## Learning objectives

This course aims to bring students to a level of knowledge in measurement methodologies that enable them to manage independently simple laboratory experiments to determine mechanical and calorimetric quantities. Also aims to provide students with a basic knowledge of the theory of errors with elements of probability theory and stochastic variables.

Knowledge and understanding

The student will learn the basic concepts of probability theory, the key statistical distributions and their properties, the main statistical methods for data treatment. The student will be able to discuss basic Physics topics in order to devise and carry out the experimental verification.

Applying knowledge and understanding

At the end of the course the student will be able to: plan simple experiments of Physics, evaluate and treat the statistical and systematic errors of a measurement. It will also have acquired a familiarity with the different methods of measurement and the ability to process and analyze statistically the results by means of suitable tools that help also their graphical representation and summarize relations within the experiments themselves.

Making judgements

Through working in the lab, discussions with the teacher and the team group, students will be led to analyze and evaluate the scientific method, also throughout observational process, experimental proof, and critical revision of results. Usage of the hypothetico-deductive model approach will be implemented, together with the ability, in devising new experiments, to differentiate between essential and marginal aspects.

Communication skills

After every activity in the lab students will produce a written work to develop abilities in description, presentation and discussion of their results. Continuous interaction with teacher and other students will develop also the oral communication skill.

Learning skills

Lab activity, necessity to provide new or different solutions to a problem, will develop both analytical and creative skills in problem-solving, thus widening the students way of thinking.

## Prerequisites

Some basic concepts of math: algebra, trigonometry, analytic geometry, differential and integral calculus.

Some basic concepts in physics: kinematics and dynamics of material point, calorimetry.

Health, safety and risk assessment are of paramount importance in the laboratory. An on-line course must be attended, otherwise access to the laboratories will be denied.

## Course unit content

Metrology: base and derived physical quantities, units of measurements in mechanics, measuring instruments, characteristic of measuring instruments (accuracy, precision, promptness, dynamic range), graph representations of data.

Uncertainty in measurements: systematic and casual errors, uncertainty propagation in indirect measurements, statistical methods in data treatment, random variables, frequency distributions, bad data treatment, weighted mean, gaussian distribution.

Correlation coefficient, best fitting and regression, chi-squared tests.

Basics of theory of probability: statistics and probability, stochastic variables, discrete and continue events, events and sample spaces, dependent and independent events, conditional probability, probability distributions (Normal, Binomial, Poisson, etc.), estimators and their properties, distribution functions and probability density functions, law of very large numbers, central limit theorem.

The laboratory experiments will be defined with reference to the topics treated in the Course of Physics 1 and will cover the following subjects:

- free body fall

- motion of rigid bodies

- motion of pendulum

- harmonic oscillations

- fluid mechanics

- waves in continuum media

- calorimetry

## Full programme

Module I

1. The measurement: direct and indirect measurements of physical quantities, units, characteristics and selection criteria of measuring instruments: accuracy, precision, promptness, dynamic range. Systematic and random errors, confidence intervals; orders of magnitude and significant figures.

2. Study of uncertainties in physical measurements: error propagation (sum, difference, product, quotient, the sum in quadrature, error as a function of one and two variables), error as differential. Measurement errors and their representation: confidence interval, significant digits, consistency / discrepancy between measurements, verification of physical laws.

3. Study of uncertainties in physical measurements: statistical treatment of data and their representation; statistical analysis of random errors: mean, variance and standard deviation, histograms and frequency distributions. Cumulative frequency. Short account on the treatment of systematic errors.

4. Study of uncertainties in physical measurements: frequency and probability, the limit distribution, probability density, normalization, mean value and standard deviation. Gaussian distribution: confidence and standard deviation, standard error integral; comparison of results. Mean as the best estimate. Population distributions.

5. Study of uncertainties in physical measurements: weighted averages, data rejection (Chauvenet criterion). Short account on the method of least squares and regression.

6. Introduction to probability theory: statistics and probability, discrete and continuous variables, the concept of event; favorable and possible cases, classical and frequentist definition of probability.

7. Combinatorics: simple distributions, distributions with repetition, simple permutations, permutations with identical objects, simple combinations, combinations with repetition. Lottery games.

8. Elements of calorimetry: definition of temperature, methods of temperature measurement, thermocouples, specific heat and heat capacity. Mechanisms of heat transfer, calorimeters, measurement of the specific heat.

The experiences in the Laboratory will cover:

• Basic measurements of physical quantities

• Free body fall

• Composition of Forces

• One-dimensional harmonic motion

• Motion of simple pendulum

• Bernoulli and Poisson distributions

• The adiabatic calorimeter

Unit II

1. Basics of theory of probability: statistics and probability. Short account on the axiomatic theory of probability: axioms of Kolmogorov.

2. Fundamental theorems of the theory of probability: addition and multiplication of events; complement of an event; dependent and independent events; conditional probability. Addition and multiplication rules for independent and dependent events; total probability theorem; Bayes’ formula. Repeated trials: Bernoulli trials, binomial law. Short account on the deduction of the theorems in the frame of axiomatic theory of probability.

3. Probability distributions: distribution laws, cumulative distribution functions and probability density; estimators and their properties: mean, median, mode; moments of a distribution, asymmetry and kurtosis. Chebishev inequality.

4. Discrete probability distributions: discrete uniform distribution; binomial distribution: moments, recurrence relations; Poisson distribution: moments. Radioactive decays.

5. Continue probability distributions: continue uniform distribution; Gauss distribution; standardized gaussian distribution; moments; gaussian approximation of binomial and Poisson distributions. Central limit theorem. Chi-squared distribution. Cauchy distribution.

6. Gaussian distribution: maximum likelihood criterion: mean as the best estimate, standard deviation, standard deviation of the mean, weighted average. Demonstrations of relations for error propagation: basic operations, sum of squared errors, general formula.

7. Applications to data treatment: least squares fitting and regression, linear fitting, weighted least squares fitting; non-linear fitting. Multiple stochastic variables, marginal density, stochastic independence, covariance; covariance and error propagation. Correlation: linear correlation coefficient.

8. Applications to data treatment: consistency tests: significance level, chi-squared test; consistency of a distribution.

The laboratory experiments will cover the following subjects:

• motion of rigid bodies

• motion of pendulum

• torsional oscillations

• damped and forced oscillatory motion

• fluid mechanics

• waves in continuum media

• calorimetry and phase transitions

## Bibliography

- J.R. Taylor, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, University Science Books, Third edition 2022.

M. Loreti, Teoria degli errori e fondamenti di statistica, http://wwwcdf.pd.infn.it/labo/INDEX.html (2005).

- Additional material provided by the lecturer.

## Teaching methods

The didactic activities are divided into classroom lessons and practical laboratory activity. The course is 12 CFU (credits) 6 in the first semester and 6 in the second one. Classroom lessons are 4 credits (2 each semester) that correspond to a total of 28 hours of classroom activity. The practical laboratory activity is 4 credits (4 each semester), which corresponds to a total of 96 hours of laboratory activity. The slides used to support classroom lessons will be uploaded weekly on the Elly platform. To download the slides, you need to enroll in the online course. Slides are considered an integral part of teaching material. Part of the frontal lessons will be dedicated to the detailed description of the laboratory experiments. In the laboratory will also be presented both the instrumentation to be used and data acquisition and analysis programs.

The practical laboratory activity might be as partially modified to adhere the safety lineguides imposing social distancing which, unfortunately, reduces also the laboratory working capacity (SARS Cov2 pandemic).

## Assessment methods and criteria

In Itinere evaluations and an oral final examination.

The laboratory work is accounted for by written reports, one for each laboratory experiment. All the written reports will receive in each semester a final mark (0-30). At the end of the course an oral examination (0-30) and, in case of not positive evaluation of the written reports during the course, a laboratory experience might be required. The final graded will be a weighted average between the written reports (40%) and the oral examination (60%). Two oral examination each at end of each semester might substitute for the final examination.

## Other information

The course is split up into two periods: 6 CFU in the first semester and 6 CFU in the second semester. There is a single final exam at the end of the second semester.

At least 70% of all the laboratory activities have to be fulfilled. In case of particular needs (i.e., as an esample, working students) the design of personalized solutions will be evaluated.

Health, safety and risk assessment are of paramount importance in the laboratory. An on-line course must be attended, otherwise access to the laboratories will be denied.