Physics for Engineers - Fundamental Concepts and Measurements

Foundations of Physics and Engineering

• Physics is defined as one of the most fundamental sciences and serves as the foundation for all engineering and technology.

• The nature of physics is that of an experimental science, following a specific progression:

  • 1. Observe

  • 2. Patterns

  • 3. Theories

  • 4. Laws

Systematic Problem-Solving in Physics

• To solve physics problems effectively, a four-step process is utilized:

  • 1. Identify the relevant concepts.

  • 2. Set up the problem.

  • 3. Execute the solution.

  • 4. Evaluate your answer.

Units and Measurements

• Measurements of physical quantities are expressed in terms of units, which function as standardized values.

• A physical quantity is any number which describes a physical phenomenon quantitatively.

• Quantities are categorized into two types:

  • 1. Basic/Fundamental quantities: These do not depend on other quantities and are used to fully describe other quantities. Examples include length and mass.

  • 2. Derived quantities: These are expressed as an algebraic combination of basic or fundamental quantities. Examples include area and volume.

Major Systems of Units

• There are two primary systems used in measurement:

  • 1. SI units: Short for the French Système International d’Unités, commonly referred to as the metric system.

  • 2. English units: Also known as the customary or imperial system. This system is sometimes referred to as the foot–pound–second ($fps$) system.

SI Base Units (International System of Quantities)

• The SI system is built upon seven fundamental base units:

  • 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 ($cd$)

SI Derived Units

• Specific derived quantities and their corresponding SI units include:

  • Force, weight: Newton ($N$), equivalent to $kg \times m \times s^{-2}$

  • Work, Energy, Heat: Joule ($J$), equivalent to $N \times m$

  • Pressure: Pascal ($Pa$), equivalent to $N/m^{2}$

  • Power, Radiant Flux: Watt ($W$), equivalent to $J/s$

  • Electric Charge: Coulomb ($C$), equivalent to $A \times s$

  • Voltage: Volt ($V$), equivalent to $W/A$

  • Magnetic Flux: Weber ($Wb$), equivalent to $V \times s$

  • Inductance: Henry ($H$), equivalent to $Wb/A$

  • Capacitance: Farad ($F$), equivalent to $C/V$

  • Resistance: Ohm ($\Omega$), equivalent to $V/A$

Metric Prefixes

• Metric prefixes are used to denote powers of 10:

  • yotta- ($Y$): $10^{24}$

  • zetta- ($Z$): $10^{21}$

  • exa- ($E$): $10^{18}$

  • peta- ($P$): $10^{15}$

  • tera- ($T$): $10^{12}$

  • giga- ($G$): $10^{9}$

  • mega- ($M$): $10^{6}$

  • kilo- ($k$): $10^{3}$

  • hecto- ($h$): $10^{2}$

  • deka- ($da$): $10^{1}$

  • deci- ($d$): $10^{-1}$

  • centi- ($c$): $10^{-2}$

  • milli- ($m$): $10^{-3}$

  • micro- ($\mu$): $10^{-6}$

  • nano- ($n$): $10^{-9}$

  • pico- ($p$): $10^{-12}$

  • femto- ($f$): $10^{-15}$

  • atto- ($a$): $10^{-18}$

  • zepto- ($z$): $10^{-21}$

  • yocto- ($y$): $10^{-24}$

English Unit Equivalents

• The following mappings relate metric units to English (Imperial) units:

  • Length: meter ($m$) vs Foot ($ft$)

  • Mass: kilogram ($kg$) vs Slugs

  • Temperature: $^{\circ}C$ vs Degree Fahrenheit ($^{\circ}F$)

  • Force: Newton ($N$) vs Pound Force ($lb$)

  • Volume: Liter ($L$) or $m^{3}$ vs Gallon ($gal$)

  • Area: Hectare ($ha$) vs Acre

  • Pressure: Pascal ($Pa$) vs $lb/in^{2}$ ($psi$)

• Specific conversion factors:

  • $1\,m = 3.28\,ft$

  • $2.54\,cm = 1\,in$

  • $1\,km = 1.609\,mile$

  • $1\,kg = 0.0685\,Slugs$

  • $1\,kg = 2.2046\,lb$

  • $3.785\,L = 1\,gal$

  • $1\,ha = 2.47\,acres$

  • $1\,kPa = 0.145\,psi$

• Traditional English conversions:

  • $1\,ft = 12\,in$

  • $1\,yd = 3\,ft$

  • $1\,mile = 5280\,ft$

  • $1\,gal = 4\,quarts$

  • $1\,quart = 2\,pints$

  • $1\,pint = 16\,oz$

Uncertainty and Accuracy

• Uncertainty is defined as the maximum difference between the measured value and the true value.

• Accuracy describes how close a measurement is to the true value.

• Examples of accuracy and uncertainty notation:

  • $40.86 \pm 0.09\,cm$

  • $1.879(25)\,kg$

  • $63\,ohms \pm 10\%$

Precision vs Accuracy

• Accuracy: Describes how close your measurements are to the true value.

• Precision: Measures how close your measured values are to each other.

Significant Figures and Rounding

• Significant figures refer to the number of meaningful digits in a value.

• Rules for Significant Figures:

  • 1. Non-zero digits are always significant.

  • 2. Any zeros between two significant figures are significant.

  • 3. A final zero or trailing zeros are significant only in the decimal portion.

• Significant Figure Count Examples:

  • $5.64\,cm$: 3 significant figures

  • $6008\,m$: 4 significant figures

  • $0.99\,g$: 2 significant figures

  • $0.430\,in$: 3 significant figures

  • $5000\,yd$: 1 significant figure

  • $5000.0\,m$: 5 significant figures

  • $0.00350\,mm$: 3 significant figures

  • $0.002\,lb$: 1 significant figure

• Operations with Significant Figures:

  • Multiplication and Division: The result can have no more significant figures than the factor with the fewest significant figures. Example: $0.453 \times \frac{3.3}{6.592} = 0.23$

  • Addition and Subtraction: The number of significant figures is determined by the term with the largest uncertainty (i.e., the fewest digits to the right of the decimal point). Example: $34.678 + 45.8 - 71.32 = 9.2$

• Rounding Off Numbers:

  • 1. If the digit to be dropped is less than 5, the number is written without the digit (e.g., $78.374$ becomes $78.37$).

  • 2. If the digit to be dropped is exactly 5, the nearest even number is used for the preceding digit (e.g., $78.375$ becomes $78.38$; $78.385$ becomes $78.38$).

  • 3. If the digit to be dropped is greater than 5, the preceding digit is increased by 1 (e.g., $78.386$ becomes $78.39$).

Scientific Notation

• Scientific notation is used to express very large or very small numbers concisely.

• Examples:

  • $355,000,000 = 3.55 \times 10^{8} = 355 \times 10^{6}$

  • $200 = 2 \times 10^{2}$

  • $4 \times 10^{-6} = 0.000004$

Unit Conversion Factors and Processes

• Unit conversion is the process of changing a measurement from one unit to another while maintaining the same quantity. This is performed by multiplying or dividing by a conversion factor.

• Detailed Conversion Table:

  • Length:

    • $1\,m = 100\,cm = 1000\,mm = 10^{6}\,\mu m = 10^{9}\,nm$

    • $1\,km = 1000\,m = 0.6214\,mi$

    • $1\,m = 3.281\,ft = 39.37\,in$

    • $1\,cm = 0.3937\,in$

    • $1\,in = 2.540\,cm$

    • $1\,ft = 30.48\,cm$

    • $1\,yd = 91.44\,cm$

    • $1\,mi = 5280\,ft = 1.609\,km$

    • $1\,\text{\AA} = 10^{-10}\,m = 10^{-8}\,cm = 10^{-1}\,nm$

    • $1\,nautical\,mile = 6080\,ft$

    • $1\,light-year = 9.461 \times 10^{15}\,m$

  • Volume:

    • $1\,liter = 1000\,cm^{3} = 10^{-3}\,m^{3} = 0.03531\,ft^{3} = 61.02\,in^{3}$

    • $1\,ft^{3} = 0.02832\,m^{3} = 28.32\,liters = 7.477\,gallons$

    • $1\,gallon = 3.788\,liters$

  • Area:

    • $1\,cm^{2} = 0.155\,in^{2}$

    • $1\,m^{2} = 10^{4}\,cm^{2} = 10.76\,ft^{2}$

    • $1\,in^{2} = 6.452\,cm^{2}$

    • $1\,ft^{2} = 144\,in^{2} = 0.0929\,m^{2}$

  • Time:

    • $1\,min = 60\,s$

    • $1\,h = 3600\,s$

    • $1\,d = 86,400\,s$

    • $1\,y = 365.24\,d = 3.156 \times 10^{7}\,s$

  • Angle:

    • $1\,rad = 57.30^{\circ} = \frac{180^{\circ}}{\pi}$

    • $1^{\circ} = 0.01745\,rad = \frac{\pi}{180}\,rad$

    • $1\,revolution = 360^{\circ} = 2\pi\,rad$

    • $1\,rev/min\,(rpm) = 0.1047\,rad/s$

  • Speed:

    • $1\,m/s = 3.281\,ft/s$

    • $1\,ft/s = 0.3048\,m/s$

    • $1\,mi/min = 60\,mi/h = 88\,ft/s$

    • $1\,km/h = 0.2778\,m/s = 0.6214\,mi/h$

    • $1\,mi/h = 1.466\,ft/s = 0.4470\,m/s = 1.609\,km/h$

    • $1\,furlong/fortnight = 1.662 \times 10^{-4}\,m/s$

  • Acceleration:

    • $1\,m/s^{2} = 100\,cm/s^{2} = 3.281\,ft/s^{2}$

    • $1\,cm/s^{2} = 0.01\,m/s^{2} = 0.03281\,ft/s^{2}$

    • $1\,ft/s^{2} = 0.3048\,m/s^{2} = 30.48\,cm/s^{2}$

    • $1\,mi/h \cdot s = 1.467\,ft/s^{2}$

  • Mass:

    • $1\,kg = 10^{3}\,g = 0.0685\,slug$

    • $1\,g = 6.85 \times 10^{-5}\,slug$

    • $1\,slug = 14.59\,kg$

    • $1\,u = 1.661 \times 10^{-27}\,kg$

    • $1\,kg\,\text{weight at globe surface} = 2.205\,lb\,\text{where}\;g = 9.80\,m/s^{2}$

  • Force:

    • $1\,N = 10^{5}\,dyn = 0.2248\,lb$

    • $1\,lb = 4.448\,N = 4.448 \times 10^{5}\,dyn$

  • Pressure:

    • $1\,Pa = 1\,N/m^{2} = 1.450 \times 10^{-4}\,lb/in^{2} = 0.0209\,lb/ft^{2}$

    • $1\,bar = 10^{5}\,Pa$

    • $1\,lb/in^{2} = 6895\,Pa$

    • $1\,lb/ft^{2} = 47.88\,Pa$

    • $1\,atm = 1.013 \times 10^{5}\,Pa = 1.013\,bar = 14.7\,lb/in^{2} = 2117\,lb/ft^{2}$

    • $1\,mm\,Hg = 1\,torr = 133.3\,Pa$

  • Energy:

    • $1\,J = 10^{7}\,ergs = 0.239\,cal$

    • $1\,cal = 4.186\,J$

    • $1\,ft \cdot lb = 1.356\,J$

    • $1\,Btu = 1055\,J = 252\,cal = 778\,ft \cdot lb$

    • $1\,eV = 1.602 \times 10^{-19}\,J$

    • $1\,kWh = 3.600 \times 10^{6}\,J$

  • Mass-Energy Equivalence:

    • $1\,kg = 8.988 \times 10^{16}\,J$

    • $1\,u = 931.5\,MeV$

    • $1\,eV = 1.074 \times 10^{-9}\,u$

  • Power:

    • $1\,W = 1\,J/s$

    • $1\,hp = 746\,W = 550\,ft \cdot lb/s$

    • $1\,Btu/h = 0.293\,W$

Dimensional Analysis

• Dimension refers to a physical property described by words: time, length, or mass. This property remains the same regardless of units.

• Dimensional analysis is the study of relationships between physical quantities by identifying these fundamental dimensions.

• Dimensional Symbols:

  • Length: $L$

  • Time: $T$

  • Mass: $M$

  • Electrical Current: $I$

  • Thermodynamic Temperature: $\theta$

  • Amount of Substance: $N$

  • Luminous Intensity: $J$

• Principles of Dimensional Consistency:

  • 1. For addition and subtraction, quantities must have identical dimensional units.

  • 2. For division and multiplication, quantities may have different dimensional units.

  • 3. For an equation to hold true, it must have the same dimensional units on both sides.

  • 4. Note: Constants and angles (e.g., $\cos(\theta), \sin(\theta), \tan(\theta), 1, 2, \frac{1}{2}, \frac{4}{3}, \exp, \pi, \ln, \log$) have dimensional units equal to $1$.

Dimensional Formulas for Standard Quantities

• Displacement: $M^{0}LT^{0}$

• Area: $L \times L = L^{2}$ ($M^{0}L^{2}T^{0}$)

• Volume: $L \times L \times L = L^{3}$ ($M^{0}L^{3}T^{0}$)

• Velocity/Speed: $L/T = LT^{-1}$ ($M^{0}LT^{-1}$)

• Momentum: $MLT^{-1}$

• Acceleration: $LT^{-2}$ ($M^{0}LT^{-2}$)

• Force: $MLT^{-2}$

• Impulse: $MLT^{-1}$

• Work / Energy / Torque: $ML^{2}T^{-2}$

• Power: $ML^{2}T^{-3}$

• Density ($mass/volume$): $ML^{-3}T^{0}$

• Pressure ($F/A$) / Stress / Young's Modulus: $ML^{-1}T^{-2}$

• Angular Displacement: $M^{0}L^{0}T^{0}$

• Angular Velocity: $M^{0}L^{0}T^{-1}$

• Angular Acceleration: $M^{0}L^{0}T^{-2}$

• Moment of Inertia: $ML^{2}T^{0}$

• Angular Momentum: $ML^{2}T^{-1}$

• Frequency: $M^{0}L^{0}T^{-1}$

• Strain: $M^{0}L^{0}T^{0}$

• Surface Tension / Force Constant (spring): $ML^{0}T^{-2}$

• Coefficient of Viscosity: $ML^{-1}T^{-1}$

• Gravitational Constant ($G$): $M^{-1}L^{3}T^{-2}$

• Gravitational Potential: $M^{0}L^{2}T^{-2}$

• Temperature ($\theta$): $M^{0}L^{0}T^{0}\theta^{+1}$

• Heat: $ML^{2}T^{-2}$

• Specific Heat: $M^{0}L^{2}T^{-2}\theta^{-1}$

• Latent Heat: $M^{0}L^{2}T^{-2}$

• Coefficient of Thermal Conductivity: $MLT^{-3}\theta^{-1}$

• Universal Gas Constant ($R$): $ML^{2}T^{-2}\theta^{-1}$

• Mechanical Equivalent of Heat ($J$): $M^{0}L^{0}T^{0}$

• Charge ($Q$): $M^{0}L^{0}TA$

• Current ($I$): $M^{0}L^{0}T^{0}A$

• Electric Potential ($V$): $ML^{2}T^{-3}A^{-1}$

• Electric Permittivity ($\epsilon_{0}$): $M^{-1}L^{-3}T^{4}A^{2}$

• Intensity of Electric Field ($E$): $MLT^{-3}A^{-1}$

• Capacitance ($C$): $M^{-1}L^{-2}T^{4}A^{2}$

• Dielectric Constant: $M^{0}L^{0}T^{0}$

• Resistance ($R$): $ML^{2}T^{-3}A^{-2}$

• Conductance: $M^{-1}L^{-2}T^{3}A^{2}$

• Specific Resistance / Resistivity ($\rho$): $ML^{3}T^{-3}A^{-2}$

• Conductivity: $M^{-1}L^{-3}T^{3}A^{2}$

• Magnetic Induction ($B$): $MT^{-2}A^{-1}$

• Magnetic Flux ($\phi$): $ML^{2}T^{-2}A^{-1}$

• Magnetic Intensity ($H$): $M^{0}L^{-1}T^{0}A$

• Magnetic Permeability: $MLT^{-2}A^{-2}$

• Coefficient of Self/Mutual Inductance: $ML^{2}T^{-2}A^{-2}$

• Electric Dipole Moment ($p$): $M^{0}LTA$

• Magnetic Dipole Moment ($M$): $M^{0}L^{2}AT^{0}$

Sample Mathematical Problems

• Sample Problem 1 (Rest Energy Calculation):

  • Formula: $E = mc^{2}$

  • Given: electron mass $m = 9.11 \times 10^{-31}\,kg$

  • Required answer precision: 3 significant figures.

  • Unit: Joules ($J$).

• Sample Problem 2 (Density Calculation):

  • Formula: $Density = \frac{mass}{volume}$

  • Given: mass = $1.80\,kg$, volume = $6.0 \times 10^{-4}\,m^{3}$

  • Task: Find density in units of $kg/m^{3}$.

• Uncertainty: Defined as the maximum difference between the measured value and the true value. Examples of notation include:

  • $40.86 \, ext{cm} \pm 0.09 \, ext{cm}$

  • $1.879(25) \, ext{kg}$

  • $63 \, ext{ohms} \pm 10\%$

• Accuracy: Describes how close measurements are to the true value. This includes examples like:

  • A thermometer reads $98.6 \, ext{°F}$ for your body temperature, which is accurate if the true temperature is also $98.6 \, ext{°F}$.

• Precision: Measures how close repeated measurements are to each other. An example would be:

  • If you measure the length of a table five times and get $100.0 \, ext{cm}, 100.1 \, ext{cm}, 100.0 \, ext{cm}, 99.9 \, ext{cm}, 100.1 \, ext{cm}$, this indicates high precision, even if the true length is $101 \, ext{cm}.$

Uncertainty Problems

1. A scale reads $12.4 \, kg \pm 0.1 \, kg$. What is the true value if the measurement is accurate?

   • Solution: The range of true values is:

     • Calculate lower limit: $12.4 - 0.1 = 12.3 \, kg$

     • Calculate upper limit: $12.4 + 0.1 = 12.5 \, kg$

     • Answer: True value could be between $12.3 \, kg$ and $12.5 \, kg$.

2. A thermometer reads $37.0 \, ^{\circ}C \pm 0.2 \, ^{\circ}C$. What is the range of possible true temperatures?

   • Solution:

     • Lower limit: $37.0 - 0.2 = 36.8 \, ^{\circ}C$

     • Upper limit: $37.0 + 0.2 = 37.2 \, ^{\circ}C$

     • Answer: Between $36.8 \, ^{\circ}C$ and $37.2 \, ^{\circ}C$.

3. A measurement is reported as $5000 \, m \pm 50 \, m$. Is this a precise measurement?

   • Solution:

     • Uncertainty in $5000 \, m$ is significant compared to its magnitude.

     • Answer: No, it is not precise due to a significant uncertainty relative to the measurement.

4. An engineer measures a component length to be $150 \, mm \pm 2 \, mm$. Is the uncertainty acceptable?

   • Solution:

     • Uncertainty is small compared to the measurement size.

     • Answer: Yes, the uncertainty is acceptable for engineering applications.

5. The measured distance is $75.50 \, m \pm 0.5 \, m$. Is this measurement accurate?

   • Solution:

     • Accuracy check requires comparison with true value; true value should be around $75.50 \, m$.

     • Answer: Accuracy can only be evaluated against a standard; if known, check within the uncertainty range.

6. An instrument has an uncertainty of $0.01 \, kg$. If it measures $35.5 \, kg$, what is the potential range of error?

   • Solution:

     • Range is $35.5 - 0.01 = 35.49 \, kg$ to $35.5 + 0.01 = 35.51 \, kg$.

     • Answer: From $35.49 \, kg$ to $35.51 \, kg$.

7. A length of $25.0 \, cm \pm 0.2 \, cm$ is recorded. If the true length is $24.8 \, cm$, classify if the measurement is accurate.

   • Solution:

     • Check if $24.8 \, cm$ falls within $25.0 \pm 0.2$.

     • Range: $24.8 \, cm$ falls in between $24.8 \, cm$ and $25.2 \, cm$.

     • Answer: Yes, the measurement is accurate since $24.8 \, cm$ is within the uncertainty range.

8. An electronic balance gives readings of $200.0 \, g \pm 0.05 \, g$. How would you assess its precision?

   • Solution:

     • Compare uncertainty against measurement; small uncertainty signifies high precision.

     • Answer: The measurement is precise due to a small uncertainty relative to the whole.

9. A reading of $1.5 \, L \pm 0.1 \, L$ indicates good precision. True?

   • Solution:

     • Good precision is characterized by low uncertainty.

     • Answer: Yes, it is precise; accuracy cannot be confirmed without a reference.

10. A height of $1.72 \, m \pm 0.03 \, m$ is recorded. Is the uncertainty magnitude reasonable for such a measurement?

    • Solution:

      • Evaluate if $0.03 \, m$ is acceptable for $1.72 \, m$.

      • Answer: Yes, the uncertainty is appropriate given the certainty of the measuring device.

Accuracy and Precision Problems

1. If a dart lands on the bullseye, what can you say about your accuracy and precision?

   • Solution:

     • Definitions: Accurate = close to true value; precise = closely grouped measurements.

     • Answer: Both accuracy and precision are high.

2. All darts hit close together but not on the bullseye indicate:

   • Solution:

     • Close together = precise; not on target = not accurate.

     • Answer: High precision, low accuracy.

3. A student repeatedly measures $10 \, cm$ but gets a range from $9.0 \, cm$ to $11.0 \, cm$. Classify the measurement's precision.

   • Solution:

     • Measure the spread of results.

     • Answer: Low precision due to a wide spread.

4. If several measurements of a known length yield an average shorter than expected, what does this indicate about accuracy?

   • Solution:

     • Average is less than true = measurement is inaccurate.

     • Answer: The measurement is inaccurate.

5. A gauge consistently gives $15.0 \, mm$ when measuring a part that is $15.5 \, mm$ indicates:

   • Solution:

     • Consistency = precise; difference from true value = inaccurate.

     • Answer: High precision, low accuracy.

6. A set of measurements lands all over a range but averages correctly reflects the true value. What does this indicate?

   • Solution:

     • Wide distribution means low precision; average being correct indicates high accuracy.

     • Answer: Low precision, high accuracy.

7. Measuring devices, which yield varying results but one average value, indicate:

   • Solution:

     • Variability means low precision; validity of average relates to accuracy.

     • Answer: Low precision and could be inaccurate as well.

8. Five measurements yield results of $10.5 \, m, 10.6 \, m, 10.5 \, m, 10.7 \, m, 10.5 \, m$. Assess precision and accuracy without a reference.

   • Solution:

     • Close together indicates precision; reference needed for accuracy.

     • Answer: Good precision; accuracy cannot be determined.

9. If a thermometer reads consistently $20 \, ^{\circ}C$ for an ice-water mix, what is its accuracy?

   • Solution:

     • Check against known melting point of ice.

     • Answer: Accurate for the ice-water mixture.

10. If measurement variability remains within $0.01 \, m$ around a standard but is incorrect, what could be inferred?

    • Solution:

      • Good precision but being consistently off means inaccurate.

      • Answer: Good precision but low accuracy.

Significant Figures Problems

1. How many significant figures are in $0.0042 \, m$?

   • Solution:

     • Identify non-zero digits = 4 and 2.

     • Answer: 2 significant figures (4 and 2).

2. Calculate the result of $3.00 \, J \div 2.0 \, g$. How many significant figures should the answer have?

   • Solution:

     • Perform division: $3.00 \, J \div 2.0 \, g = 1.5 \, J/g$.

     • Consider significant figures: minimum between 3 and 2.

     • Answer: Answer should have 2 significant figures. Result: $1.5 \, J/g$.

3. What is the significant figure count for $0.00056$?

   • Solution:

     • Identify non-zero digits = 5 and 6; leading zeros are not counted.

     • Answer: 2 significant figures (5 and 6).

4. The number $5.600$ has how many significant figures?

   • Solution:

     • Count all: non-zero digits and trailing zeros due to decimal confirm count.

     • Answer: 4 significant figures (5, 6, and two trailing zeros).

5. Round $78.39$ to 3 significant figures.

   • Solution:

     • Round: Observe the fourth digit (3), so it remains.

     • Answer: $78.4$.

6. Evaluate how many significant figures are in $3300 \, m$ if there is no decimal.

   • Solution:

     • Trailing zeros without decimal do not count as significant.

     • Answer: 2 significant figures (3 and 3).

7. Calculate $12.2 + 3.4$. How many significant figures should the answer use?

   • Solution:

     • Perform addition: $12.2 + 3.4 = 15.6$.

     • Determine by the number with least decimal places (1 decimal).

     • Answer: $15.6$ with 1 decimal place.

8. If $0.0053$ is multiplied by $1200$, how many significant figures will the result have?

   • Solution:

     • Calculate: $0.0053 \times 1200 = 6.36$.

     • Minimal significant figures from numbers involved.

     • Answer: 2 significant figures; provides a result of $6.4$.

9. What is $0.00450$ rounded to 3 significant figures?

   • Solution:

     • Zeros after a decimal point count when they follow significant figures.

     • Answer: $0.00450$ retains 3 significant figures.

10. In $50000$, how many significant figures should we consider if it's ambiguous?

    • Solution:

      • Without clarification, analyze significant nature of zeros.

      • Answer: Not determined without clarification; can be considered 1 or 5 significant figures.

Unit Conversion Problems

1. Convert $10 \, km$ to meters.

   • Solution:

     • Use conversion: $10 \, km \times 1000 \, m/km = 10000 \, m$.

     • Answer: $10000 \, m$.

2. How many liters are in $5.0 \, gallons$?

   • Solution:

     • Conversion: $5.0 \times 3.785 \, L/gallon = 18.93 \, L$.

     • Answer: $18.93 \, L$.

3. Convert $20 \, in$ to centimeters.

   • Solution:

     • Change: $20 \, in \times 2.54 \, cm/in = 50.8 \, cm$.

     • Answer: $50.8 \, cm$.

4. Change $45 \, lbs$ to kilograms.

   • Solution:

     • Convert: $45 \, lbs \times 0.4536 \, kg/lb = 20.41 \, kg$.

     • Answer: $20.41 \, kg$.

5. If a velocity of $60 \, mi/h$ is needed in km/h, what is the conversion?

   • Solution:

     • Velocity change: $60 \, mi/h \times 1.609 \, km/mi = 96.56 \, km/h$.

     • Answer: $96.56 \, km/h$.

6. Convert $150 \, m$ to feet.

   • Solution:

     • Perform conversion: $150 \, m \times 3.281 \, ft/m = 492.13 \, ft$.

     • Answer: $492.13 \, ft$.

7. How many milliliters are in $3.0 \, L$?

   • Solution:

     • Convert: $3.0 \, L \times 1000 \, mL/L = 3000 \, mL$.

     • Answer: $3000 \, mL$.

8. Transform $30 \, g$ to ounces.

   • Solution:

     • Change: $30 \, g \times \frac{1 \, oz}{28.35 \, g} = 1.06 \, oz$.

     • Answer: $1.06 \, oz$.

9. Convert $5.0 \, yards$ to meters.

   • Solution:

     • Use conversion: $5.0 \, yards \times 0.9144 \, m/yard = 4.57 \, m$.

     • Answer: $4.57 \, m$.

10. If you have $100 \, ^{\circ}F$, what is it in degrees Celsius?

    • Solution:

      • Apply formula: $$^{\circ}C = (^{\circ}F - 32) \times