $\text{Mass} \times \text{Acceleration}$

• Moment of force: $\text{Force} \times \text{Perpendicular Distance}$

• Pressure / Stress: $\frac{\text{Force}}{\text{Area}}$

SI Derived Units with Special Names:

For convenience, over 20 SI derived units have special names. Key examples include:

• Frequency: Name: hertz ($\text{Hz}$); Base unit expression: $\text{s}^{-1}$.

• Force: Name: newton ($\text{N}$); Base unit expression: $\text{m} \times \text{kg} \times \text{s}^{-2}$.

• Pressure: Name: pascal ($\text{Pa}$); Other SI units: $\text{N/m}^2$; Base unit expression: $\text{m}^{-1} \times \text{kg} \times \text{s}^{-2}$.

• Energy / Work: Name: joule ($\text{J}$); Other SI units: $\text{N} \times \text{m}$; Base unit expression: $\text{m}^2 \times \text{kg} \times \text{s}^{-2}$.

• Power: Name: watt ($\text{W}$); Other SI units: $\text{J/s}$; Base unit expression: $\text{m}^2 \times \text{kg} \times \text{s}^{-3}$.

Measuring Instruments and Precision

Precision refers to the smallest unit an instrument can measure. Selecting the correct instrument is vital for accurate data collection.

• Metre rule: Precision of $0.1\,\text{cm}$.

• Vernier calipers: Precision of $0.01\,\text{cm}$. They are used for more accurate measuring than is possible with a standard measuring rule.

• Micrometer screw gauge: Precision of $0.01\,\text{mm}$.

Using the Micrometer Screw Gauge

The micrometer screw gauge is used for high-precision length measurements.

• The Ratchet: This component is used to ensure the pressure exerted on the object is consistent and within tolerance. Users should stop turning once the clicks are heard to avoid over-tightening the thimble.

• Cleaning: The ends of the anvil and spindle must be cleaned before measurement, as any dirt on the surfaces can affect the reading.

• Zero Error Check: The micrometer should be closed with nothing between the anvil and spindle to check for zero error. Ideally, the reading should be zero.

Correcting for Zero Error

If a zero error exists, the final measurement must be adjusted using the following formula:

$\text{Corrected reading} = \text{observed reading} - \text{zero error}$

Calculation Examples:

• Case 1 (No Zero Error): If observed reading is $3.50 + 0.11$, the result is $3.61\,\text{mm}$.

• Case 2 (Positive Zero Error): Observed reading = $2.37\,\text{mm}$; Zero error = $+0.02\,\text{mm}$. Corrected reading = $2.37 - (+0.02) = 2.35\,\text{mm}$.

• Case 3 (Negative Zero Error): Observed reading = $2.87\,\text{mm}$; Zero error = $-0.03\,\text{mm}$. Corrected reading = $2.87 - (-0.03) = 2.90\,\text{mm}$.

Error Analysis, Accuracy, and Precision

An error is defined as the deviation of a measurement from the true value. Errors arise from instrument sensitivity limitations or imperfections in experimental design and techniques.

Types of Errors

1. Systematic Errors:

   • Caused by the experimenter or the instruments.

   • Result in all readings being consistently above or below the accepted value.

   • Cannot be eliminated by repeating measurements or averaging.

   • Can only be reduced by improving experimental techniques.

   • Examples: Zero errors on a micrometer, wrongly calibrated scales, and the reaction time of an experimenter.

2. Random Errors:

   • Caused by the observer through estimation.

   • Result in readings being scattered around the accepted value.

   • May be reduced by repeated readings and averaging, or by plotting a graph and drawing a best-fit line.

   • Examples: Parallax error and timing oscillations (where timing may not start/stop at the exact same point).

Accuracy vs. Precision

• Precision: The degree to which repeated measurements under unchanged conditions show the same results (consistency).

• Accuracy: The degree of closeness of measurements to the actual (true) value of the quantity.

• Measurement errors influence both accuracy and precision. High accuracy combined with high precision represents the ideal measurement scenario.

Uncertainty Analysis and Propagation

A measurement recorded as $y$ with a maximum uncertainty $\Delta y$ is written as $(y \pm \Delta y)$.

• Consistency Rule: The measurement must have the same number of decimal places as the quoted maximum uncertainty. For example, a length is written as $(10.0 \pm 0.1)\,\text{cm}$, but never $(10 \pm 0.1)\,\text{cm}$.

• Significance: An uncertainty of $\pm 1\,\text{mm}$ is negligible for a $100\,\text{cm}$ length but significant for a $1\,\text{cm}$ length.

Uncertainty Ratios

• Maximum Fractional Uncertainty: $\frac{\Delta y}{y}$

• Percentage Uncertainty: $\frac{\Delta y}{y} \times 100\%$

Rules for Propagation of Error

If $z$ is the result of operations on quantities $x$ and $y$:

• Addition ($z = x + y$): $\Delta z = \Delta x + \Delta y$. Final form: $z = [(x + y) \pm (\Delta x + \Delta y)]$

• Subtraction ($z = x - y$): $\Delta z = \Delta x + \Delta y$. Final form: $z = [(x - y) \pm (\Delta x + \Delta y)]$

• Multiplication ($z = xy$): $\frac{\Delta z}{z} = \frac{\Delta x}{x} + \frac{\Delta y}{y}$

• Division ($z = \frac{x}{y}$): $\frac{\Delta z}{z} = \frac{\Delta x}{x} + \frac{\Delta y}{y}$

• Product of Powers ($z = x^m y^n$): $\frac{\Delta z}{z} = |m|\frac{\Delta x}{x} + |n|\frac{\Delta y}{y}$

Rounding and Scientific Notation Rules

• Significant Figures in Uncertainty: The uncertainty should be rounded to one or two significant figures. If the leading figure is a $1$, use two significant figures; otherwise, use one.

• Matching Decimal Places: The final answer must be rounded to match the decimal place of the uncertainty.

• Scientific Notation: When using scientific notation, the uncertainty must use the same power of ten as the value. For example, if $z = 1.43 \times 10^6\,\text{s}$ and $\Delta z = 2 \times 10^4\,\text{s}$, the correct form is $(1.43 \pm 0.02) \times 10^6\,\text{s}$.

Worked Examples in Error Propagation

Example 1: Addition and Subtraction Given $w = (4.52 \pm 0.02)\,\text{cm}$, $x = (4.0 \pm 0.2)\,\text{cm}$, and $y = (3.0 \pm 0.6)\,\text{cm}$. Find $z = x + y - w$.

1. Calculate value: $z = 4.0 + 3.0 - 4.52 = 2.48\,\text{cm}$.

2. Calculate uncertainty: $\Delta z = \Delta x + \Delta y + \Delta w = 0.2 + 0.6 + 0.02 = 0.82\,\text{cm}$.

3. Apply rounding: Round $\Delta z$ to one significant figure ($0.8$). Match $z$ to that decimal place ($2.5$).

4. Result: $(2.5 \pm 0.8)\,\text{cm}$.

Example 2: Complex Propagation Given $w = (4.52 \pm 0.02)\,\text{cm}$, $A = (2.0 \pm 0.2)\,\text{cm}^2$, and $y = (3.0 \pm 0.6)\,\text{cm}$. Find $z = \frac{wy}{A^2}$.

1. Calculate value: $z = \frac{4.52 \times 3.0}{2.0^2} = 28.765\,\text{cm}^{-1}$.

2. Calculate fractional uncertainty: $\frac{\Delta z}{z} = \frac{0.02}{4.52} + \frac{0.6}{3.0} + 2\frac{0.2}{2.0} = 0.454$.

3. Calculate absolute uncertainty: $\Delta z = 0.454 \times 28.765 = 13.06\,\text{cm}^{-1}$.

4. Apply rounding: Since uncertainty starts with $1$, use two significant figures ($13$). Round $z$ to match ($29$).

5. Result: $(29 \pm 13)\,\text{cm}^{-1}$.

Example 3: Sum of Powers Given $w = (4.52 \pm 0.02)\,\text{cm}$, $x = (2.0 \pm 0.2)\,\text{cm}$, and $y = (3.0 \pm 0.6)\,\text{cm}$. Find $z = wx + y^2$.

1. Let $a = wx$ and $b = y^2$, so $z = a + b$.

2. Calculate $a = 4.52 \times 2.0 = 9.04$. $\Delta a = a(\frac{\Delta w}{w} + \frac{\Delta x}{x}) = 9.04(\frac{0.02}{4.52} + \frac{0.2}{2.0}) = 0.944$.

3. Calculate $b = 3.0^2 = 9.0$. $\Delta b = 2b(\frac{\Delta y}{y}) = 2(9.0)(\frac{0.6}{3.0}) = 3.6$.

4. Calculate total: $z = 9.04 + 9 = 18.04$. $\Delta z = 0.944 + 3.6 = 4.544$.

5. Apply rounding: Round $\Delta z$ to one significant figure ($5$). Round $z$ to match ($18$).

6. Result: $(18 \pm 5)\,\text{cm}$.

Scalar and Vector Quantities

• Scalars: Quantities that possess magnitude only.

• Vectors: Quantities that possess both magnitude and direction.

Classification of Quantities:

• Scalars: distance, speed, mass, time, pressure, energy, volume, density, temperature.

• Vectors: displacement, velocity, weight, acceleration, force, momentum.

Vector Representation and Multiplication

A vector is represented by an arrow. The length of the arrow is proportional to the vector's magnitude, and the orientation of the arrow indicates its direction.

Multiplying a vector by a scalar changes the magnitude of the vector. If the scalar is negative, the direction of the vector is reversed.

Vector Addition and Subtraction

Graphical Method (Head-to-Tail)

1. Alignment: All given vectors must be joined head-to-tail.

2. Order: The order in which vectors are added does not affect the final result.

3. Resultant Vector: The magnitude and direction of the resultant vector are measured from the tail of the first vector to the head of the last vector.

4. Vector Subtraction: To subtract vector $B$ from vector $A$, add the negative of vector $B$. Mathematically: $\mathbf{A} - \mathbf{B} = \mathbf{A} + (-\mathbf{B})$.

Analytical Method (Resolving Vectors)

It is often convenient to split a single vector into two perpendicular components (usually horizontal, $x$, and vertical, $y$).

For a force $F$ inclined at an angle $\theta$ to the horizontal:

• Vertical Component: $F_y = F \sin(\theta)$

• Horizontal Component: $F_x = F \cos(\theta)$

Example Application: Two forces $F_1 = 15.0\,\text{N}$ and $F_2 = 10.0\,\text{N}$ are both inclined at $50^{\circ}$ to the x-axis at point O.

1. Calculate individual components:

   • $F_{1x} = 15.0 \cos(50^{\circ})$

   • $F_{1y} = 15.0 \sin(50^{\circ})$

   • $F_{2x} = 10.0 \cos(50^{\circ})$

   • $F_{2y} = 10.0 \sin(50^{\circ})$

2. The vector sum of the x-components is the sum of $F_{1x}$ and $F_{2x}$.

3. The vector sum of the y-components is the sum of $F_{1y}$ and $F_{2y}$.