Topic 1:Working as a physicist

1. Physical quantities, base and derived units:

• Physical quantities are characteristics of objects or systems that can be measured and quantified. Here are some common physical quantities:
  • Mechanical Quantities (length, mass, time, velocity, acceleration, force)
  • Thermal Quantities (temperature, heat, energy)
  • Electromagnetic Quantities (electric charge, electric potential, electric current, resistance, inductance, capacitance)
  • Radiation Quantities (Frequency, wavelength, energy)
  • Some other quantities (pressure, volume, density, viscosity, surface tension)
• These physical quantities can be measured using various instruments and techniques, allowing us to understand and describe the physical world around us.
• All these physical quantities are divided into Base Quantities and Drive Quantities.

• ⇒ SI units:

• A system of measurement is needed so that a comparison of the sizes can be made with the values of other people.
• Over the years, many different systems of units have been used. In the UK and US, pounds and ounces, degrees Fahrenheit and miles are still common measurements of mass, temperature and length, respectively.
• Scientists have devised an international system that uses agreed base units for the seven base quantities.
• These are termed SI units (abbreviated from the French Système International d’Unités).
• “The International System of Units (SI) is the modern A-level system, widely used in science, technology, and everyday life. It’s based on seven base quantities and their respective units”.
• Using SI units ensures clarity, consistency, and precision in scientific communication and everyday applications.

• ⇒ Base Quantities:

• “Base quantities are the fundamental physical quantities that are used to define all other physical quantities”. They are the building blocks of measurement.
• In the International System of Units (SI), the base units are the fundamental units of measurement that define the seven base quantities. Here are the seven base units, along with their symbols and definitions.
• Table 1 Base quantities, units, symbols, and definitions
• 
• These base units are the foundation of the SI system, and all other units are derived from them.

• ⇒ Drive Quantities:

• Derived quantities are physical quantities that can be expressed in terms of the seven base quantities, Table 2 illustrate “derive quantities” because they are derived from the base quantities through various mathematical operations, such as multiplication, division, and exponentiation.
• Derived units are units of measurement that are derived from the base units through various mathematical operations, such as multiplication, division, and exponentiation.
• Table 2 Drive quantities, definition, drive units with base units
• 
• These derived units are used to express various physical quantities and are essential in science, engineering, and everyday life.

• ⇒ Prefixes:

• In the International System of Units (SI), prefixes are used to denote multiples and submultiples of units.
• They are cleverly designed to make calculations easier and more intuitive.
• Numerous measures deviate significantly from the SI base unit in size.
• A human hair’s thickness can be measured in micrometers, whereas an X-ray tube’s voltage may reach hundreds of thousands of volts.
• Writing them as multiples or sub-multiples of the underlying unit is frequently helpful.
• Table 3 prefixes, symbols, multiple with examples
• 

2. Practical skill:

• Practical skills are abilities that are used to apply knowledge and expertise to real-world situations. Here are some examples of practical skills:
  1. Measurement skills: ability to use instruments like thermometers, spectrophotometers, and micropipettes.
  2. Laboratory skills: ability to conduct experiments, collect data, and analyze results.
  3. Technical skills: proficiency in using software, programming languages, and tools like CAD, GIS, or Python.
  4. Problem-solving skills: ability to identify, analyze, and solve complex problems.
  5. Communication skills: ability to effectively convey ideas, results, and findings through writing, presenting, and collaborating.
  6. Data analysis skills: ability to collect, organize, and interpret data to draw meaningful conclusions.
  7. Critical thinking skills: ability to evaluate information, identify biases, and make informed decisions.
  8. Time management skills: ability to prioritize tasks, manage time, and meet deadlines.
  9. Collaboration skills: ability to work effectively with others, build teams, and lead projects.
  10. Adaptability skills: ability to adjust to new situations, learn quickly, and evolve with changing circumstances.
• Practical skills are essential in various fields, including science, technology, engineering, mathematics (STEM), and beyond.
• They help individuals apply theoretical knowledge to real-world problems, making them valuable assets in the workplace and everyday life.
• In practical work, we need to be clear about certain terminology used in physics, particularly precision, accuracy and errors.

3. Uncertainty:

• Uncertainty is a fundamental concept in physics, acknowledging that our knowledge and measurements are imperfect.
• It’s a quantitative measure of the doubt or uncertainty associated with a physical quantity, such as a measurement, calculation, or prediction.
• Types of uncertainty:
  1. Measurement uncertainty: Errors in instrument calibration, reading, or procedure.
  2. Statistical uncertainty: Variability in data due to random fluctuations.
  3. Systematic uncertainty: Biases in measurement or calculation methods.
  4. Theoretical uncertainty: Limitations in understanding or modeling physical phenomena.
  5. Computational uncertainty: Errors in numerical simulations or algorithms.
• Quantifying uncertainty:
  1. Standard deviation (σ): A measure of spread or dispersion.
  2. Variance: The square of the standard deviation (σ²).
  3. Confidence intervals: Range of values within a certain probability (e.g., 95%).
  4. Error bars: Visual representations of uncertainty in graphs.
  5. Uncertainty budgets: Detailed breakdowns of individual uncertainty sources.
• Managing uncertainty:
  1. Error analysis: Identifying and quantifying sources of uncertainty.
  2. Sensitivity analysis: Studying how uncertainty affects results.
  3. Uncertainty propagation: Calculating uncertainty in derived quantities.
  4. Robustness analysis: Evaluating the impact of uncertainty on conclusions.
  5. Improving measurements: Enhancing precision, accuracy, and reliability.
• Embracing uncertainty in physics:
  1. Acknowledges limitations: Recognizes the imperfections in our knowledge.
  2. Fosters skepticism: Encourages critical evaluation of results.
  3. Drives improvement: Motivates refinement in measurements and models.
  4. Facilitates collaboration: Encourages sharing and discussion of uncertainty.
  5. Enhances credibility: Demonstrates transparency and rigor in research.
• By confronting and quantifying uncertainty, physicists can strengthen their findings, refine their understanding, and advance our knowledge of the physical world.

⇒ For addition and subtraction uncertainty:

• Absolute uncertainties are added, for example the distance x determine by the difference between two separate position measurements
• [math] X_1 = 10.5 \pm 0.1 \, \text{cm}, \quad X_2 = 26.8 \pm 0.1 \, \text{cm} \, \text{is recorded as} [/math]
• [math] X = X_1 – X_2 = (10.5 – 26.8 ) \pm 0.1 \, \text{cm} [/math]

⇒ For multiplication and division uncertainty:

• Percentage uncertainty are added. For example, the maximum possible uncertainty in the value of resistance R of a conductor determined from the measurements of potential difference V and resulting current flow I by using R=V/I is found as follows
• [math]  V = 5.2 \pm 0.1 V \\ I = 0.84 \pm 0.05 A [/math]
• The %age uncertainty for V is [math] = \frac{0.1V}{0.05} * 100 = \text{about} \, 2 % [/math]
• The %age uncertainty for I is [math] = frac{0.05A}{0.84A} * 100 = \text{about} \, 6% [/math]
• Hence total uncertainty in the value of resistance R when V is divided by I is 8%. The result is thus quoted as
• [math] R = \frac{5.2 \, \text{V}}{0.84 \, \text{A}} = 6.19 VA^{-1} \\
  R = 6.19 \, \text{ohm with a percentage uncertainty of} \, 8\% [/math]
• [math] \text{That is} \, ( 6.19 * \frac{ 8}{100} = \frac{49}{100} = 0.49 = 0.5 ) \, \text{so} \, [/math]
• [math]R = 6.2 \pm 0.5 \, \text{ohm} [/math]
• The result is rounded off to two significant digits because both V and R have two significant figures and uncertainty, being an estimate only, is recorded by one significant figure.

⇒ For power factor uncertainty:

• Multiply the percentage uncertainty by that power. For example, in the calculation of the volume of a sphere using
• [math] V = \frac{4}{3} \pi r^3[/math]
• %age uncertainty in V= 3 * %age uncertainty in radius r. As uncertainty is multiplied by power factor, it increases the precision demand of measurement. If the radius of a small sphere is measured as 2.25 cm by a vernier calipers with least count 0.01 cm, thenThe radius r is recorded as
• [math] r = 2.25 \pm 0.01 \, \text{cm} \\
  \text{Absolute uncertainty} = \text{least count} = \pm 0.01 \, \text{cm} \\
  \text{Percentage uncertainty in} \, r = \left( \frac{0.01 \, \text{cm}}{2.25 \, \text{cm}} \right) * 100 = 0.4\% \\
  \text{Total percentage uncertainty in} \, V = 3 * 0.4 = 1.2\% [/math]
• Thus volume
• [math] V = \frac{4}{3} \pi r^3 \\
  V = \frac{4}{3} (3.14) (2.25 \, \text{cm})^3 = 47.69 \, \text{cm}^3 \\
  V = 47.69 \, \text{cm}^3 \, \text{with a 1.2\% uncertainty, so} \\
  = 47.69 * \frac{ 1.2}{100} \\
  \text{Round of value:} \, 50 * \frac{12}{1000} = \frac{5 * 12}{100} = \frac{60}{100} = 0.6 [/math]
• So,
• Thus, the result should be recorded as
• [math] V = 47.7 \pm 0.6 cm^3 [/math]

⇒ For uncertainty in the average value of many measurements:

1. Find the average value of measured values.
2. Find deviation of each measured value from the average value.
3. The mean deviation is the uncertainty in the average value.
4. For example, the six readings of the micrometer screw gauge to measure the diameter of a wire in mm are
5. 1.20, 1.22, 1.23, 1.19, 1.22, 1.21.
6. Then
7. [math] \text{Average} \, = \frac{1.20+1.22+1.23+1.19+1.22+1.21}{6} \\ \text{Average} \, = 1.21 mm[/math]
8. The deviation of the readings, which are the difference without regards to sign, between each reading and average value are (1.21-1.20=0.01, 1.22-1.21=0.01, 1.23-1.21=0.2, 1.21-1.19=0.02, 1.22-1.21=0.01, 1.21-1.21=0) so
9. [math] \text{Mean of deviations} \,=\frac{0.01+0.01+0.02+0.02+0.01+0}{6} \\ \text{Mean of deviations} \, = 0.01 mm [/math]
10. Thus, likely uncertainty in the mean diameter 1.21 mm is 0.01mm recorded as
11. [math] 1.21 \pm 0.01 mm [/math]

⇒ For the uncertainty in a timing experiment:

• The uncertainty in the time period of a vibration body is found by dividing the least count of timing device by the number of vibrations. For example, the time of 30 vibrations of a simple pendulum recorded by a stopwatch accurate up to one tenth of a second is 54.6 s. the period
• [math] T = \frac{54.6 \, \text{s}}{30} = 1.82 \, \text{s} \, \text{with uncertainty} \, \frac{0.1 \, \text{s}}{30} = 0.003 \, \text{s} [/math]
• Thus, period T is quoted as [math] T = 1.820 \pm 0.003 s [/math]
• Hence, it is advisable to count large number of swings to reduce timing uncertainty.

4. Errors: accuracy and precision:

• Accuracy and precision are related but distinct concepts in physics:

• ⇒ Accuracy:

  – Refers to how close a measurement or result is to the true value.
  – Describes the degree of closeness to the actual value.
  – A measurement can be accurate but not precise, meaning it’s close to the true value but not repeatable.

• ⇒ Precision:

  – Refers to the consistency or repeatability of measurements.
  – Describes the degree of agreement among multiple measurements.
  – A measurement can be precise but not accurate, meaning it’s consistent but not close to the true value.

• To illustrate the difference:
  – Accuracy is like hitting the center of a target.
  – Precision is like hitting the same spot on the target repeatedly.
• In physics, both accuracy and precision are crucial. You want measurements to be:
• 1. Accurate: Close to the true value.
• 2. Precise: Consistent and repeatable.
• Ideally, you want to achieve high accuracy and high precision, like hitting the center of the target repeatedly.

⇒ Errors:

• Errors are deviations from the expected or true value.
• They can arise from various sources, including:
  1. Human error: Mistakes made by individuals, such as calculation mistakes or incorrect measurements.
  2. Instrumental error: Limitations or flaws in instruments or equipment, like calibration issues or sensor malfunctions.
  3. Systematic error: Reproducible errors due to faulty assumptions, methods, or equipment.
  4. Random error: Unpredictable fluctuations in measurements, like thermal noise or quantum fluctuations.
  5. Statistical error: Errors due to insufficient data or sampling size.
• Types of errors:
  1. Absolute error: The difference between the measured and true values.
  2. Relative error: The ratio of the absolute error to the true value.
  3. Percentage error: The relative error expressed as a percentage.
  4. Mean absolute error: The average absolute error over multiple measurements.
• Error propagation:
  1. Adding quantities: Errors add linearly.
  2. Multiplying quantities: Errors multiply by the relative error.
  3. Dividing quantities: Errors divide by the relative error.
• Error reduction strategies:
  1. Improve instrumentation: Upgrade or calibrate instruments.
  2. Increase precision: Enhance measurement techniques or data analysis.
  3. Averaging: Combine multiple measurements to reduce random errors.
  4. Error analysis: Identify and quantify error sources.
  5. Quality control: Implement rigorous data collection and analysis protocols.
• By understanding and addressing errors, physicists can ensure the accuracy and reliability of their results, leading to robust conclusions and breakthrough discoveries.

5. Planning, making measurements and recording data:

⇒Planning an Experiment

• Objective Definition: Clearly state the aim or hypothesis of the experiment.
• Independent Variable: The variable you change or control in the experiment.
• Dependent Variable: The variable you measure in response to changes in the independent variable.
• Control Variables: Variables that must be kept constant to ensure a fair test.

⇒ Methodology:

• Procedure: Develop a step-by-step plan outlining how to conduct the experiment.
• Equipment and Materials: List all necessary tools and substances required for the experiment.
• Risk Assessment: Identify potential hazards and outline safety measures.

⇒ Making Measurements:

• Choosing Instruments: Select appropriate instruments based on the required precision and the range of measurements.
• Accuracy and Precision: Aim for accurate and precise measurements by using calibrated instruments and consistent techniques.
• Zero Errors: Ensure that instruments are correctly zeroed before use to avoid systematic errors.
• Reading Scales: Properly interpret readings, including estimating between scale divisions if necessary.

⇒ Recording Data

• Data Tables: Organize data in tables with clear headings, units, and appropriate significant figures.
• Significant Figures: Record data with the correct number of significant figures based on the precision of the measuring instrument.
• Uncertainty: Include uncertainty estimates for each measurement to indicate the range within which the true value is likely to lie.
• Repeated Measurements: Take multiple readings where possible and calculate the mean to improve reliability.
• Anomalous Data: Identify and exclude anomalous results, providing reasons for their exclusion.

⇒ Examples:

(1)

• In the tennis ball experiment, the student identified the distance s and the time t as the two variables and measured them appropriately with a meter rule and digital stopwatch, respectively.
• She identified the acceleration, a, as being constant.
• To control this, she should ensure that the angle of the slope is constant.

(2)

• The student sensibly presented her data in a table and then plotted a suitable graph.
• The data were recorded with appropriate units, expressed in the conventional way.
• The symbol of the quantity being measured, followed by a forward slash and then the unit.
• For example, t/s means the time, measured in seconds.
• o The graph enabled an appropriate value for the acceleration to be determined by averaging several values, thereby reducing random error.
• We could also interpret from the graph that there appeared to be a systematic error in the measurements.

(3)

• If at all feasible, measurements should be made again to ensure that there is no misinterpretation and to enable the decrease of random errors by the average of two or more data.
• For instance, be sure there is zero error on the micrometer (or digital calipers) before using them to measure the diameter of a wire length.
• Next, in order to check for taper, the diameter should be measured at both ends and the middle of the wire.
• At each location, measurements should be taken at right angles to one another in order to ensure uniformity of cross-section. Figure 8 illustrates this.

Figure 8 Measuring the diameter of a wire.

(4)

• In some experiments, such as finding the current–voltage characteristics of a filament lamp.
• It would be completely wrong to take repeat readings, as the lamp will heat up over time and its characteristics will change.
• In such a situation you should plan for this by making sure that you take a sufficient number of readings the first time around.

(5)

• The degree of error in the length s in the tennis ball experiment is probably related to how precisely the ball can be positioned on the scale reading of the meter rule.
• The precision of the scale (calibrated in millimeters) and human error in determining the point at which the ball contacts the rule will control this.
• Most likely, the uncertainty is ±2mm. The stopwatch’s resolution of 0.01 seconds is most definitely not the cause of the timing inaccuracy.
• The human response time for starting and stopping the stopwatch controls the uncertainty.
• Even if there is a tendency for these effects to cancel each other out, the uncertainty would still be estimated to be about 0.1 s.

6. Analysis and evaluation:

• Analysis and evaluation are crucial steps in the scientific process, allowing physicists to extract insights and meaning from data and experiments. Here’s a breakdown of these steps:
• ⇒Analysis:
  1. Data analysis: Extracting relevant information from data, using statistical methods and visualizations.
  2. Pattern identification: Recognizing patterns, trends, and correlations within the data.
  3. Model fitting: Comparing data to theoretical models or hypotheses.
  4. Uncertainty quantification: Estimating errors and uncertainties in the data and analysis.
• ⇒ Evaluation:
  1. Interpretation: Drawing conclusions from the analysis, considering the research question and hypotheses.
  2. Validation: Verifying the results through additional experiments or data analysis.
  3. Comparison: Contrasting findings with existing research, theories, or models.
  4. Implication: Considering the broader significance, applications, and potential impact.
• By rigorously analyzing and evaluating data, physicists can:
  1. Test hypotheses: Determine whether data supports or refutes theoretical predictions.
  2. Refine models: Improve theoretical frameworks based on new insights.
  3. Identify limitations: Recognize areas for future research or experimental improvement.
  4. Inform decisions: Provide evidence-based guidance for practical applications or policy-making.
• Analysis and evaluation are essential for:
  1. Advancing knowledge: Expanding our understanding of the physical world.
  2. Improving technologies: Developing innovative solutions and applications.
  3. Addressing challenges: Tackling complex problems, like climate change or energy security.

⇒Eample

(1)

• When the student shows her teacher the results of the tennis ball experiment, he suggests that she develops the investigation further by comparing her value for the acceleration, a, with the theoretical value. He tells her that this is:
• [math] \vec{a} = \frac{3}{5} \vec{g} \sin \theta [/math]
• where θ is the angle between the slope and the horizontal bench top.
• 1. Show that the angle should be about 6°.
  2. Estimate the percentage uncertainty in attempting to measure this angle with a protractor and comment on your answer.
  3. Draw a diagram to show how you could use a trigonometric method to determine sinθ and comment on the advantage of this technique.
• Solution
• 1 if
• [math] \vec{a} = \frac{3}{5} \vec{g} \sin \theta \\ \sin \theta = \frac{5 \vec{a}}{3 \vec{g}} \\
  \sin \theta = \frac{5 * 0.62}{3 * 9.8} = 0.105 \\
  \theta = \sin^{-1}(0.105) \\
  \theta = 6^\circ [/math]
• 2 Using a protractor, the angle could probably not be measured to better than an uncertainty of ±1°, which gives a percentage uncertainty of:
• [math] \frac{1^\circ}{6^\circ} * 100\% = 17\% [/math]
• This is a large uncertainty, which suggests that using a protractor is not a suitable technique.
• In the tennis ball experiment figure 9, we quantitatively examined the mistake that would probably arise from measuring the slope’s angle with a protractor.
• 3 After coming to the conclusion that an uncertainty of about 17% was intolerable, we modified the experiment by applying a trigonometric technique.
• Based on the measurement of the slope’s angle, we may draw the following conclusions about the tennis ball experiment:
• Figure 9 A tennis ball roll down on the slop
• [math] \vec{a} = \frac{3}{5} \vec{g} \sin \theta \\
  \vec{a} = \frac{3}{5} * 9.8 * 0.102 = 0.60 \, \text{m/s}^2 [/math]
• The experimental value found for the acceleration was [math] 0.62ms^{-2}[/math] . The two values differ by
• [math] \frac{(0.62 – 0.60)  ms^{-2}}{0.61  ms^{-2}} * 100\% = 3\% [/math]
• This is a very acceptable experimental error and so we can conclude that, within experimental error, the acceleration of a tennis ball rolling down a slope is given by the formula
• [math] \vec{a} = \frac{3}{5} \vec{g} \sin \theta [/math]
• Therefore, the average value of the acceleration ([math] 0.61ms^{-2}[/math] ) was used as the denominator.