Physics

Chapter 1: Units and Measurement

Imron Rosyadi

1.1 The Scope and Scale of Physics

Learning Objectives

By the end of this section, you will be able to:

• Describe the scope of physics.
• Calculate the order of magnitude of a quantity.
• Compare measurable length, mass, and timescales quantitatively.
• Describe the relationships among models, theories, and laws.

What is Physics?

Physics is dedicated to understanding all natural phenomena.
It explores the physical world from subatomic particles to the entire universe.
Despite its breadth, subfields share a common core.

Note

Definition of Physics:

From Greek “phúsis”, meaning “nature,” physics describes the interactions of energy, matter, space, and time to uncover fundamental mechanisms.

Physics is a fundamental science that underpins many other scientific and engineering disciplines. Emphasize that it’s about understanding the “why” and “how” of natural processes at all scales.

Physics in the Real World

Physics principles are everywhere, from galaxies to everyday tech.

Cosmic Scale:

• Whirlpool Galaxy: billions of stars, gas, dust.
• Forces like gravity apply universally.

Terrestrial Scale:

• Hydroelectric dams use gravity.
• Rocket launches, home construction.

Use the Whirlpool Galaxy example to illustrate the vast scale of physics and how the same fundamental laws apply across different scales. Reinforce the interconnectedness of phenomena.

Physics and Technology

Physics is the foundation for technological advancements.

Historical Impact:

• Stone, Bronze, Iron Ages: knowledge of material physics.
• Early steel production in South India and Sri Lanka.

Modern Innovations:

• MOSFET (1959): basis for all modern electronics (computers, smartphones, etc.).
• Pioneered by Mohamed M. Atalla and Dawon Kahng.

Highlight how physics concepts translate into practical applications. The MOSFET is a great example of a seemingly small innovation with massive global impact.

Modern Technologies Driven by Physics

Cutting-edge tech relies on physics principles.

• Levitating Trains: Magnetic forces and superconductivity.
• Invisibility Cloaks: Bending light (metamaterials).
• Microscopic Robots: Nanophysics and biomechanics.

Tip

Physics concepts are applied daily by engineers, pilots, physicians, physical therapists, electricians, and programmers.

Connect these exciting technologies to physics, demonstrating the relevance and potential of the field. This can help motivate students.

Physics and Other Sciences

Physics provides fundamental laws that connect diverse phenomena.

Interconnectedness of Nature:

• Conservation of energy: ties together food calories, batteries, heat, light.

Cross-Disciplinary Importance:

• Chemistry: Atomic and molecular physics.
• Engineering: Designing new technologies within physical laws.
• Architecture: Structural stability, acoustics, heating, lighting.
• Geology: Radioactive dating, earthquake analysis, heat transfer.
• Biology/Medicine: Cell properties, human body mechanics, diagnostics (X-rays, MRI), therapies (radiotherapy).

Caption: Apple iPhone with GPS function

Emphasize that physics is the foundational science, providing principles used across all STEM fields. Give examples to make this concrete for students.

The Scale of Physics: Order of Magnitude

To grasp the vastness of phenomena, we use “order of magnitude.”

Definition: The power of 10 that most closely approximates a number. It refers to the scale or size of a value.

How to find:

1. Write the number in scientific notation (\(N \times 10^x\)).
2. If \(N < \sqrt{10} \approx 3.16\), the order of magnitude is \(10^x\).
3. If \(N \ge \sqrt{10} \approx 3.16\), the order of magnitude is \(10^{x+1}\).

Examples:

• Order of magnitude of 800: \(8 \times 10^2\). Since \(8 \ge 3.16\), it’s \(10^{2+1} = 10^3\).
• Order of magnitude of 250: \(2.5 \times 10^2\). Since \(2.5 < 3.16\), it’s \(10^2\).

Introduce order of magnitude as a tool for quick estimations and comparing quantities across vast scales. Provide the rule for determining it from scientific notation.

Vast Ranges of Length, Mass, and Time

Physics deals with phenomena across immense scales.

Caption: Orders of magnitude for length, mass, and time.

Use the provided image (Figure 1.4) to illustrate the incredible range. Encourage students to formulate their own comparative questions to engage with the data.

Building Models, Theories, and Laws

Science aims to uncover the laws of nature through observation.

Models:

• Representations of complex systems.
• Simplified to explain certain aspects.
• E.g., Bohr model of an atom (electron orbits nucleus).
• Useful even if not perfectly accurate for all phenomena.

Theories:

• Testable explanations for patterns in nature.
• Supported by scientific evidence, verified multiple times.
• Broader than models, should describe all aspects within its domain.
• E.g., Newton’s theory of gravity, theory of evolution.

Caption: The Bohr model of a single-electron atom

Clearly distinguish between models, theories, and laws. Emphasize that a scientific theory is not a “guess” but a well-substantiated explanation.

Building Models, Theories, and Laws (Cont.)

Laws:

• Concise descriptions of generalized patterns in nature.
• Supported by scientific evidence and repeated experiments.
• Often expressed as a single mathematical equation.
• E.g., Law of conservation of energy, Newton’s second law (\(F = ma\)).

Warning

Laws are intrinsic to the universe; humans discover, not create, them.

If experiments contradict a law/theory, it must be modified or overthrown.

Reinforce the idea that laws are fundamental and universal. Stress the dynamic nature of science where theories and laws are continually tested and refined.

1.2 Units and Standards

Learning Objectives

By the end of this section, you will be able to:

• Describe how SI base units are defined.
• Describe how derived units are created from base units.
• Express quantities given in SI units using metric prefixes.

The Need for Standardized Units

Measurements are expressed in units, which are standardized values.

• Essential for scientists to express and compare values meaningfully.

Major Systems of Units:

• SI Units (Système International d’Unités): Metric system, used by scientists worldwide and nearly every country (except U.S.).
• English Units: Customary/imperial system, mainly used in the U.S. (foot–pound–second system, SAE units).

Caption: Distances in unknown units are useless

Emphasize the importance of standardized units for clear communication in science. Introduce SI units as the primary system used in physics.

SI Units: Base and Derived

Base Quantities: Physical quantities defined through a measurement process. Base Units: The units for base quantities.

Derived Quantities: Physical quantities expressed as algebraic combinations of base quantities. Derived Units: The units for derived quantities.

Note

The choice of base quantities is somewhat arbitrary, but they must be independent and measurable to high precision.

Seven SI Base Quantities and Units:

ISQ Base Quantity	SI Base Unit
Length	meter (m)
Mass	kilogram (kg)
Time	second (s)
Electrical current	ampere (A)
Thermodynamic temperature	kelvin (K)
Amount of substance	mole (mol)
Luminous intensity	candela (cd)

Explain the distinction between base and derived quantities/units. Highlight the seven SI base units as the foundation for all other measurements in physics.

Examples of Derived Units

Derived units are combinations of base units.

• Area: Length \(\times\) Length \(\rightarrow\) square meters (\(m^2\))
• Volume: Length \(\times\) Length \(\times\) Length \(\rightarrow\) cubic meters (\(m^3\))
• Speed: Length / Time \(\rightarrow\) meters per second (\(m/s\))
• Density: Mass / Volume \(\rightarrow\) kilograms per cubic meter (\(kg/m^3\))

• Angle: Ratio of arc length to radius \(\rightarrow\) radian (dimensionless)
• Mass Flow Rate: Mass / Time \(\rightarrow\) kilograms per second (\(kg/s\))
• Electric Charge: Current \(\times\) Time \(\rightarrow\) ampere-second (A·s)

Provide concrete examples of derived units. This helps students understand how all physical quantities can be traced back to the seven base units.

Units of Time, Length, and Mass

These three base units are fundamental to mechanics.

The Second (s):

• Historically: 1/86,400 of a mean solar day.
• Current definition (1967): Time required for 9,192,631,770 vibrations of a cesium-133 atom.
• High precision is crucial (e.g., GPS).

Caption: An atomic clock

The Meter (m):

• Historically: 1/10,000,000 of distance from equator to North Pole.
• Current definition (1983): Distance light travels in a vacuum in 1/299,792,458 of a second.
• Speed of light is defined as exactly 299,792,458 m/s.

Caption: The meter defined by speed of light

Detail the definitions of the second and meter, highlighting their evolution towards greater precision and reliance on fundamental physical constants.

The Kilogram

The Kilogram (kg):

• Historically (1795-2018): Mass of a platinum–iridium cylinder (International Prototype of the Kilogram, “Le Grand K”).
• Current definition (May 2019): Based on the Planck constant (\(h\)), speed of light (\(c\)), and cesium-133 frequency (\(\Delta v_{Cs}\)).
• Measured using a Kibble balance.

Note

The new definition ensures a stable and universally reproducible standard, as physical constants do not change.

Caption: The U.S. National Institute of Standards and Technology’s Kibble balance

Explain the historical and modern definitions of the kilogram. Emphasize the shift to a definition based on fundamental constants for stability and precision.

Metric Prefixes

Metric prefixes simplify expressing very large or very small quantities.

Units are categorized by factors of 10.

Prefix	Symbol	Meaning	Prefix	Symbol	Meaning
yotta-	Y	\(10^{24}\)	yocto-	y	\(10^{-24}\)
zetta-	Z	\(10^{21}\)	zepto-	z	\(10^{-21}\)
exa-	E	\(10^{18}\)	atto-	a	\(10^{-18}\)
peta-	P	\(10^{15}\)	femto-	f	\(10^{-15}\)
tera-	T	\(10^{12}\)	pico-	p	\(10^{-12}\)
giga-	G	\(10^9\)	nano-	n	\(10^{-9}\)
mega-	M	\(10^6\)	micro-	\(\mu\)	\(10^{-6}\)
kilo-	k	\(10^3\)	milli-	m	\(10^{-3}\)
hecto-	h	\(10^2\)	centi-	c	\(10^{-2}\)
deka-	da	\(10^1\)	deci-	d	\(10^{-1}\)

Important

Do not “double up” prefixes (e.g., no “megagigameters”).

For mass, prefixes are applied to the gram (g), not the kilogram (kg).

Example: \(10^3\text{ kg} = 10^6\text{ g} = 1\text{ Mg}\) (megagram).

Present the table of metric prefixes and explain their use. Highlight the “no doubling up” rule and the special case of applying prefixes to the gram for mass.

Example: Using Metric Prefixes

Restate the mass \(1.93 \times 10^{13}\text{ kg}\) using a metric prefix such that the numerical value is between 1 and 1000.

Strategy:

1. Convert to grams (since prefixes apply to grams).
2. Find the closest prefix from Table 1.2 that results in a coefficient between 1 and 1000.

Solution:

1. Convert to grams: \(1.93 \times 10^{13}\text{ kg} = 1.93 \times 10^{13} \times 10^3\text{ g} = 1.93 \times 10^{16}\text{ g}\)

2. Find suitable prefix:

   • \(10^{16}\) is between peta- (\(10^{15}\)) and exa- (\(10^{18}\)).
   • Using peta- (\(10^{15}\)): \(1.93 \times 10^{16}\text{ g} = 1.93 \times 10^{1} \times 10^{15}\text{ g} = 19.3\text{ Pg}\)
   • Using exa- (\(10^{18}\)): \(1.93 \times 10^{16}\text{ g} = 1.93 \times 10^{-2} \times 10^{18}\text{ g} = 0.0193\text{ Eg}\)

Since the problem asks for a numerical value between 1 and 1000, \(19.3\text{ Pg}\) is the correct answer.

Walk through the example step-by-step, explaining the reasoning behind each conversion and prefix selection. Emphasize checking the final answer’s magnitude.

1.3 Unit Conversion

Learning Objectives

By the end of this section, you will be able to:

• Use conversion factors to express the value of a given quantity in different units.

Converting Between Units

It is often necessary to convert units (e.g., liters to cups, feet to miles).

Method: Using Conversion Factors

1. Identify current units and desired units.
2. Find a conversion factor (a ratio equal to 1) that relates the two units.
3. Multiply your quantity by the conversion factor such that the unwanted units cancel out.

Example: Convert 80 m to kilometers.

• We have: meters (m). Want: kilometers (km).
• Conversion factor: \(1\text{ km} = 1000\text{ m}\). So, \(\frac{1\text{ km}}{1000\text{ m}}\) or \(\frac{1000\text{ m}}{1\text{ km}}\).
• Calculation: \(80\text{ m} \times \frac{1\text{ km}}{1000\text{ m}} = 0.080\text{ km}\)

Introduce unit conversion as a necessary skill. Explain the method of using conversion factors, demonstrating with a simple metric conversion.

Example: Converting Nonmetric to Metric

The distance from the university to home is 10 miles. It takes 20 minutes to drive. Calculate average speed in meters per second (m/s).

Strategy:

1. Calculate average speed in given units (miles/minute).
2. Multiply by conversion factors to convert miles to meters and minutes to seconds.

Solution:

1. Average speed = \(\frac{\text{Distance}}{\text{Time}} = \frac{10\text{ mi}}{20\text{ min}} = 0.50\text{ mi/min}\)

2. Convert to m/s: \(0.50\text{ mi/min} \times \frac{1609\text{ m}}{1\text{ mi}} \times \frac{1\text{ min}}{60\text{ s}}\) \(= \frac{(0.50)(1609)}{60}\text{ m/s}\) \(= 13\text{ m/s}\)

Significance Check:

• Units cancel correctly (mi and min cancel, leaving m/s).
• Final units match desired units.

Walk through the example, emphasizing careful selection of conversion factors and unit cancellation. Stress the importance of checking units.

Example: Converting Between Metric Units

The density of iron is \(7.86\text{ g/cm}^3\) under standard conditions. Convert this to \(\text{kg/m}^3\).

Strategy:

We need to convert grams to kilograms and cubic centimeters to cubic meters.

• Conversion factors: \(1\text{ kg} = 10^3\text{ g}\) and \(1\text{ cm} = 10^{-2}\text{ m}\).
• For cubic centimeters, we cube the length conversion factor: \((1\text{ cm})^3 = (10^{-2}\text{ m})^3 = 10^{-6}\text{ m}^3\).

Solution:

\(7.86\text{ g/cm}^3 \times \frac{1\text{ kg}}{10^3\text{ g}} \times \left(\frac{1\text{ cm}}{10^{-2}\text{ m}}\right)^3\)

\(= 7.86\text{ g/cm}^3 \times \frac{1\text{ kg}}{10^3\text{ g}} \times \frac{1\text{ cm}^3}{10^{-6}\text{ m}^3}\)

\(= \frac{7.86 \times 10^{-6}}{10^3}\text{ kg/m}^3 = 7.86 \times 10^3\text{ kg/m}^3\)

Significance Check:

• Units cancel correctly (g and \(\text{cm}^3\) cancel, leaving \(\text{kg/m}^3\)).
• Final units match desired units.

This example shows converting a derived unit and the need to cube conversion factors for volume. Reiterate the importance of unit checks.

Consequences of Unit Conversion Errors

Failure to pay attention to unit conversions can be very costly.

Caution

Mars Climate Orbiter (1999):

• NASA lost contact with the probe while guiding it into Mars orbit.
• Software error: one team used English units (pound-force-seconds) for thruster data, while another expected SI units (newton-seconds).
• Caused probe to follow an incorrect trajectory, likely burning up or escaping into space.
• Cost: hundreds of millions of dollars.

Use the Mars Climate Orbiter example to emphasize the critical importance of careful unit conversion in real-world applications, especially in engineering and space exploration.

1.4 Dimensional Analysis

Learning Objectives

By the end of this section, you will be able to:

• Find the dimensions of a mathematical expression involving physical quantities.
• Determine whether an equation involving physical quantities is dimensionally consistent.

Dimensions of Physical Quantities

The dimension of a physical quantity expresses its dependence on the base quantities (length, mass, time, etc.) as a product of their symbols.

Base Quantities and Their Dimensions:

Base Quantity	Symbol for Dimension
Length	L
Mass	M
Time	T
Current	I
Thermodynamic temperature	\(\Theta\)
Amount of substance	N
Luminous intensity	J

Examples:

• Length: [r] = L
• Area: \(L^2\)
• Volume: \(L^3\)
• Speed: L/T or \(LT^{-1}\)
• Density: M/\(L^3\) or \(ML^{-3}\)

Note

Square brackets [ ] are used to denote the dimension of a quantity (e.g., [r] = L).

Dimensionless quantities (or “pure numbers”) have dimension \(L^0M^0T^0I^0\Theta^0N^0J^0 = 1\).

Define “dimension” and provide the symbols for base quantities. Give examples of how dimensions are derived. Explain the concept of dimensionless quantities.

Dimensional Consistency

Any mathematical equation relating physical quantities must be dimensionally consistent.

Rules for Dimensional Consistency:

1. Terms must have the same dimensions: You cannot add or subtract quantities with different dimensions. Both sides of an equation must have the same dimensions. (e.g., you can’t add meters and seconds).
2. Arguments of functions must be dimensionless: Arguments of trigonometric (sin, cos), logarithmic, or exponential functions must be pure numbers (dimensionless).

Important

If an equation violates these rules, it cannot be a correct physical law.

Dimensional analysis is useful for:

• Checking for typos or algebraic mistakes.
• Recalling physical laws.
• Suggesting forms of new laws.

Clearly state the two rules for dimensional consistency. Emphasize that this is a powerful tool for validating equations and detecting errors.

Example: Using Dimensions to Remember an Equation

Suppose you recall two expressions for a circle, \(\pi r^2\) and \(2\pi r\). One is area, the other circumference. Which is which?

Strategy:

Use dimensional analysis. Area has dimension \(L^2\). Circumference has dimension \(L\).

Solution:

1. Dimension of \(\pi r^2\):

   \([\pi r^2] = [\pi] \cdot [r]^2 = 1 \cdot L^2 = L^2\)

   (Since \(\pi\) is a dimensionless constant and \(r\) is length).

   This has the dimension of area.

2. Dimension of \(2\pi r\):

   \([2\pi r] = [2] \cdot [\pi] \cdot [r] = 1 \cdot 1 \cdot L = L\)

   (Since 2 and \(\pi\) are dimensionless, and \(r\) is length).

   This has the dimension of length.

Conclusion: \(\pi r^2\) is the area formula, and \(2\pi r\) is the circumference formula.

This simple example demonstrates how dimensional analysis can resolve common memory errors. Reinforce that it helps with the form of equations, not dimensionless constants.

Example: Checking Equations for Dimensional Consistency

Given quantities \(s, v, a, t\) with dimensions \([s]=L\), \([v]=LT^{-1}\), \([a]=LT^{-2}\), and \([t]=T\).

Determine if each equation is dimensionally consistent:

1. \(s = vt + 0.5at^2\)
2. \(s = vt^2 + 0.5at\)
3. \(v = \sin(at^2/s)\)

Solution:

1. Check dimensions of each term:

• \([s] = L\)
• \([vt] = [v] \cdot [t] = (LT^{-1}) \cdot T = LT^0 = L\)
• \([0.5at^2] = [a] \cdot [t]^2 = (LT^{-2}) \cdot T^2 = LT^0 = L\) All terms have dimension \(L\). Dimensionally consistent.

2. Check dimensions of each term:

• \([s] = L\)
• \([vt^2] = [v] \cdot [t]^2 = (LT^{-1}) \cdot T^2 = LT\)
• \([0.5at] = [a] \cdot [t] = (LT^{-2}) \cdot T = LT^{-1}\) Terms have different dimensions. Not dimensionally consistent.

3. Check argument of sine function and then terms:

• Argument: \([at^2/s] = ([a] \cdot [t]^2) / [s] = (LT^{-2} \cdot T^2) / L = L/L = 1\) (Dimensionless - OK)
• Terms:
  • \([v] = LT^{-1}\)
  • \([\sin(at^2/s)] = 1\) (Sine function returns a dimensionless number)

Terms have different dimensions. Not dimensionally consistent.

Guide students through each part of the example, applying the rules of dimensional consistency. Highlight how to handle functions like sine.

Calculus and Dimensions

Dimensions also obey rules under calculus operations.

Derivative: The dimension of a derivative is the ratio of the dimensions of the quantities.

For quantities \(v\) and \(t\):

\[ \left[\frac{dv}{dt}\right] = \frac{[v]}{[t]} \]

Integral: The dimension of an integral is the product of the dimensions of the quantities.

For quantities \(v\) and \(t\):

\[ \left[\int v dt\right] = [v] \cdot [t] \]

Tip

These rules are analogous for units as well.

For example, if velocity \(v\) is in m/s and time \(t\) is in s, then acceleration \(dv/dt\) is in \(\text{(m/s)/s} = \text{m/s}^2\).

Introduce how dimensions behave with derivatives and integrals. This prepares students for concepts like velocity and acceleration, and work and impulse, where these operations are common.

1.5 Estimates and Fermi Calculations

Learning Objectives

By the end of this section, you will be able to:

• Estimate the values of physical quantities.

The Art of Estimation

Estimates (or guesstimates, order-of-magnitude approximations, Fermi calculations) are rough ideas of a quantity’s value.

• Not random guesses, but based on prior experience and physical reasoning.
• Develops physical intuition.
• Useful for “sanity checks” on calculations or policy proposals.

Note

Enrico Fermi was famous for his ability to make surprisingly precise estimates.

Strategies for Estimation:

1. Break down complex quantities: Get big lengths from smaller lengths (e.g., building height from number of floors and person height).
2. Use simple models for geometry: Get areas/volumes from lengths (e.g., approximate Earth as a sphere).
3. Relate mass, volume, and density: Get masses from volumes and densities (e.g., density of water \(\approx 10^3\text{ kg/m}^3\)).

Explain the purpose of estimation and its value in developing intuition. Introduce Fermi calculations and the basic strategies for making estimates.

Strategies for Estimation (Cont.)

4. Bound the unknown: If direct intuition is lacking, establish upper and lower bounds.
   • For example, mass of a moose: between a person (\(10^2\text{ kg}\)) and a car (\(10^3\text{ kg}\)).
   • Geometric mean: \((10^2 \times 10^3)^{0.5} = 10^{2.5} \approx 3 \times 10^2\text{ kg}\).
5. One “sig. fig.” is fine: Keep arithmetic simple, usually only the order of magnitude is needed.
6. Ask: Does this make sense? Compare with known values, check units, magnitude, and sign.

Important

If an estimate yields a “wacky” answer (e.g., ocean mass > Earth mass), check your units, arithmetic, and logical reasoning.

Continue with the remaining estimation strategies, particularly bounding and the importance of a sanity check. Encourage students to embrace risk in estimation.

Example: Mass of Earth’s Oceans

Estimate the total mass of the oceans on Earth.

Strategy:

1. Mass = Density \(\times\) Volume.
2. Volume = Surface Area \(\times\) Average Depth.
3. Surface Area of oceans \(\approx\) Surface Area of Earth (approximate as sphere, \(A = \pi d^2\)).
4. Estimate Earth’s diameter (\(10^7\text{ m}\) from Fig. 1.4).
5. Estimate average ocean depth (bound it: between 1 km and 10 km, so \(\approx 3 \times 10^3\text{ m}\)).
6. Density of water \(\approx 10^3\text{ kg/m}^3\).
7. Use one significant figure for calculations.

Solution:

1. Earth’s surface area: \(A = \pi d^2 \approx 3 \times (10^7\text{ m})^2 = 3 \times 10^{14}\text{ m}^2\).
2. Ocean volume: \(V = A \times D = (3 \times 10^{14}\text{ m}^2) \times (3 \times 10^3\text{ m}) = 9 \times 10^{17}\text{ m}^3\).
3. Ocean mass: \(M = \rho V = (10^3\text{ kg/m}^3) \times (9 \times 10^{17}\text{ m}^3) = 9 \times 10^{20}\text{ kg}\).

Conclusion: The order of magnitude of Earth’s oceans is \(10^{21}\text{ kg}\).

Guide students through the estimation process for ocean mass, applying the strategies discussed. Show how to break a big problem into smaller, estimable parts.

Significance Check for Ocean Mass Estimate

Does it make sense?

• Mass of atmosphere: \(10^{19}\text{ kg}\) (from Fig. 1.4).
• Mass of Earth: \(10^{25}\text{ kg}\) (from Fig. 1.4).
• Our estimate (\(10^{21}\text{ kg}\)) falls between these, which is reasonable.

Actual Value: Web search for “mass of oceans” yields \(\approx 1.4 \times 10^{21}\text{ kg}\).

Our estimate is of the correct order of magnitude.

Tip

Practicing estimation helps build confidence and critical thinking, which can be valuable skills in various fields, including tech interviews.

Emphasize the “sanity check” step by comparing the estimate to known values. Reinforce that estimation is a practical and valuable skill.

1.6 Significant Figures

Learning Objectives

By the end of this section, you will be able to:

• Determine the correct number of significant figures for the result of a computation.
• Describe the relationship between the concepts of accuracy, precision, uncertainty, and discrepancy.
• Calculate the percent uncertainty of a measurement, given its value and its uncertainty.
• Determine the uncertainty of the result of a computation involving quantities with given uncertainties.

Accuracy and Precision

Accuracy: How close a measurement is to the accepted reference value. Precision: How close repeated independent measurements are to each other (under the same conditions).

Example: Paper Length

• Accepted value: 11.0 in.
• Measurements: 11.1 in., 11.2 in., 10.9 in.
• Accurate: Yes, close to 11.0 in.
• Precise: Yes, measurements are close to each other (range = 0.3 in.).

Caption: (a) High accuracy, low precision. (b) Low accuracy, high precision.

Define accuracy and precision clearly, using the paper length example to illustrate. Use the dartboard analogy (Figure 1.12) to further differentiate.

Uncertainty and Discrepancy

Uncertainty (\(\delta A\)): A quantitative measure of how much measured values deviate from one another (related to precision).

• Can be expressed as a range or standard deviation.
• Example: 11.1 \(\pm\) 0.15 in. for paper length.

Discrepancy (Measurement Error): The difference between the measured value and a given standard/expected value (related to accuracy).

• Example: 11.1 in. (average) - 11.0 in. (accepted) = 0.1 in. (discrepancy).

Factors contributing to uncertainty:

• Limitations of the measuring device.
• Skill of the person taking the measurement.
• Irregularities in the object being measured.
• Environmental factors.

Introduce uncertainty and discrepancy, linking them to precision and accuracy, respectively. Provide examples of factors that contribute to uncertainty.

Percent Uncertainty

Percent uncertainty expresses uncertainty as a percentage of the measured value.

\[ \text{Percent uncertainty} = \frac{\delta A}{A} \times 100\% \]

Example: Calculating Percent Uncertainty

A bag of apples has an average weight of \(A = 5.1\text{ lb}\) with an uncertainty of \(\delta A = 0.3\text{ lb}\).

\[ \text{Percent uncertainty} = \frac{0.3\text{ lb}}{5.1\text{ lb}} \times 100\% = 5.9\% \approx 6\% \]

So, the weight is \(5.1\text{ lb} \pm 6\%\).

Tip

Percent uncertainty is dimensionless.

Define and illustrate percent uncertainty with an example. Emphasize that it’s dimensionless and provides a relative measure of precision.

Uncertainties in Calculations

When calculating quantities from measurements with uncertainties:

1. Multiplication and Division:
   • The percent uncertainty in the result is the sum of the percent uncertainties in the quantities used in the calculation.
   • Example: Area = Length \(\times\) Width. If length has 2% uncertainty and width has 1% uncertainty, area has 3% uncertainty.
2. Addition and Subtraction:
   • The result can contain no more decimal places than the least-precise measurement.
   • Example: \(7.56\text{ kg} - 6.052\text{ kg} + 13.7\text{ kg} = 15.208\text{ kg}\).
   • Least precise is 13.7 kg (one decimal place). Result rounds to \(15.2\text{ kg}\).

Provide the rules for propagating uncertainty through multiplication/division and addition/subtraction. Give clear examples for each.

Significant Figures

Significant figures indicate the precision of a measuring tool.

• The last digit written down in a measurement is the first digit with some uncertainty.

Rules for Counting Significant Figures:

1. Non-zero digits: Always significant (e.g., 36.7 cm has 3 sig figs).
2. Zeros:
   • Leading zeros: Not significant (placeholders for decimal point, e.g., 0.053 has 2 sig figs).
   • Captive zeros: Always significant (between non-zero digits, e.g., 10.053 has 5 sig figs).
   • Trailing zeros: Significant ONLY if the number contains a decimal point (e.g., 1300. has 4 sig figs, 1300 has 2-4 ambiguous).
   • Ambiguity: Use scientific notation to clarify (e.g., \(1.3 \times 10^3\) for 2 sig figs, \(1.300 \times 10^3\) for 4 sig figs).

Define significant figures and explain how they relate to measurement precision. Go through the rules for counting significant figures, with special attention to zeros.

Significant Figures in Calculations

When combining measurements with different precision:

1. Multiplication and Division:

   • Result has the same number of significant figures as the quantity with the least number of significant figures.

   • Example: Area of circle with \(r = 1.2\text{ m}\) (2 sig figs).

     \(A = \pi r^2 = \pi (1.2\text{ m})^2 = 4.5238934\text{ m}^2\).

     Round to 2 sig figs: \(4.5\text{ m}^2\).

2. Addition and Subtraction:
   • Result contains no more decimal places than the least-precise measurement.
   • Example: \(7.56\text{ kg} - 6.052\text{ kg} + 13.7\text{ kg} = 15.2\text{ kg}\). (Because \(13.7\text{ kg}\) has one decimal place, the final answer is rounded to one decimal place).

Note

In this text: Most numbers are assumed to have three significant figures. Exact numbers (e.g., 2 in \(C=2\pi r\)) and conversion factors (e.g., 100 cm/1 m) do not limit significant figures.

Reiterate the rules for significant figures in calculations for both multiplication/division and addition/subtraction. Explain conventions used in the textbook.

1.7 Solving Problems in Physics

Learning Objectives

By the end of this section, you will be able to:

• Describe the process for developing a problem-solving strategy.
• Explain how to find the numerical solution to a problem.
• Summarize the process for assessing the significance of the numerical solution to a problem.

Problem-Solving in Physics

Physics requires analytical skills to apply broad physical principles to specific situations.

This is more powerful than memorizing facts.

Three-Stage Process:

1. Strategy
2. Solution
3. Significance

Caption: Problem-solving skills are essential to your success in physics.

Introduce the structured problem-solving approach. Emphasize that physics is about understanding, not just computation.

Stage 1: Strategy

This stage is about understanding the problem and planning how to solve it.

1. Examine the situation:
   • Determine which physical principles are involved.
   • Draw a simple sketch (useful for coordinate systems, directions).
2. Identify “Knowns”:
   • List what is given or can be inferred.
   • Example: “Stopped” means velocity is zero.
3. Identify “Unknowns”:
   • List exactly what needs to be determined.
4. Select physical principles/equations:
   • Find equations that relate knowns and unknowns.
   • Aim for equations with only one unknown initially.
   • You may need multiple equations for complex problems.

Detail the steps involved in the strategy phase. Stress the importance of conceptual understanding and planning before jumping into calculations.

Stage 2: Solution

This is where the math happens.

• Substitute the knowns (with units) into the appropriate equations.
• Perform the necessary algebraic, calculus, geometric, or arithmetic operations.
• Obtain numerical solutions, complete with units.

Tip

Always carry units through your calculations. This helps catch errors if units don’t work out correctly in the end.

Explain that the solution phase is about execution. Remind students to include units throughout their calculations.

Stage 3: Significance

After getting a numerical answer, assess its meaning and validity.

1. Check your units:
   • If units are incorrect, an error has been made.
   • Correct units don’t guarantee a correct numerical answer, but incorrect units guarantee an incorrect answer.
   • Use dimensional consistency checks.
2. Check for reasonableness:
   • Does the magnitude make sense? (e.g., is the speed of a car faster than light?)
   • Does the sign make sense? (e.g., is acceleration in the expected direction?)
   • Compare with rough estimates or known quantities.
3. Interpret the meaning:
   • What does the answer tell you about the world?
   • Does it confirm or challenge your understanding?
   • Can the method be adapted for other interesting questions?

Emphasize that physics doesn’t end with a numerical answer. The significance stage is crucial for developing deeper understanding and critical thinking.

Key Takeaways

• Physics explains all natural phenomena across vast scales, from subatomic to cosmic.
• Order of magnitude is a powerful tool for estimating and comparing quantities.
• Models, theories, and laws are the building blocks of scientific understanding, constantly refined by observation and experiment.
• SI units (International System of Units) are the global standard, built upon seven base units.
• Unit conversions are critical and must be done carefully to avoid errors (e.g., Mars Climate Orbiter).
• Dimensional analysis ensures equations are consistent and helps identify errors or recall formulas.
• Estimates and Fermi calculations develop physical intuition and provide “sanity checks” for answers.
• Accuracy (closeness to true value) and precision (reproducibility of measurements) are distinct and quantified by discrepancy and uncertainty.
• Significant figures reflect the precision of measurements and dictate how results of calculations should be reported.
• A systematic problem-solving strategy (Strategy, Solution, Significance) is essential for success in physics.

Key Equations

Equation	Description
\(\text{Percent uncertainty} = \frac{\delta A}{A} \times 100\%\)	Defines the relative uncertainty of a measurement \(A\) with absolute uncertainty \(\delta A\).
\([A] = L^a M^b T^c I^d \Theta^e N^f J^g\)	General form for the dimension of any physical quantity \(A\), where \(L, M, T, I, \Theta, N, J\) are the base dimensions.
\([\frac{dv}{dt}] = \frac{[v]}{[t]}\)	Dimension of a derivative (ratio of dimensions).
\([\int v dt] = [v] \cdot [t]\)	Dimension of an integral (product of dimensions).

Key Terms

Term	Definition
Accuracy	The degree to which a measured value agrees with the correct or accepted value.
Base quantity	One of a small set of physical quantities from which all other physical quantities can be derived.
Base unit	The unit of a base quantity.
Dimensional analysis	A procedure used to check the consistency of physical equations by comparing the dimensions of physical quantities.
Dimensionless quantity	A physical quantity whose dimension is \(L^0M^0T^0I^0\Theta^0N^0J^0\), meaning it has no units associated with it.
Dimensions	The expression of a physical quantity in terms of its dependence on base quantities (e.g., length L, mass M, time T).
Discrepancy	The difference between the measured value and a given standard or expected value (measurement error).
Derived quantity	A physical quantity that can be expressed as an algebraic combination of base quantities.
Derived unit	The unit of a derived quantity.
English units	A system of measurement primarily used in the United States (also known as customary or imperial units).
Estimate	A rough or approximate calculation or judgment of the value, number, quantity, or extent of something.
Kibble balance	A device used to define the kilogram based on fundamental physical constants.
Law	A concise statement of a generalized pattern in nature, supported by scientific evidence and repeated experiments.
Metric prefixes	Prefixes based on powers of 10 used with SI units (e.g., kilo-, milli-, micro-).
Model	A representation of something that is often too difficult or impossible to display directly, used to explain certain aspects of a physical system.
Order of magnitude	The power of 10 that most closely approximates a number, referring to its scale or size.
Percent uncertainty	The ratio of the uncertainty in a measurement to the measured value, expressed as a percentage.
Physics	The science concerned with describing the interactions of energy, matter, space, and time to uncover the fundamental mechanisms that underlie every phenomenon.
Precision	The degree to which repeated measurements under the same conditions show the same results.
Scientific notation	A way of writing numbers that are too large or too small to be conveniently written in decimal form, using powers of 10.
Significant figures	The digits in a measurement that carry meaning contributing to its precision.
SI units	The International System of Units (metric system), the globally accepted standard for measurement.
Theory	A testable explanation for patterns in nature, supported by scientific evidence and verified multiple times.
Uncertainty	A quantitative measure of how much measured values deviate from one another (related to precision).
Unit conversion	The process of converting a quantity from one unit of measurement to another.